ATTRACTORS AND ATTRACTING MEASURESpetersen.web.unc.edu/files/2018/04/M261S972018_01_23.pdf · Dynamical systems arise from systems of diﬀerential equations that describe some types

ATTRACTORS AND ATTRACTING MEASURES

KARL PETERSEN

Mathematics 261, Spring 1997University of North Carolina at Chapel Hill

Copyright c©1997 Karl Petersen

NOTETAKERS

Mark Anderson (MA)Russell Jackson (RJ)Kim Johnson (KJ)Lorelei Koss (LK)Natalie Priebe (NP)Kennan Shelton (KS), Coordinating EditorSujin Shin (SS)Paul Strack (PS)

Date: January 23, 2018.

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2 KARL PETERSEN

Contents

1. Introduction 12. Plan of the course. January 8, 1997 (Notes by LK) 13. Plan of the course, continued. January 10 (Notes by LK) 33.1. Smale’s horseshoe 44. Smale’s horseshoe. January 13 (Notes by MA) 65. Some topological and differentiable dynamics. January 15 (Notes by MA) 85.1. Some terminology from differential geometry 96. Some more smooth dynamics. January 17 (Notes by MA) 117. Hartman-Grobman and Stable Manifold theorems. January 22 (Notes by NP) 137.1. Stable and unstable sets. 137.2. Properties of hyperbolic fixed points. 148. Homoclinic points. January 24 (Notes by NP) 168.1. Homoclinic points and the Smale Homoclinic Point Theorem. 168.2. A sketch of the proof of Smale’s Homoclinic Point Theorem 168.3. The homoclinic mesh. 189. Axiom A systems. January 27 (Notes by MA) 1910. Structural stability. January 29 (Notes by MA) 2111. Smale’s solenoid. January 31 (Notes by PS) 2311.1. Smale’s solenoid, an Axiom A attractor 2412. Symbolic dynamics in action. February 3 (Notes by PS) 2613. Topological ergodicity. February 5 (Notes by PS) 2714. Stable and unstable manifolds. February 7 (Notes by MA) 2815. Expansiveness, canonical coordinates, basic sets. February 10 (Notes by RJ) 2916. Proof of Spectral Decomposition Theorem. February 12 (Notes by RJ) 3117. Topological mixing on basic sets. February 14 (Notes by RJ) 3418. Shadowing. February 17 (Notes by SS) 3619. Specification. February 19 (Notes by SS) 3820. Specification in shifts of finite type. February 21 (Notes by SS) 3921. Specification in Axiom A systems. February 24 (Notes by KJ) 4122. Consequences of specification. February 26 (notes by KJ) 4222.1. More consequences of the pseudo-orbit shadowing property 4423. Anosov Closing Lemma. February 28 (Notes by KJ) 4523.1. More consequences of the pseudo-orbit shadowing property 4524. More consequences of pseudo-orbit shadowing. Markov partitions. March 3 (Notes by LK) 4624.1. Markov partitions 4725. Exercises on solenoid and Markov partitions. March 5 (Notes by LK) 4826. Existence of Markov partitions. March 7 (Notes by LK) 4927. Start the proof of existence of Markov partitions. March 17 (Notes by NP) 5228. More of the proof. March 19 (Notes by NP) 5429. Proof continued. March 24 (Notes by MA) 5729.1. A candidate for a Markov partition 5830. Near the end of the proof. March 31 (Notes by PS) 6031. End of the proof. April 2 (Notes by PS) 6232. More end of the proof. April 4 (Notes by PS) 6433. Back to coding 6534. Obtaining symbolic dynamics. April 7 (Notes by RJ) 6835. Symbolic dynamics. Entropy. April 9 (Notes by RJ) 7035.1. Entropy, pressure, equilibrium states, Gibbs states 7136. Entropy, pressure, Gibbs measures. April 11 (Notes by RJ) 7236.1. Bowen’s definitions of topological entropy 7236.2. Gibbs Measures 73

Attractors and Attracting Measures 1

37. Motivations from physics. April 14 (Notes by SS) 7438. Maximizing entropy or free energy. April 16 (Notes by SS) 7539. Existence and uniqueness of equilibrium states. April 18 (Notes by SS) 7640. Ruelle’s Operator Perron-Frobenius Theorem, g-measures. April 21, 1997 (Notes by KJ) 7840.1. From last time. . . 7840.2. Infinite-dimensional extension of the Perron-Frobenius Theorem for nonnegative matrices 7941. Symbolic dynamics yields existence of equilibrium states on basic sets. April 23, 1997 (Notes

by KJ) 8141.1. Sketch of proof of the existence of equilibrium states on basic sets 8141.2. Theorems 2 and 3 8242. Finding attractors in Axiom A systems. April 24 (Notes by LK) 8343. Attracting measures. April 28 (Notes by KS) 88


1. Introduction

These are notes from a graduate course on symbolic dynamics given at the University ofNorth Carolina, Chapel Hill, in the spring semester of 1997. The course began with somebackground on smooth dynamics and then mainly worked through R. Bowen’s EquilibriumStates and the Ergodic Theory of Anosov Diffeomorphisms, dealing with the construction ofMarkov partitions, entropy, pressure, equilibrium states, and equilibrium (SRB) measureson attractors. The aim was to see one important source of symbolic dynamics, which wasstudied in its own right in the following course the next spring. The author thanks allthe students who took notes, wrote them up, and typed them, and Kennan Shelton formanaging the entire project.

2. Plan of the course. January 8, 1997 (Notes by LK)

In this course, we intend to study the dynamical aspects of Axiom A attractors; specifi- Introductioncally, we want to identify such attractors and any corresponding attracting measures.

The required texts for this course are out of print but can be purchased in a coursepack available at the bookstore. They consist of Bowen, Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms, Springer-Verlag LNM 470, 1975 and sections of Denker,Grillenberger, and Sigmund, Ergodic Theory on Compact Spaces, Springer-Verlag LNM 527,1976.

The following sources provide a more detailed background to ergodic theory and are onreserve in the library.

(1) Katok and Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,Cambridge Univ. Press, 1995.

(2) Mane, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, 1983.(3) Petersen, Ergodic Theory, Cambridge Univ. Press, 1983.(4) Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982.(5) Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC

Press, 1995.(6) Smale, Differentiable Dynamical Systems, Bulletin of the AMS, vol. 73 (1967),

747-817 (an older summary of the theory of dynamical systems).

As an illustration of how we can find attracting behavior in seemingly chaotic systems, webegin with an example from H. Abarbanel’s On the Analysis of Chaotic Dynamical Systems.The example on pp 3-12 describes a cutting tool, as in machining some type of part with alathe. This type of work needs to be precisely controlled to obtain a finished product withinthe specified parameters. Even so, there is still some variation in the accuracy of the cuts.A graph of the the displacement of the lathe versus the time can be found on page 3. Sincethis is a time series signal, we might try to use the Fourier transform to study it. However,as the graph on page 4 demonstrates, harmonic analysis does not provide much insight intothe behavior of this system.

Nevertheless, we can examine the pseudo-phase space (called pseudo because it is obtai-ned from a numerical approximation) by plotting the vector (x(t), x(t+τ), x(t+2τ)), wherex(t) is the displacement at time t and τ is chosen in some appropriate manner. As we seefrom the graph on page 5, we obtain an object with some structure. Although it is not yetclear to us what this new graph signifies, it may help us to understand qualitative aspects

2 Karl Petersen

U

φU

Λ

Figure 1. The attractor Λ

of our original system and therefore possibly make predictions about it or even control it.In fact, this procedure works with many systems and is being used currently in many partsof science. Generally, we take one-dimensional data, space it out nicely and plot it in aconvenient dimension to obtain a simpler object with some obvious structure. Hopefully,that new structure will tell us something about the original system.

A dynamical system is a space X and a family S of maps φ : X → X, where we usuallyassume that S is at least a semi-group (if φ,ψ ∈ S then φ ψ ∈ S). The space X can be atopological space, a measure space, or a (compact) manifold.

Dynamical systems arise from systems of differential equations that describe some types ofphysical, biological or abstract systems. We want the system of equations to be autonomous,meaning that the laws of the system do not change over time. The system gives a flow inphase space described by (position, momentum) or (y, y′). We letX be some closed invariantset (for example, the constant energy manifold) and we let φ : X → X denote the time-onemap of the flow. Then we have a family of maps S = φn where n can be an integer or ncan be restricted to the positive integers.

We pause to note that the closed invariant setX will turn out to be a manifold in a naturalway and thus will have a natural measure coming from Lebesgue measure. However, thismeasure may not see the dynamics, as the dynamically interesting part of the space maybe a null set with respect to this measure class. In part, this course will focus on findingmeasures that are dynamically interesting.

If X is a topological space (manifold) and φ : X → X is a homeomorphism (diffeomor-phism) then we call a set Λ ⊂ X an attractor if there is an open set U containing Λ with

φU ⊂ U and Λ =⋂

n≥0

φnU , as in Figure 1.

If x ∈ U then the set of limit points of the iterates φnx is contained in Λ. The basin

of attraction of Λ is the set⋃

n≥0

φ−nU . If Λ is anything more complicated than a periodic

orbit, we call Λ a strange attractor.Sinai, Ruelle, and Bowen proved that (hyperbolic) attractors exist for Axiom A systems.

We will define these terms later in the course and for now just give a few examples. Anosovsystems, where Λ is the entire manifold, are Axiom A systems. For example, there areone-to-one and onto maps of the torus which are Anosov because the action is hyperbolic


and complicated everywhere. Solenoids are examples of Axiom A attractors which are notAnosov. On the basis of numerical studies, attractors are suspected to exist in many othercases, such as for Henon maps and Lorentz systems.

Further, we will study the proof of the existence of attracting measures in Bowen’s book.We will examine the SRB measure (Sinai - Ruelle - Bowen), a probability measure µ on Λsuch that for all continuous functions f : U → R,

1

n

n−1∑

k=0

f(φkx) →

∫

Λf dµ

for a.e. x ∈ U with respect to the Lebesgue measure class. We note that this is trueeven though the attractor probably has measure 0 with respect to Lebesgue measure, themeasure that includes experimental observations in a laboratory. In other words, there isa set of Lebesgue measure zero that is determining the long-term behavior of what we aretrying to observe. In the proof, we will see that the SRB measure is found as a Gibbsmeasure using equilibrium states.

3. Plan of the course, continued. January 10 (Notes by LK)

There are at least two reasons for studying mathematical models of the kind we discussed Introductioncont.in the last class. First, we may satisfy our intellectual curiosity, as these examples are

interesting from a purely mathematical point of view. Second, these models mirror in somerespect what is going on in the real world. The exact nature of this connection with physicalreality (if any), however, is in dispute.

Last class, we defined an attractor Λ and noted that such attractors have been provedto exist in Axiom A systems. We also want to study attracting measures such as the SRB-measure defined last class. We will begin with an outline of our coverage of this topic andpostpone definitions until later.

Supposing we have Λ, we will proceed through the following steps to find attractingmeasures. First, we will find a Markov partition of Λ into sets of arbitrarily small diameter.Roughly, a Markov partition is a way to cut Λ into sets that are mapped by φ in a nicemanner; φ maps a set of the partition to a finite union of other sets of the partition.

Second, we will use the Markov partition to obtain symbolic dynamics. Namely, we codethe orbit of x , O(x) = φnx : n ∈ Z according to which atom of the partition φnx is in.Define

ξ : x→ ξ(x) ∈ 0, 1, 2, . . . , r − 1Z = Σr

by (ξ(x))j = m if and only if φjx is in the m-th cell of the partition. Applying φ or φ−1

will shift the sequence in one direction or the other. Therefore, if we define σ : Σr → Σr by(σ(ω))j = ωj+1 where ω ∈ Σr, then clearly ξ(φx) = σ(ξx).

We note that since the mapping on Λ usually has some restrictions, we often don’t usethe full shift. Instead, we use a shift of finite type. We find a ΣA ⊂ Σr such that there is amap π : ΣA → Λ which is a “tight coding”, meaning that π is one-to-one (except on a firstcategory set) and the following diagram is commutative:

4 Karl Petersen

ΣAσ

−−−−→ ΣA

π

y

yπ

Λφ

−−−−→ Λ

In fact, what we get is almost a dynamical isomorphism, so studying ΣA is almost thesame as studying Λ. Both will have the same topological entropy and both will be intrin-sically ergodic, meaning they have a unique measure of maximal entropy. Also, we will seethat under the correct conditions they both will be topologically mixing as well.

The process described above demonstrates one of the major justifications for studyingsymbolic dynamics. Information theory constitutes another important application of thisfield.

Third, we use the Shannon-Parry measure on ΣA and map it down to the SRB-measureµ on Λ. The first two chapters of Bowen’s book deal with finding the unique measure ofmaximal entropy on ΣA, showing it is an equilibrium state for the constant function, anddemonstrating that it is a Gibbs measure.

Finally, we will see that for other Holder continuous functions f : Λ → R, we obtain thatf π : ΣA → R is a Holder continuous function and has a unique equilibrium state whichprojects to one on Λ.

In our attempts to understand these steps we will learn about expansiveness, specifica-tion, pseudo-orbit shadowing, stable and unstable manifolds, and canonical coordinates. Tostudy the dynamical aspects of (Λ, φ, µ), we will discuss Lyapunov exponents, the multiplica-tive and subadditive ergodic theorems, topological and measure-theoretic entropy, Hausdorffdimension, and various formulas (Pesin, Young) relating these topics.

We note that there are approaches to the SRB-measure that do not use Markov partitions.Often there is numerical evidence (and sometimes a proof) that SRB-measures exist evenwhen Markov partitions don’t. In any case, Markov partitions are an extremely useful tool,as well as being historically important.

3.1. Smale’s horseshoe. We are now ready for our first concrete example, the SmaleSmale’shorseshoe horseshoe map. This example is useful for illustrating how chaotic dynamics can arise in

a deterministic dynamical system. In this example we will observe that the attractor Λlooks like a 2-shift. Further, we will see that this map is also found in more complicateddynamical systems.

We begin with a rectangle R ⊂ R2 and define a map φ : R→ R2 as in Figure 2.We can see the hyperbolicity as a strong uniform contraction in one direction and a

strong uniform expansion in the other direction. As defined, this is not a map of manifolds.However, we could put this map onto a 2-sphere and extend it to a mapping of the entire2-sphere (except one point) and study it there; this was the approach that Smale used.

We will use R0 and R1 as our basic partition; these labels also provide a coding.We are looking for the largest φ-invariant set Λ =

⋂

n∈Z φnR. We include sketches of

R ∩ φR and R ∩ φR ∩ φ2R in Figures 3 and 4 respectively.If we continue this process, we see that R ∩ φR ∩ φ2R . . . = C × I, a Cantor set cross an

interval. Similarly, Figure 5 shows an illustration of R ∩ φ−1R.


top

top

top

0 1R R

R

R

φ

Figure 2. Smale’s horseshoe

R0

R 1

Figure 3. R ∩ φR

Figure 4. R ∩ φR ∩ φ2R

We also obtain that R ∩ φ−1R ∩ φ−2R . . . = I × C ′, an interval cross a Cantor set.Therefore,

Λ =∞⋂

n=−∞

φnR = (C × I) ∩ (I × C ′) = C × C ′.

6 Karl Petersen

-1R

0

-1R1

φ

φ

Figure 5. R ∩ φ−1R

Thus, Λ is a compact, totally disconnected, perfect set and is topologically a Cantor set.In the next lecture, we will show that (Λ, φ|Λ) is topologically conjugate to (Σ2, σ); that

is, there exists a homeomorphism h : Σ2 → Λ such that hσ = φh.

4. Smale’s horseshoe. January 13 (Notes by MA)

We continue our discussion of Smale’s horseshoe and show its connection with symbolicSmale’shorseshoecont.

dynamics. See Section 3.1 for a description of how Smale’s horseshoe is constructed.Let Σ2 = 0, 1Z be the space of bi-infinite sequences of zeros and ones. We can make

Σ2 into a metric space with the distance function defined by

d(ω, η) =1

1 + k, where k = inf|j| : ωj 6= ηj.

Remarks 4.1.

(1) In this metric, two points (sequences) of Σ2 are close if they agree on a long centralblock. Basic open sets are cylinder sets centered about 0; i.e., sets of the form

U = [u−j . . . u0 . . . uj] = ω ∈ Σ2 : ωi = ui for |i| ≤ j.

(2) Σ2 is a compact, totally disconnected metric space.(3) The shift map σ : Σ2 → Σ2 is a homeomorphism.

Define the map π : Σ2 → Λ by

π(ω) =⋂

n∈Z

φ−nRωn .

Theorem 4.1. The map π is a well-defined, one-to-one, onto, continuous map which con-jugates σ and φ; i.e., the following diagram commutes:

Σ2σ

−−−−→ Σ2

π

y

yπ

Λφ

−−−−→ Λ

Remark 4.1. We say that (Σ2, σ) and (Λ, φ) are topologically conjugate and write (Σ2, σ) ≈(Λ, φ).


U

V

n

V

φ

Figure 6. Topological Mixing

Unσ

0

V

Figure 7. σnU and V

The proof of this theorem is left to the reader.Because (Σ2, σ) and (Λ, φ) are topologically conjugate, we can use the dynamics of the

shift on Σ2 to tell us about the behavior of φ on Λ. In particular, we will show that φ istopologically mixing and topologically transitive.

Definition 4.1. A system (X,φ) is called topologically mixing if for any nonempty opensets U and V in X, there exists an N such that φnU ∩ V 6= ∅ for all n ≥ N .

A system (X,φ) is called topologically transitive (or topologically ergodic) if there existsa dense orbit. Equivalently, if every closed invariant proper subset of X is nowhere dense(has empty interior). See Exercise 1 (Section 5).

If (X,φ) is topologically mixing, then the images of U will eventually ‘fill’ the entire spaceX. See Figure 6.

Corollary 4.2. The map φ restricted to Λ (φ|Λ) is topologically mixing, and hence topolo-gically transitive. Further, the periodic points of φ are dense in Λ.

Proof. We will actually show that (Σ2, σ) is topologically mixing (and hence topologicallytransitive) with dense periodic points. Since π conjugates σ and φ, we will then have thecorollary.

Let U and V be open subsets of Σ2. Without loss of generality, we may consider U andV to be basic open sets: U = [u−j . . . uj] and V = [v−k . . . vk]. Take N > j + k. Then forn > N , σnU will be centered at −n, and σnU will not specify any coordinates also specifiedby V (see Figure 7).

Thus there is some ω in Σ2 such that ω−j−n . . . ωj−n = u−j . . . uj , and ω−k . . . ωk =v−k . . . vk, and so σnU ∩ V 6= ∅. Therefore (Σ2, σ) is topologically mixing.

8 Karl Petersen

Figure 8. Variations of Smale’s horseshoe

That the periodic points of σ are dense in Σ2 is easy to see: given ω ∈ Σ2, let n be a largeinteger and set ωn to be the finite sequence (ω−n . . . ωn). Then define ω′ = (. . . ωnωnωn . . . ).ω′ will be a periodic sequence (and thus a periodic point for σ) that is close to ω.

Note that once we have topological mixing (and thus topological transitivity), we havesome sort of nontrivial recurrence. This is the first sign of complicated dynamics.

Remark 4.2. The horseshoes will persist, at least under C1 perturbations. Thus we willhave the same topological dynamics even if the map φ is “wiggled” a bit. Besides perturbingφ, other variations are also possible. For instance, we could have the image of R intersect Rin several places. Examples of possible variations are given in Figure 8. Common propertiesof the variations include strict hyperbolicity and complete strips for φ(R) and φ−1(R).

Remark 4.3. Horseshoes are found in actual systems. For example:

(1) Henon map. This is a map h (and h−1) defined on R2 by

h(x, y) = (a− by − x2, x)

h−1(x, y) =

(

y,a− x− y2

b

)

where a and b are parameters. For b = −0.3, a = 1.4, experimental evidence seemsto indicate the existence of a strange attractor.

One can show (Devaney-Nitecki) that for b 6= 0 and a large enough (say, a =5, b = −0.3), there exists a square Rab which has a horseshoe attractor Λ. Theaction of h is indicated in Figure 9.

Benedicks-Carleson showed there are strange attractors for some (even many)a, b. Other recent work on the Henon map by John Smillie and Zhongguo Yang(UNC Ph.D.) explores the complicated dynamics of this system for various valuesof the parameters.

(2) Smale’s homoclinic point theorem. We will cover this next time.

5. Some topological and differentiable dynamics. January 15 (Notes by MA)

We start with a list of definitions, then an exercise.

Definition 5.1. Let (X,φ) be a dynamical system. Then


h(R ab)

R ab

Figure 9. Henon’s horseshoe

(1) (X,φ) is topologically mixing if for U, V nonempty open subsets of X, there existsN such that for n ≥ N , φnU ∩ V 6= ∅;

(2) (X,φ) is regionally transitive if for U, V nonempty open subsets of X, there existsat least one n such that φnU ∩ V 6= ∅;

(3) (X,φ) is topologically ergodic if every proper closed invariant subset of X is nowheredense;

(4) (X,φ) is topologically transitive if there exists a dense orbit.

We note that if (X,φ) is topologically transitive, then the set of points with dense orbitswill be residual (the complement of a union of countably many nowhere dense sets).

Exercise 1. Let X be a second countable, compact Hausdorff space, φ : X → X a homeo-morphism. Show that

(X,φ) is topologically mixing ⇒ (X,φ) is regionally transitive⇔ (X,φ) is topologically ergodic⇔ (X,φ) is topologically transitive.

The main thrust of the next set of lectures will be to state and explain Smale’s Homoclinic HomoclinicPointThm.

Point Theorem. First we state the theorem, then give the background and definitionsrequired to understand the statement. Finally, we will give an idea of the proof.

Theorem 5.1 (Smale’s Homoclinic Point Theorem). Let M be a compact C∞ manifold andφ : M → M a C1 diffeomorphism. Suppose that p ∈ M is a hyperbolic periodic point for φwhich has a transverse homoclinic point x. Then there is an r > 0 such that φr has a closedinvariant hyperbolic set Λ which contains p and x and such that (Λ, φ|Λ) is topologicallyconjugate to the two-shift (Σ2, σ). In fact, homoclinic points y and corresponding sets Λcan be found in every neighborhood of p.

5.1. Some terminology from differential geometry.

Definition 5.2. A topological manifold M is a connected second countable Hausdorff space Differentialgeometrybackground

such that for each x in M , there exists a neighborhood U of x and homeomorphism h fromU to an open ball in Rd. We call (U, h) a chart (or system of local coordinates) about x,and we say that M has dimension d. The set of charts (U, h) on M is called an atlas.

A Ck manifold is a manifold such that for any two charts (U1, h1) and (U2, h2), the maph2 h

−11 on h1(U1 ∩U2) is a C

k map (see Figure 10). In this case, a maximal atlas is called

a Ck differentiable structure.

10 Karl Petersen

U1 U2

h1 h2

h h2 1-1

Figure 10

Definition 5.3. Let M and N be Ck manifolds and f : M → N . We say that f isdifferentiable (or Ck) if for every pair of charts (U, h) for M and (V, g) for N , the mapg f h−1 is differentiable (or Ck) on h(U ∩ f−1V ).

Definition 5.4. A Ck diffeomorphism of manifolds M and N is a 1 : 1, onto, Ck map fromM to N whose inverse is also Ck. We denote the space of all Ck diffeomorphisms from Mto N by Diffk(M,N).

We can define a topology on Diffk(M,N) — two maps f and g are Ck-close if for somecoordinate charts (U1, h1) on M and (U2, h2) on N , the maps

F = h2 f h−11 and G = h2 g h

−11

are close in the Ck topology on Rd, i.e., the norms of the derivatives of F and G up to orderk are close.

Let p be a point in the manifold M and γ any (C∞) curve from (−1, 1) to M withγ(0) = p. Then γ acts on C∞(M) by f 7→ (f γ)′(0). We denote this map by γ′(0). Notethat f γ will be a map from (−1, 1) to R, so the regular notion of the derivative at 0 makessense.

We will say that two curves γ and α are equivalent if γ(0) = α(0) = p and γ′(0) = α′(0)as maps on C∞(M).

Definition 5.5. If M is a manifold and p is a point in M , then we define the tangent spaceto M at p (denoted TpM) to be the set of all equivalence classes of curves γ : (−1, 1) →Mwith γ(0) = p. Each equivalence class of curves is called a tangent vector at p.

Proposition 5.2. Let M be a d-dimensional manifold. Then for each p in M , TpM ∼= Rd.

We won’t prove this proposition, but will give some indication of its proof by defining thestandard basis for TpM . For p ∈ M and chart (U, h) about p, set γi(t) = h−1(h(p) + tei),

where ei is the ith basis element in the standard basis for Rd (note that we may have torescale t to keep h(p) + tei in the image of h for t between −1 and 1, but that’s ok).

Then the standard basis for TpM is the set of vectors

γ′i(0) :=∂

∂xi

.


Definition 5.6. The tangent bundle of M , denoted TM , is defined to be the disjoint union

over p ∈ M of TpM ; i.e., TM =•∪p∈MTpM . We think of TM as the set of ordered pairs

(p, v) where p ∈M and v ∈ TpM .

The tangent bundle TM is in fact a manifold itself. Let (U, h) be any chart on M anddefine the map H : TM → Rd ×Rd by

H(p, v) = (h(p), (v1, v2, . . . , vd))

where the vi are the coefficients of v ∈ TpM with respect to the standard basis

∂

∂xj

.

This gives us the chart (U ×∪p∈UTpU,H) on TM , and we see that TM is a manifold (ofdimension 2d).

6. Some more smooth dynamics. January 17 (Notes by MA)

Now that we know what it means for a function φ : M → N to be differentiable (see Differentialgeometryback-ground,cont.

Definition 5.3), we define the derivative of φ.

Definition 6.1. If φ : M → N is differentiable, then the derivative of φ is the mapDφ : TM → TN (sometimes also denoted Tφ) given by

(1) (Dφ)(p, γ′(0)) = (φ(p), (φ γ)′(0))

where γ′(0) is a tangent vector in Tp(M), p ∈ M , and (φ γ)′(0) ∈ Tφ(p)M . We can speakof the derivative of φ at a point p, Dpφ, which acts on TpM by restricting Dφ to TpM . Dpφis a linear transformation from TpM to Tφ(p)N . (The fact that Dφ is well-defined followsfrom the differentiability of φ). See Figure 11.

Definition 6.2. A smooth vector field X on M is a map from M to TM such that X(p) =(p, v), where v ∈ TpM , and which is smooth as a map between manifolds.

Remark 6.1. Recall that we can think of X(p) ∈ TpM as operating on C∞(M) (Definition5.5). Then the vector field X is smooth if for each p, X(p) sends smooth functions tosmooth functions.

p

TpM

M

(p)

T N

Dp

N

vDp v

γ γφ

φ

φ

φ

(p)

φ

Figure 11. (Dpφ)(v) for v = γ′(0)

12 Karl Petersen

Definition 6.3. Let M be a differentiable manifold. We say that M is a Riemannianmanifold if there is an inner product gp(·, ·) =< ·, · >p defined on each tangent spaceTpM for p ∈ M such that for any smooth vector fields X and Y on M , the functionp 7→ gp(X(p), Y (p)) is a smooth function of p.

Equivalently, we could require that when we express the function gp in terms of localcoordinates, each coordinate function will be a smooth function of p.

On a manifold M , there is a natural measure class (in the sense of absolute continuity)called the Lebesgue measure class: Given (U, h) a chart on M , we say that A ⊂ U hasmeasure 0 if and only if h(A) ⊂ h(U) has d-dimensional Lebesgue measure 0.

IfM is orientable, then we can construct elements in the Lebesgue measure class by usingnondegenerate d-forms, called volume forms.

Definition 6.4. A volume form ω is a map that assigns to each p an alternating d-tensor onthe vector space TpM . That is, ωp is an alternating multilinear map from TpM × · · · × TpM

︸︷︷︸

d timesto R. We again require that this assignment depend smoothly on p.

It is a basic result that there exists a nondegenerate volume form onM iff oriented charts

can consistently be chosen forM iff the space∧d T ∗M is one-dimensional (T ∗

pM is the spaceof linear maps from TpM to R).

Let M be a Riemannian (orientable) manifold. For each p, let u1, . . . , ud be the stan-dard orthonormal, positively oriented basis for TpM . Then define the volume form ω byωp(u1, . . . , ud) = 1.

We will use ω to define a measure µω on M in the following way: If (U, h) is a chart andA ⊂ U , define µω(A) by

(2) µω(A) =

∫

h(A)

∣∣ωh−1x

(Dxh

−1(e1), . . . ,Dxh−1(ed)

)∣∣ dx1 . . . dxd,

where (e1, . . . , ed) is the standard basis for Rd, h−1 is a map from (a subset of) Rd to M ,and so Dxh

−1 is a map from TxRd to Th−1xM .

The idea is to define the volume of a small box-like subset of M which approxima-tes the volume of the box spanned by (ǫu1, . . . , ǫud) in Th−1xM , for small ǫ, to be aboutωh−1x(ǫu1, . . . , ǫud). Then, instead of integrating over A inM , we integrate over h(A) ⊂ Rd,where integration makes sense (compare (2) with the usual change of variables formula).See Figure 12.

Definition 6.5. A hyperbolic periodic point of φ :M →M is a point p ∈M such thatDynamicsbackground (1) φm(p) = p for some m > 0 (when m is the smallest possible, we say that p has

period m);(2) Dp(φ

m) : TpM → TpM is a hyperbolic linear map (no eigenvalues with modulusequal to 1).

When p is a hyperbolic periodic point, we can decompose TpM into a stable subspaceEsp (the eigenvalues of the restriction of Dp(φ

m) to Esp have modulus less than 1) and anunstable subspace Eup (the eigenvalues of the restriction of Dp(φ

m) have modulus greaterthan 1). Thus TpM = Esp ⊕ Eup .


A

U

M h

B=h(U)

h(A)

Figure 12. The map h takes U ⊂M to B ⊂ Rd

7. Hartman-Grobman and Stable Manifold theorems. January 22 (Notes byNP)

We will start off today’s notes with some topological definitions which lead up to anothercondition equivalent to topological ergodicity. Assume thatX is a second countable compactHausdorff space.

Definition 7.1. A set G ⊂ X is residual if it contains the countable intersection of denseopen sets. A subset of the complement of a dense open set is called nowhere dense. A setF ⊂M is said to be first category if it is the countable union of nowhere dense sets. A setE ⊂M is said to have the property of Baire if E = (G∪M1)\M2 with G open and M1,M2

first category.

Exercise 2. Show that the following condition can be added to the list of conditionsequivalent to topological ergodicity studied in Exercise 1:

If E is an invariant set for φ (i.e. φ(E) = E) with the property of Baire,then either E or Ec is first category.

This condition should be compared to the measure-theoretic definition of ergodicity: φis metrically (measure-theoretically) ergodic for an invariant measure µ if and only if X isindecomposable, i.e., φ(E) = E ⇒ µ(E) = 0 or µ(Ec) = 0. The equivalence of these variousforms of topological ergodicity is stated in an article by John Oxtoby in the Proceedings ofthe National Academy of Sciences, 1937.

7.1. Stable and unstable sets. Let’s get back to defining the terms we need in order Dynamicsbackgroundcont.

to understand the Smale Homoclinic Point Theorem. To this point we have defined thetopological space everything is taking place on (a C∞ Riemannian manifold M), whatkind of action is taking place (a C1 diffeomorphism), and what it means for p ∈ M tobe a hyperbolic periodic point for φ. In order to study the ideas in a notationally andconceptually simpler fashion we will restrict our attention to hyperbolic fixed points of φ;this makes sense because if p is a periodic point with φm(p) = p then it is a fixed point ofthe diffeomorphism φm.

Given any p ∈ M we can consider the behavior under φ of other points in M throughcomparison to the behavior of p under φ. The stable or forward asymptotic set of p is theset of all points whose forward iterates approach the forward iterates of p; formally:

W s(p) = x ∈M : d(φnx, φnp) → 0 as n→ ∞.

14 Karl Petersen

p Ws(p)

Wu(p)

Figure 13. The stable and unstable sets of p

Similarly, we can look at the backwards iterates of points to define the unstable or backwardasymptotic set of p; it is the set of all points whose iterates under φ−1 approach those of p:

W u(p) = x ∈M : d(φ−nx, φ−np) → 0 as n→ ∞.

Note that since we are on a C∞ manifold it is possible to define a metric on M which isconsistent with the ambient differentiable structure and this is why it makes sense to lookat the distance between the iterates of points in M . Figure 13 shows the movement underφ of points in the stable and unstable sets of p.

7.2. Properties of hyperbolic fixed points. The following very important theoremshows that near its hyperbolic fixed points a function is locally linearizable; it acts justlike its derivative tells you it should. That is, there is a topological conjugacy between φand Dpφ on a neighborhood U of p.

Theorem 7.1 (Hartman-Grobman). Let p be a hyperbolic fixed point of a Ck diffeomor-phism φ : M → M . Then locally φ is the same as Dpφ up to a continuous change ofcoordinates, i.e. there are a neighborhood U of p and a homeomorphism h : U → TpM suchthat

(Dpφ) h = h φ

where both are defined.

The situation is depicted in this commutative diagram.

Uφ

−−−−→ φ(U)

h

y

yh

TpMDpφ

−−−−→ TpM

At hyperbolic fixed points the tangent space splits into contracting and expanding sub-spaces; in fact these subspaces are copies of the stable and unstable sets of the fixed point.It is also true that the stable and unstable sets are copies of Euclidean spaces inside M ,


p

EspW

W u(p)

u

s(p)

E

Figure 14. The tangent space to M at p.

although they are not actually submanifolds of M . (By abuse of terminology, W s(p) andW u(p) are sometimes referred to as the stable and unstable manifolds of p.) The topologicalconjugacy is now made precise in the Stable Manifold Theorem.

Definition 7.2. Given a manifold M and a set N ⊂ M we say that N is injectivelyimmersed in M if there is a manifold N and a 1:1 smooth map ψ : N → M with Dφinjective and ψ(N ) = N . We say that N is tangent at p = ψ(x) to the subspace E of TpM

if Dxψ : TxN → TpM has image E.

Theorem 7.2 (Stable Manifold Theorem). Let p be a hyperbolic fixed point of a Ck dif-feomorphism φ : M → M with splitting TpM = Esp ⊕ Eup into Dpφ-invariant contractingand expanding subspaces. Then W s(p) and W u(p) are injectively immersed images of Euc-lidean spaces in M which are tangent to Esp and Eup and thus have dimensions dim(Esp) anddim(Eup ) respectively.

Elements of the statement and proof of this theorem go back to Poincare, Hadamard,Perron, and others, but the modern statement presented here was given by Hirsch andPugh.

Figure 14 is a diagram intended to depict the stable and unstable manifolds of p alongwith its tangent space decomposition. Remember that the manifolds and subspaces involvedhere need not be just one-dimensional.

Putting the Stable Manifold Theorem together with the Hartman-Grobman Theorem wesee that a neighborhood of p in M is compressed in the direction of the stable set andstretched in the direction of the unstable set. Points near the stable set move closer to pwhile those near the unstable set move away from it. If there is a point x which is in bothsets, how do points near x behave under φ? Such a point x is called a homoclinic point andthe existence of such a point causes interesting dynamics. The big idea is:

16 Karl Petersen

hyperbolicity + recurrence⇒ very complicated behavior.

8. Homoclinic points. January 24 (Notes by NP)Dynamicsbackgroundcont.

8.1. Homoclinic points and the Smale Homoclinic Point Theorem. Assume that pis a hyperbolic fixed point for φ and let x ∈W s(p)∩W u(p), so that x is a homoclinic pointfor p. We say that x is a transverse homoclinic point for p if the tangent spaces to W s(p)andW u(p) at x span TxM . See Figure 15, where x is a transverse homoclinic point of p butthe point y, while homoclinic, has only a partial tangency and is therefore not transverse.

In the presence of a transverse homoclinic point x for a hyperbolic fixed point p we find φto be very unpredictable. In fact, it is just as unpredictable as Smale’s horseshoe mapping(a.k.a. the full 2-shift!). In fact, if one examines the behavior of points in a neighborhoodU of p (such as the neighborhood of p containing x in Figure 16,) one sees that φ stretchesout U along the unstable set according to the linear map Dp(φ).

For k sufficiently large φk(U) will stretch around to contain x again. In fact, this ishow the horseshoe mapping and coding onto the full 2-shift is seen in this system at thehyperbolic point p, and it is the idea of the proof of Smale’s Homoclinic Point Theorem(Theorem 5.1), which we will now outline.

8.2. A sketch of the proof of Smale’s Homoclinic Point Theorem. A referenceProof ofSmale’sHomocli-nic PointThm.

for this proof is found in S. Newhouse’s Lectures on Dynamical Systems (CIME lectures,Bressanone, 1978, Birkhauser, 1980, pages 1 - 114).

Without loss of generality we can assume that the hyperbolic periodic point p is actuallya fixed point for φ (if not, replace φ with φm where m is the period of p). Construct aneighborhood U of p as (in local coordinates) the product of a small disk in W u(p) anda small open ball in W s(p). This neighborhood contains a homoclinic point x since all

Ws(p)

p x

Wu(p)

y

Figure 15. A hyperbolic point p and a transverse homoclinic point for p.


p x

U

x

(U)

p

Figure 16. The action of φ on a neighborhood of a hyperbolic point.

U

k(U)

xp

A10A

Figure 17. Under repeated iteration of φ there is recurrence.

homoclinic points are in W s(p) and therefore have forward iterates arbitrarily close to p;any image under φ of a homoclinic point is also homoclinic.

Since p is a hyperbolic point for φ we know that the neighborhood U is being stretchedalong W u(p) and will eventually stretch enough to contain x again. Choose the first kfor which φk(U) ∩ U is nonempty and for which x ∈ φk(U). Looking at Figure 17, onecan see the Smale horseshoe mapping by restricting attention to the behavior of φ in theneighborhood U . (The disk in W u(p) is taken small enough that what these pictures showactually happens, except perhaps with other intersections of φk(U) and U also “between”p and x.)

18 Karl Petersen

x2

1

Ws(p)

(x)

(x)

Figure 18. W s(p) accumulates back on itself.

Label A0 the connected component of φk(U) ∩ U containing p, and A1 the one whichcontains x, and call Λ the largest φ-invariant set in A0 ∪A1. That is,

Λ =

∞⋂

j=−∞

(φk)j(A0 ∪A1).

Now we can code Λ over to the full 2-shift via a map h; for any ω ∈ Σ2 let

h(ω) =

∞⋂

j=−∞

(φk)−jAωj

This gives the commutative diagram:

Σ2σ

−−−−→ Σ2

h

y

yh

Λφ

−−−−→ ΛThe mapping treats a point ω ∈ Σ2 as the itinerary of some point u ∈ U as it travels

via φ through the sets A0 and A1. The map h is one to one since the sets A0 and A1 aredisjoint, and it is clear that h intertwines σ and φ. The surjectivity and bicontinuity of themapping should also be checked, as well as the fact that Λ is a hyperbolic set, a notion tobe defined in the next lecture.

8.3. The homoclinic mesh. Since it is true that the stable and unstable sets of a hyper-bolic fixed point are preserved under φ, we can investigate their structure by looking at theforward and backward iterates of a transverse homoclinic point x (if there is one). It iseasy to see that each φj(x) is also a homoclinic point for p; it is also transverse since φ is adiffeomorphism. So, as we watch the iterates of x under φ march through the stable set ofp we see that W s(p) must behave as in Figure 18.


(x)

(x)

x

-1

-2Wu(p)

Figure 19. W u(p) accumulates back on itself.

Figure 20. The homoclinic mesh.

A similar phenomenon occurs when we examine the behavior of the backwards iteratesof x, and look at the unstable set (see Figure 19).

When we combine the information from these pictures we get a feeling for how chaoticthe set of homoclinic points must actually be (see Figure 20).

9. Axiom A systems. January 27 (Notes by MA)

We now start our study of Axiom A diffeomorphisms. These form a large class of diffeo- Axiom Asystemsmorphisms that contain many interesting examples, such as hyperbolic toral automorphisms

and time one maps of gradient flows. More generally, Axiom A diffeomorphisms include twoimportant classes of diffeomorphisms: Anosov and Morse-Smale (definitions below). Forrelated material from the coursepack, see Bowen, Chap. 3 (p. 68) and DGS, Chap. 23 (p.224).

20 Karl Petersen

p

T Mp

Figure 21. Automorphism of a torus, showing the splitting of TpM

Definition 9.1. A closed φ-invariant subset Λ of a (compact, connected, C∞, Riemannian)manifold is a hyperbolic set for φ if there is a (continuous) splitting of the tangent bundleTΛ = Eu ⊕ Es (TpM = Eup ⊕ Esp for p ∈ Λ) such that

(1) (Dpφ)Eup = Euφ(p) and (Dpφ)E

sp = Esφ(p)

(2) there are constants c > 0 and λ < 1 (independent of p) such that

||Dp(φn)v|| ≤ cλn ||v|| for v ∈ Esp, n ≥ 0

||Dp(φn)v|| ≤ cλ−n ||v|| for v ∈ Eup , n ≤ 0.

Note that c and λ may depend on the Riemannian norm. If the Riemannian metric issuch that c = 1, then the metric is called adapted. It is always possible to find an adaptedRiemannian metric, so we really only need consider n = ±1. Also, that the splitting variescontinuously (in the topology on the tangent bundle) is actually a consequence of conditions(1) and (2).

Remarks 9.1.

(1) The attractor Λ in the Smale horseshoe is an example of a hyperbolic set. In thiscase, we have uniform hyperbolicity.

(2) The horseshoe we found in Smale’s Homoclinic Point Theorem (5.1) is a hyperbolicset. Figure 20 indicates hyperbolicity at each point, and in fact estimates can bemade precise using the hyperbolicity at the fixed (periodic) point p.

(3) You can prove hyperbolicity by finding a field of cones Cp ⊂ TpΛ such that (Dpφ)Cp =Cφ(p), and there is an m such that Dpφ

m expands on Cp and Dpφ−m expands on

TpM\Cp. This is sometimes easier than finding an exact splitting of the tangentspaces.

An important class of diffeomorphisms are Anosov diffeomorphisms:

Definition 9.2. The dynamical system (M,φ) is Anosov if all of M is a hyperbolic set forφ.

Example 9.1 (Hyperbolic automorphisms of the torus). Let φ be the map on the torusR2/Z2 given by the matrix

(2 11 1

). Then it is easy to see that each point of the torus is in

the hyperbolic set for φ. See Figure 21.

Definition 9.3. The nonwandering set for φ, denoted Ω(φ), is the set of all x ∈ M forwhich given any neighborhood U of x, there exists an n > 0 such that φnU ∩ U 6= ∅.


0

z

2z

Figure 22. The map z 7→ 2z on S2

Near a point in the nonwandering set, the dynamics of p exhibits a weak form of recur-rence.

Exercise 3. Show that Ω(φ) is closed and φ-invariant.

At the other ‘extreme’ of diffeomorphisms are Morse-Smale diffeomorphisms:

Definition 9.4. The dynamical system (M,φ) is Morse-Smale if

(1) Ω(φ) is finite (hence there are finitely many periodic points);(2) each periodic point is hyperbolic;(3) if x, y ∈ Ω(φ), then W s(x) and W u(y) intersect transversally, i.e., at each point z

in the intersection, the tangent spaces (from the immersions) span TzM .

Example 9.2. Time one maps of gradient flows, such as the North-South map on S2 =Riemann sphere (z 7→ 2z, see Figure 22) are Morse-Smale. In this case, Ω(φ) = 0,∞.The point 0 is a repelling fixed point (a source) while the point ∞ is an attracting fixedpoint (a sink), so we have that

W s(0) = 0 W u(0) = S2 − ∞

W s(∞) = S2 − 0 W u(∞) = ∞.

It’s clear that the points 0 and ∞ are hyperbolic, and for any point z in the intersection ofW u(0) and W s(∞), Tz(W

u(0)) + Tz(Ws(∞)) = TzS

2, i.e., the intersection is transverse.

10. Structural stability. January 29 (Notes by MA)

To continue the discussion of Morse-Smale systems, we note that Condition 3 of Definition Axiom Asystemscont.

9.4 is sometimes referred to as strong transversality, but the definition may vary.More examples of Morse-Smale systems include time one maps of gradient flows on a

torus with n ‘holes’. In this case, there will be 2(n+1) points in the nonwandering set Ω(φ)(n+ 1 sources and n+ 1 sinks). See Figure 23.

Anosov and Morse-Smale systems represent two ‘extremes’ of classes of diffeomorphisms.In an effort to combine these (and other) classes, Smale introduced the notion of Axiom A:

Definition 10.1. The dynamical system (M,φ) is called Axiom A if

(1) Ω(φ) is a hyperbolic set (note: Ω(φ) will be a potential attractor);(2) the periodic points for φ are dense in Ω(φ).

Remarks 10.1.

22 Karl Petersen

(1) It is clear that Morse-Smale systems are Axiom A.(2) It is also true that Anosov systems are Axiom A — this follows from the Anosov

Closing Lemma (Theorem 22.3).(3) In dimension 2, condition 1 of the definition implies 2 (Newhouse-Palis).(4) In dimensions greater than 2, condition 1 need not imply 2 (Dankner).

In Axiom A systems, the hyperbolicity and recurrence combine to give us complicateddynamics (on Ω(φ)). There are three things to discuss: existence of complicated dyna-mics, the persistence of qualitative dynamic behavior under perturbation, and typicality(genericity).

To make the notion of persistence more precise, we have

Definition 10.2. The dynamical system (M,φ) is (C1) structurally stable if there exists aneighborhood N of φ in the C1 topology such that each ψ ∈ N is topologically conjugateto φ (i.e., there exists a homeomorphism h :M →M such that h φ = ψ h).

Physically, structurally stable systems are the ones that are useful. Since we are not ableto make precise measurements, it is necessary to know that even when our numbers (say,the coefficients in a system of differential equations) are a bit off, the overall behavior ofour observed system is the same as for one with nearby values of the parameters.

In particular, the topological dynamics of (M,φ) and (M,ψ) will be the same: Perio-dic points of one correspond to periodic points of the other; both are either topologicallytransitive (or mixing) or not; invariant Borel measures of one correspond to invariant Borelmeasures of the other; closed invariant sets of one correspond to closed invariant sets of theother; and, their topological entropies will be equal.

Theorem 10.1 (Robbin, Robinson). Axiom A with strong transversality implies structuralstability.

Converses to this theorem have been conjectured and proved for some cases. The followingtwo corollaries were proved before Theorem 10.1

O

OO

O

O II

I

I

I

Figure 23. The gradient flow on a torus of genus 4. Points marked O aresources; points marked I are sinks.


Corollary 10.2 (Palis-Smale). Morse-Smale systems are structurally stable.

For example, in the time one map of the gradient flow φ indicated by Figure 23, anysmall perturbation of φ would simply move the ‘holes’ of M slightly. No new holes wouldbe created, or old ones removed.

Corollary 10.3 (Anosov). Anosov systems are structurally stable.

To discuss how ‘typical’ Axiom A systems are, we need the notion of generic:

Definition 10.3. A subset E of a complete metric space is called generic if it is thecomplement of a first category set. Equivalently, E is generic if it contains a dense Gδ set(a Gδ set is a countable intersection of open sets).

Theorem 10.4 (Kupka-Smale). Conditions (2) and (3) of Definition 9.4 (hyperbolic peri-odic points and strong transversality) are generic in the C1 topology on D1(M) (the spaceof C1 diffeomorphisms from M to M).

Corollary 10.5. The existence of homoclinic points and hence horseshoes is generic.

Remark 10.1. For an explicit method for finding homoclinic points, refer to the Poincare-Melnikov-Arnold method detailed in Robinson.

Remark 10.2. Are there Axiom A systems that do not have strong transversality? Pro-bably not — there could be some tangencies that do not persist under perturbations.

As an example of an Axiom A system satisfying strong transversality, consider the Smale’shorseshoe map extended to S2 (see Newhouse, p. 43). In this case, the nonwandering setis the attractor, plus a few fixed points needed to fit it onto S2. Hyperbolicity on thenonwandering set is easy to see, as is the fact that periodic points are dense (since they aredense in the 2-shift). And, while it is not so easy to see the stable and unstable manifolds forindividual points in the nonwandering set, it is true that they satisfy strong transversality.We conclude then that this map is structurally stable.

11. Smale’s solenoid. January 31 (Notes by PS)

Recall the definition of Axiom A (Definition 10.1):

Definition. A dynamical system (M,φ) is called Axiom A if

(1) the nonwandering set Ω(φ) is hyperbolic, and(2) Ω(φ) is the closure of the periodic points for φ, i.e. the periodic points for φ are

dense in Ω(φ).

Axiom A systems include Anosov and Morse-Smale systems. Different pieces of thenonwandering set Ω(φ) can be sources, sinks or attracting/repelling in different tangentdirections.

One noninvertible example is the map f(z) = z2 on the complex sphere. The nonwan-dering set consists of the attracting fixed points 0 and ∞, as well as the equator |z| = 1 ,where the dynamics are complicated.

24 Karl Petersen

11.1. Smale’s solenoid, an Axiom A attractor. Let M = S1 ×D1 = (θ, z) : 0 ≤ θ < Smale’s so-lenoid2π, |z| ≤ 1 be a solid torus with boundary.

Define the map φ : M → M by wrapping around twice in the θ direction and shrinkingby a factor of 1

4 in the z direction, with a twist:

φ(θ, z) =

(

2θ,1

4z +

1

2eiθ

)

.

See Figure 26.

Remarks 11.1.

(1) φ(M) wraps around twice.(2) z is shrunk by 1

4 and translated by at most 12 , so φ(M ) ⊂M .

(3) φ is injective.

To see 3, consider the cross-section of φ(M) for a fixed θ:

z

4+

1

2ei(θ/2) : |z| ≤ 1

∪

z

4+

1

2ei(π+θ/2) : |z| ≤ 1

.

We have two disks of radius 14 centered at 1

2ei(θ/2) and 1

2ei(π+θ/2).

Figure 24. The complex sphere. The north pole (∞) and the south pole(0) are attracting fixed points for f(z) = z2.

Figure 25. The solid torus M .


Figure 26. The image of φ :M →M .

Figure 27. A cross-section of φ(M ) for a fixed θ.

Figure 28. A cross-section of φ2(M) for a fixed θ.

We define Λ = ∩n≥0φn(M ), a closed, φ-invariant set, on which φ : Λ → Λ is a homeomor-

phism. This Λ is an attractor, in the sense that for all x ∈M , the distance d(φn(x),Λ) → 0as n→ ∞.

Each cross-section of Λ is a Cantor set. To see this, consider the cross-sections of φn(M )for fixed θ. At each stage, we get a nested pair of circles, the intersection of which will bea Cantor set.

26 Karl Petersen

Note, however, that the whole of Λ is a connected set, because each φn(M ) is connected.It should be clear that the nonwandering set Ω(φ) is contained in Λ (in fact, it equals Λ).There is a proof in Hasselblatt-Katok that Λ is hyperbolic. This should be no surprise; wehave expansion in the θ direction and contraction in the z direction.

12. Symbolic dynamics in action. February 3 (Notes by PS)

To show that Smale’s solenoid is Axiom A, we need only show that Λ = Ω(φ) and that ΛSmale’ssolenoid isAxiom A

is the closure of the periodic points of φ. To demonstrate this, we first prove the following:

Theorem 12.1. There is a continuous map h : Σ2 → Λ such that φh = hσ, i.e. (Λ, φ) isa topological/dynamic factor of (Σ2, σ). If this is true, the above assertions are immediatecorollaries. For example, because (Σ2, σ) is topologically mixing, there are dense orbits inΛ as well.

Proof. We partition M into two sets:

A0 = (θ, z) : 0 ≤ θ < π and A1 = (θ, z) : π ≤ θ < 2π

and define

h(ω) =⋂

i∈Z

φi(Aωi).

Backwards intersections divide the torusM , first in two, then quarters, then eighths, etc.,eventually narrowing down exactly what angle θ our point lies at. Forward intersectionskeep track of which of the two loops, top or bottom, the point falls into.

Considering only cross-sections, forward intersections determine which of the nested cir-cles we fall into. Together, forward and backward intersections determine a single point inΛ, so our map h is well defined.

Using standard arguments, h intertwines φ and σ. To see that h is continuous, considertwo nearby points ω1 and ω2 in Σ2 that agree on a long central block. Iterating backwards,we see that their images in Λ have nearly the same θ, and iterating forwards, we see thattheir images fall into the same nested circles. Thus, the images of the two points are neareach other in Λ as well.

Figure 29. The partitions of M : A0 and A1.


Figure 30. The forward images of A0 and A1.

What do the stable and unstable manifolds W s(θ, z) and W u(θ, z) look like?ExaminingW s(θ, z) first, note that all (θ, y) in the same cross-section at θ will eventually

be asymptotic with the orbit of (θ, z), since the nested circles contract under iteration. Now,suppose you have a point (θ′, y) such that 2nθ′ = 2nθ (mod 2π) for some n. Then eventuallyφn(θ′, y) and φn(θ, z) will fall into the same cross-section, and thereafter will converge. Onthe other hand, if 2nθ′ 6= 2n (mod 2π) for any n, then θ and θ’ will never converge. Thus:

W s(θ, z) = (θ′, y) : 2nθ′ = 2nθ, for some n

13. Topological ergodicity. February 5 (Notes by PS)

Definition. Let (X,φ) be a dynamical system. Then Solutionsto Exerci-ses 1 and2

(1) (X,φ) is topologically mixing if for U, V nonempty open subsets of X, there existsN such that for n ≥ N , φn(U) ∩ V 6= ∅.

(2) (X,φ) is regionally transitive if for U, V nonempty open subsets of X, there existsat least one n such that φn(U) ∩ V 6= ∅.

(3) (X,φ) is topologically ergodic if every proper closed invariant subset of X is nowheredense.

(4) (X,φ) is topologically transitive if there exists a dense orbit. In fact (as we will showlater), the set of all points with dense orbit is residual.

Definition. A subset E in X is first category if it is the union of countably many nowheredense sets. A subset E is residual if it contains the intersection of countably many opendense sets. The complement of a first category set is residual.

Theorem 1. Topological mixing implies regional transitivity. It is also true that regionallytransitive, topologically ergodic and topologically transitive are all equivalent properties.

Proof.

(1) Assume (X,φ) is regionally transitive. Let C be a proper closed subset of X andassume C is not nowhere dense. Then there is a nonempty open set U ⊂ C. Inaddition, Cc is also open and nonempty. By regional transitivity, there exists an nfor which φn(U)∩Cc 6= ∅, contradicting the fact that C is invariant. Thus, C mustbe nowhere dense, and X is topologically ergodic.

28 Karl Petersen

(2) Assume (X,φ) is topologically ergodic. Since X is a 2nd countable space, there is acountable basis Bi for X. Define the orbit of each Bi, O(Bi) = ∪n∈Zφ

n(Bi). BothO(Bi) and its complement O(Bi)

c are invariant. Since O(Bi)c is a closed invariant

set, by topological ergodicity it is nowhere dense, and thus O(Bi) is an open denseset.

Let F = ∩n≥0O(Bi), then F is a residual set. We claim that every x ∈ F hasa dense orbit. Let U be any open set in X. Then there exists a basic open setBi ⊂ U . By the definition of F , x ∈ O(Bi), and thus there is an n for whichφn(x) ∈ O(Bi) ⊂ U . Since the orbit of x meets every open set U , the orbit is densein X. Thus (X,φ) is topologically transitive. As we claimed above, the set of allpoints whose orbit is dense (F ) is in fact residual.

(3) Suppose (X,φ) is topologically transitive. Let x be a point with dense orbit. LetU, V be nonempty open sets. Since O(x) is dense, there exist n and m such thatφn(x) ∈ U and φm(x) ∈ V . Therefore, φm−n(U) ∩ V contains at least the pointφm(x) and is nonempty. Thus, (X,φ) is regionally transitive.

Definition. A set E satisfies the property of Baire if E = (G ∪M1) \M2, where M1 andM2 are first category sets.

Definition. A dynamical system (X,φ) is Baire ergodic if any for any invariant set Esatisfying the property of Baire, either E or Ec is residual. Equivalently, either E or Ec isfirst category.

Theorem 2. Baire ergodicity is equivalent to the properties in the previous theorem.

Proof.

(1) Suppose (X,φ) is Baire ergodic. Let C be a closed invariant subset. Trivially, Cc

satisfies the property of Baire, since it is open. Thus, either C or Cc is residual. IfC is residual, it is both dense and closed, and therefore all of X. If Cc is residual,it is dense. Therefore, it is impossible for C to have an open subset, so C must benowhere dense. We have therefore shown that (X,φ) must be topologically ergodic.

(2) Suppose (X,φ) is topologically transitive. Let E be an invariant subset of X, withthe property of Baire, that is E = (G ∪ M1) \ M2, where M1 and M2 are firstcategory. Let F be the set of all points whose orbit is dense.

If G ∩ F is empty, then Ec must be residual, since F \M1 ⊂ Ec. If G ∩ F isnonempty, then its orbit O(G) is open and dense. Since E is invariant, O(G) \O(M2) ⊂ E, and thus E is residual.

14. Stable and unstable manifolds. February 7 (Notes by MA)

We will now examine more closely the structure of the nonwandering set of an Axiom Adiffeomorphism. To do this, we will need the Stable Manifold Theorem for a hyperbolic set Λin M . This theorem essentially says that the splitting of the tangent bundle TΛ = Es⊕Eu

(i.e., for p ∈ Λ, TpM = Esp ⊕ Eup ) is integrable — it arises as tangent spaces to injectivelyimmersed manifolds.


For ǫ > 0, and x ∈M , we define

W sǫ (x) = y ∈M : d(φnx, φny) ≤ ǫ for all n ≥ 0

W uǫ (x) = y ∈M : d(φnx, φny) ≤ ǫ for all n ≤ 0.

Theorem 14.1 (Hirsch-Pugh). Let Λ be a hyperbolic set for a Ck (k ≥ 1) diffeomorphism Stable Ma-nifold The-orem

φ :M →M , with a splitting TpM = Esp⊕Eup for p ∈ Λ, as in the definition of the hyperbolic

set. We assume (as usual) that M has an adapted Riemmanian metric. Then for smallǫ > 0,

(1) W sǫ and W u

ǫ are Ck disks (injectively immersed) of dimension dimEsx and dimEuxrespectively, with TxW

sǫ = Esx, TxW

uǫ (x) = Eux .

(2) There is 0 < λ < 1 such that d(φnx, φny) ≤ λnd(x, y) for y ∈ W sǫ (x), and

d(φ−nx, φ−ny) ≤ λnd(x, y) for y ∈W uǫ , for n ≥ 0.

(3) W sǫ (x) and W

uǫ (x) vary continuously with x ∈ Λ.

Some consequences of Theorem 14.1:

(1) Statement (2) implies that W sǫ (x) ⊂ W s(x) for each x in Λ, since d(φnx, φny) is

approaching 0 exponentially fast. Similarly, W uǫ (x) ⊂W u(x).

(2) φ(W sǫ (x)) ⊂W s

ǫ (φx) and φ−1(W u

ǫ (x)) ⊂W uǫ (φ

−1x).(3) W s(x) = ∪n≥0φ

−nW sǫ (φ

nx) and W u(x) = ∪n≥0φnW u

ǫ (φ−nx). These are increasing

unions.(4) So for x ∈ Λ, W s(x) andW u(x) are injectively immersed copies of Euclidean spaces

of the right dimensions.

Corollary 14.2. If Λ is a hyperbolic set in (M,φ), then (Λ, φ|Λ) is expansive, i.e., there isa δ > 0 such that if x, y ∈ Λ, x 6= y, then there is some integer n such that d(φnx, φny) ≥ δ.

Proof. The proof hinges on the claim that for ǫ small enough, W sǫ (x) ∩ W u

ǫ (x) = x.Given this, it is easy to see that if x and y are such that d(φnx, φny) < ǫ for all n, theny ∈W s

ǫ (x) ∩Wuǫ (x), and so x = y.

Thus, we need only show that W sǫ (x)∩W

uǫ (x) = x. But this is clear, since W ǫ

s (x) andW ǫu(x) are small (Ck injectively immersed) disks which intersect transversely at x, just like

their tangent spaces (which have complementary dimensions and span TxM).

15. Expansiveness, canonical coordinates, basic sets. February 10 (Notes byRJ)

Definition 15.1. A dynamical system (X,φ) is said to be expansive if there exists a δ > 0such that if x, y ∈ X, x 6= y, then there exists an integer n such that d(φnx, φny) ≥ δ.

Corollary 15.1. If Λ is a hyperbolic set for a Ck diffeomorphism φ : M →M , then (Λ, φ|Λ)is expansive.

Proof. Choose δ so that δ < ǫ where ǫ is as in the Stable Manifold Theorem for HyperbolicSets (Theorem 14.1). If d(φnx, φny) < δ for all n ∈ Z, then y ∈ W s

δ (x) ∩Wuδ (x). But this

intersection consists only of the point x.

The following theorem of Smale describes the Canonical Coordinates or Local ProductStructure of an Axiom A diffeomorphism.

30 Karl Petersen

Theorem 15.2 (Smale). Let φ : M →M be a Ck Axiom A diffeomorphism. Then for everyCanonicalcoordinates small ǫ > 0 there is a δ > 0 such that if x, y ∈ Ω(φ) and d(x, y) < δ then W s

δ (x) ∩Wuδ (y)

consists of exactly one point [x, y]. Furthermore, [x, y] is in Ω(φ) and [x, y] is a continuousfunction of (x, y).

Proof. For the first statement we choose δ small enough to say that the pictures won’t changemuch. Specifically, W s

δ (x) and W uδ (x) are transverse, so for y near x, W s

δ (x) and W uδ (y)

remain transverse and hence have a single intersection point. This, and the continuity of[x, y] in x and y, follow from the continuity assertion in the Stable Manifold Theorem.

It remains to show that [x, y] ∈ Ω(φ). We first reduce to the case in which x and y arefixed points. (In an Axiom A system, periodic points are dense in Ω(φ), so choose x and yto be periodic. Then under an appropriate power of φ, x and y are fixed.)

Let U be an arbitrary neighborhood of [x, y] in M . We claim that

W uǫ (x) ⊂ ∪n≥0φnU.

Since φ is an expansion along W u(x), if we can find a small neighborhood V containing xin W u

ǫ (x) with V ⊂ ∪n≥0φnU then the entire set W uǫ (x) ⊂ ∪n≥0φnU .

W (x)δ

u

δ

sW (x)x

Figure 31. W sδ and W u

δ are closed disks of the appropriate dimension.There is only one intersection point if we keep δ small.

uW (x)

δ

u

δ

sW (x)

[x,y]

δW (y)

x

Figure 32. [x, y] is unique for small enough δ.


Φk

U N

k

W (x)

W (x)x

U

W (x)

Φ

Nu

s

u

s

δ

W (x)

δ

δδ

ΦkU

xx

Figure 33. Starting with U we take a high enough power φkU and restrictto a small neighborhood N of x

Let N be a small neighborhood around x in M and replace U by N ∩ (φkU) for somelarge k. Now the entire picture is local, so the Hartman-Grobman Theorem applies andsays that φ = Dxφ up to a continuous change of coordinates.

On N , Dxφ acts on eigenspaces, shrinking along the contracting subspace and blowingup along the expanding subspace. In this setting it is clear that the forward images of Uaccumulate along the unstable manifold of x.

We now play “hyperbolic pinball”: see Figure 34.The point [y, x] ∈ W s

ǫ (y) ∩Wuǫ (x), so for some n1, U

′ = φn1U ∩W sǫ (y) contains a point

near [y, x]. Repeating the argument made for U with U ′ shows that the forward images of U ′

accumulate along W uǫ (y), and for some large n2, φ

n2U ′∩U is nonempty. So φkφn1+n2U ∩Uis nonempty and the point [x, y] is nonwandering.

Now we have the tools necessary to prove the following result, the first part of which isdue to Smale and the second to Bowen:

Theorem 15.3. Let φ : M →M be an Axiom A diffeomorphism. Then Spectraldecompo-sition ofthe non-wanderingSet

Ω = Ω1 ∪ Ω2 ∪ · · · ∪ Ωn,

where the “basic sets” Ωi are pairwise disjoint closed φ-invariant sets and each (Ωi, φ) istopologically transitive.

Moreover, for each basic set Ωi,

Ωi = X0i ∪X

1i ∪ · · · ∪Xni−1

i ,

where the “elementary sets” Xji are pairwise disjoint closed sets which φ maps cyclically

(i.e. φXji = X

j+1 (mod ni)i ), and, further, each (Xj

i , φni) is topologically mixing.

16. Proof of Spectral Decomposition Theorem. February 12 (Notes by RJ)

We begin proving the Spectral Decomposition Theorem (Theorem 15.3), which was statedlast time.Proof. For each periodic point p ∈ Ω, let Xp =W u(p) ∩ Ω.

In order to show that the Xp separate Ω in the manner of the theorem, we first wishto show that each such Xp is both open and closed in Ω. To this end, let δ > 0 be as in

32 Karl Petersen

1n

uW (x)

δ

uW (x)

δ

s

δ

s

δW (x)

s

δ

s

δW (x)

W (y) W (y)

(1) (2)

U

Φ U

[x,y] [x,y]x

x

y y[y,x][y,x]

u

δ

u

δW (y) W (y)

u

δW (y)

uW (x)

δ

s

δW (x)

s

δW (y)

s

δW (x)

s

δW (y)

uW (x)

δ

u

δW (y)

n2

[y,x] y [y,x]

[x,y][x,y]xx

y

Φ

U’

U’

(4)(3)

Figure 34. “Hyperbolic Pinball”

the Canonical Coordinates Theorem so that if x, y ∈ Ω and d(x, y) < δ, there is a unique[x, y] ∈W s

ǫ (x) ∩Wuǫ (x) ∩ Ω.

Claim 1: We claim that Xp = Bδ(Xp) = y ∈ Ω: d(y,Xp) < δ.

W (p)u

sW (p)

Figure 35. Xp =W u(p) ∩ Ω.


Since the periodic points are dense in Ω, it is enough to show that for q ∈ Bδ(Xp)

periodic, q ∈ Xp. Since q ∈ W u(p) ∩ Ω, we can find an x ∈ W u(p) ∩ Ω so thatd(x, q) < δ. Let x′ = [q, x]. By definition, [q, x] ∈ W s

ǫ (q) ∩Wuǫ (x) ∩ Ω and since

x ∈W u(p), [q, x] ∈W s(q) ∩W u(p) ∩ Ω. (We have both

d(φ−nx′, φ−nx) → 0 as n→ ∞ and d(φ−nx, φ−np) → 0 as n→ ∞,

so by the triangle inequality,

d(φ−nx′, φ−np) → 0 as n→ ∞.)

Let ψ = φper(p)·per(q), so ψp = p and ψq = q. Now because x′ ∈W s(q),d(ψkx′, q) = d(ψkx′, ψkq) → 0 as k → ∞.

But at the same time ψkx′ ∈W u(p)∩Ω for all k. So q ∈W u(p) ∩ Ω = Xp. Thuseach Xp is both open and closed in Ω.

Of course many periodic points may generate the same set Xp.Claim 2: We claim that if p and q are periodic and q ∈ Xp, then Xp = Xq.

First note that φXp = Xφp and ψXp = Xψp = Xp.If q ∈ Xp then trivially W u

δ (q) ⊂ Bδ(Xp) = Xp. And (recall that the first union isan increasing union)

W u(q) = ∪n≥0φnW u

δ (φ−nq)

= ∪k≥0ψkW u

δ (ψ−kq)

= ∪k≥0ψkW u

δ (q)

⊂ ∪k≥0ψkXp

= Xp.

Hence Xq =W u(q) ∩ Ω ⊂ Xp.We claim also that p ∈ Xq. This being the case, we can reverse the roles of p and

q in the above argument to obtain Xp ⊂ Xq. So it remains to show that p ∈ Xq.Proceeding approximately as before, let x′ = [q, p]. Then x′ ∈W s

ǫ (q)∩Wuǫ (p)∩Ω

and d(ψkx′, q) = d(ψkx′, ψkq) → 0 as k → ∞. Recall from 1, however, that Xq is

open in Ω, so for large k, ψkx′ ∈ Xq. Since Xq is ψ-invariant, ψ−jx′ ∈ Xq for all j.

But also d(ψ−jx′, p) → 0 as j → ∞, so p ∈ Xq = Xq. And thus Xp = Xq.

Claim 3: Therefore we claim Ω = Xp1 ∪ Xp2 ∪ · · · ∪ Xpn where the Xpi are closed,

pairwise disjoint sets and each Xpi is invariant under φper(pi).The Xp are clearly closed. Since the Xp are also open, if there were Xp and Xq

withXp∩Xq 6= ∅, then this intersection is open and contains a periodic point q′. Butthen Xp = Xq′ = Xq. So the Xp are pairwise disjoint. Moreover, since Xp formsa cover of the compact set Ω, there is a finite subcover: Ω = Xp1 ∪Xp2 ∪ · · · ∪Xpn .

Since φXpi = Xφpi , φ acts as a permutation on the equivalence classes Xpi . TheseXpi are the elementary parts. The basic sets Ωj are the unions of the cycles of thispermutation, i.e. Ωj = Xpj ∪ φXp2 ∪ · · · ∪ φni−1Xpn .

Next time we will complete the proof by showing that (Xpi , φni) is topologically mixing.

34 Karl Petersen

n 3

1

2

3Ω Ω

Ω

Φ

Φ

Φ Φ

Figure 36. Elementary and Basic Sets

17. Topological mixing on basic sets. February 14 (Notes by RJ)

We wish to complete the proof of the Spectral Decomposition of the nonwandering setby showing that (Xpi , φ

ni) is topologically mixing.

Proof (cont).

Claim 4: (Xpi , φni) is topologically mixing.

Take U and V to be nonempty open subsets of Xpi . To show that (Xpi , φni) is

mixing, we will produce a point which is in both V and φNU for some value of N .Since the periodic points are dense , we can find periodic points p ∈ U and q ∈ V .Call their periods m and n, respectively.

For any (large) t, we can write tni = kmn + j with 0 ≤ j < mn. Then φjp =φtniφ−kmnp = φtnip ∈ Xpi

Again letting δ > 0 be small enough to ensure the existence of canonical coor-dinates, for each integer j, 0 ≤ j < mn, we can choose xj ∈ W u(φjp) ∩ Ω withd(xj , q) < δ. And let x′j = [q, xj ] ∈W s(q) ∩W u(xj) ∩ Ω ⊂W s(q) ∩W u(φjp) ∩ Ω.

Now φjU is a neighborhood of φjp, so for all large enough l, φ−lmnx′j ∈ φjU . Let

yj = φ−lmnx′j. Since x′j ∈ W s(q) and W s(q) is φ-invariant, yj ∈ W s(q). So for all

large enough k, φkmnyj ∈ V . Summarizing, yj ∈ φjU and φkmnyj ∈ V . But this

implies φ−jyj ∈ U , i.e. φkmn−tniyj ∈ U .

So φkmnyj ∈ V ∩φtniU for all large enough t, and hence (Xpi , φni) is topologically

mixing.

Example 17.1 (Morse-Smale Systems). In a Morse-Smale System, Ω(φ) is finite. Eachperiodic point is an elementary part and the basic sets are the periodic orbits.

Example 17.2 (the n-torus). Consider the system (Tn, A) where Tn is the n-torus and Ais an n-by-n integer matrix with detA = ±1. If no eigenvalue has modulus 1, then (Tn, A)is hyperbolic. Since the periodic points (the points with rational coordinates) are dense


in Tn, the nonwandering set Ω(φ) is all of Tn. Therefore, since Ω(φ) is connected, Ω(φ)consists of a single elementary part. Hence, (Tn, A) is topologically mixing.

Similarly, every Anosov diffeomorphism on Tn is topologically mixing.

Definition 17.1. Let I be a (possibly infinite) interval of integers, and let α > 0. Then an α-pseudo-orbitsand β-shadowing

α-pseudo-orbit for φ is a sequence of points (xi)i∈I in X such that d(φxi, xI+1) < α for alli, i+ 1 ∈ I

Remark 17.1. An easy way to produce an α-pseudo-orbit for φ is to pick points (xi)i∈Inear an actual orbit y, φy, φ2y, . . .. By the continuity of φ, if d(x1, φ

ny) is small enough,then d(φx1, φ

n+1y) will also be small, and by the triangle inequality, if d(x2, φn+1y) is small

and d(φx1, φn+1y) is small, then d(φx1, x2) is small.

Definition 17.2. Given a sequence of points (xi)i∈I in X and a point x ∈ X, we say thatx β-shadows (xi)i∈I if d(φix, xi) < β for all i ∈ I.

Next time we will prove the following theorem of Bowen:

Theorem 17.1. Let (X,φ) be an Axiom A system. Then given β > 0, there is an α > 0such that every α-pseudo-orbit in Ω(φ) (even of infinite length) is β-shadowed by some pointin Ω(φ).

Figure 37. Basic Sets and Elementary Parts of a Morse-Smale system

x xx

xΦ

Φ Φx 2Φ

23

1

1

Figure 38. (xi)i∈I is an α-pseudo-orbit for φ

36 Karl Petersen

x

x

x

Φ

17

18

19

Φ

Φ

x

xx17

18

19

ΦΦ

Figure 39. x β-shadows (xi)i∈I

18. Shadowing. February 17 (Notes by SS)

Example 18.1. (Basic sets and elementary parts of Axiom A diffeomorphisms)

(1) For a Morse-Smale system, the basic sets are the periodic orbits and the elementaryparts are the individual periodic points.

(2) For an Axiom A automorphism φ of the n-torus Rn/Zn, which is given by an n×ninteger matrix with det = ±1, no eigenvalue has modulus 1, since it is hyperbolic.Then the periodic points = π(Qn), where π : Rn → Rn/Zn is the natural projection.For, if x ∈ π(Qn), then x = (p1/q1, . . . , pn/qn), pi, qi ∈ Z, 0 ≤ pi < qi, for all i. Letq = lcm (q1, . . . , qn). Then x = (r1/q, . . . , rn/q), where ri ∈ 0, 1, . . . , q − 1 ∼= Zq,for all i. Now φ induces a one to one, onto map φq : (Zq)

n → (Zq)n of a finite set and

for this map every orbit is finite, i.e., every point is a periodic point. Conversely,since φ has no eigenvalue with modulus 1, det(φk − I) 6= 0 for all k ≥ 1. So, ifx ∈ Rn/Zn and φkx = x, i.e., (φk − I)x ∈ Zn, then by Cramer’s rule, x has rationalcoordinates, i.e., x ∈ π(Qn). Thus the nonwandering set Ω(φ) is Rn/Zn, sincethe periodic points are dense in Rn/Zn. Then Ω(φ) = Rn/Zn consists of a singleelementary part, since it is connected. Hence (Rn/Zn, φ) is topologically mixing.

Theorem 18.1. The nonwandering set of every Axiom A diffeomorphism has the pseudo-Pseudo-orbitShadowingTheorem

orbit shadowing property: Given β > 0, there is α > 0 such that every α-pseudo-orbit in Ωis β-shadowed by the orbit of some point in Ω.

Proof. Fix a very small ǫ > 0. Choose δ, 0 < δ < ǫ, small enough for the existence ofcanonical coordinates on Ω, i.e., if x, y ∈ Ω with d(x, y) < δ, then there exists [x, y] ∈W sǫ (x) ∩W

uǫ (y) ∩ Ω. Pick K with λKǫ < δ/2, where λ, 0 < λ < 1, is the hyperbolicity

constant. Choose α > 0 so that a tight enough α-pseudo-orbit of length (K + 1) in Ω isδ/2-shadowed by iterates of its starting point:

(yi)0≤i≤K is an α-pseudo-orbit ⇒ d(φiy0, yi) < δ/2 for i = 0, 1, . . . ,K.

It is possible. For, choose α0 > 0 such that if d(x, y) < α0, then d(φix, φiy) < δ/4K for

all i = 1, . . . ,K − 1. Let α = min α0,δ4. If (yj)0≤j≤K is an α-pseudo orbit in Ω, then

d(φi(φyj), φi(yj+1)) < δ/4K, for all i = 1, . . . ,K − 1, for all j = 0, 1, . . . ,K − 1, and so for


i = 0, 1, . . . ,K,

d(φiy0, yi) ≤i−2∑

l=0

d(φi−(l+1)(φyl), φi−(l+1)yl+1) + d(φyi−1, yi)

≤i−2∑

l=0

δ

4K+ α <

δ

4KK +

δ

4=δ

2.

We break the proof into steps:

(1) Try to β-shadow a finite α-pseudo-orbit of the form (xi)0≤i≤rK .

Define x′0 = x0 and x′K = [xK , φKx′0] ∈ W s

ǫ (xK) ∩Wuǫ (φ

Kx′0) ∩ Ω (it is possible

because d(xK , φKx′0) = d(xK , φ

Kx0) < δ/2 by choice of α).Inductively, define

x′iK = [xiK , φKx′(i−1)K ] ∈W s

ǫ (xiK) ∩W uǫ (φ

Kx′(i−1)K) ∩Ω

for i = 2, . . . , r. (It is possible. For, by choice of α, d(xiK , φKx(i−1)K) < δ/2,

and since x′(i−1)K ∈ W sǫ (x(i−1)K), we have d(φKx(i−1)K , φ

Kx′(i−1)K) < λKǫ <

δ/2. So d(xiK , φKx′(i−1)K) < δ.) Now x′rK = [xrK , φ

Kx′(r−1)K ] ∈ W sǫ (xrK) ∩

W uǫ (φ

Kx′(r−1)K) ∩ Ω.

Then x = φ−rKx′rK β-shadows (xi)0≤i≤rK , i.e., d(φix, xi) < β for i = 0, 1, . . . , rK.

For suppose sK ≤ i < (s+ 1)K, 0 ≤ s ≤ r. Then,

d(φix, φi−sKx′sK) = d(φi−rKx′rK , φi−sKx′sK)

≤r∑

l=s+1

d(φi−lKx′lK , φi−(l−1)Kx′(l−1)K)

=

r∑

l=s+1

d(φi−lKx′lK , φi−lK(φKx′(l−1)K))

≤r∑

l=s+1

λ−(i−lK)ǫ

≤ǫλ

1− λ,

since x′lK ∈W uǫ (φ

Kx′(l−1)K) and i− lK ≤ 0 for all l = s+ 1, . . . , r.

Also, d(φi−sKx′sK , φi−sKxsK) ≤ ǫ, since x′sK ∈ W s

ǫ (xsK) and i − sK ≥ 0, and

d(φi−sKxsK , xi) < δ/2 by choice of α.Thus

d(φix, xi) ≤ d(φix, φi−sKx′sK) + d(φi−sKx′sK , φi−sKxsK)

+ d(φi−sKxsK , xi)

<ǫλ

1− λ+ ǫ+ δ/2.

Set ǫ at the beginning such that the right side is less than β.

38 Karl Petersen

(2) To shadow a finite α-pseudo-orbit of the form (xi)0≤i≤n (n may be not a multiple

of K), just extend it to a new α-pseudo-orbit (xi)o≤i≤rK , where xi = φi−nxn, for

i = n+ 1, . . . , rK and rK ≤ n < (r + 1)K. Then, an x that β-shadows (xi)0≤i≤rKwill also β-shadow (xi)0≤i≤n.

(3) For an arbitrary finite α-pseudo-orbit (xi)s≤i≤s+l, set x′i = xi+s for i = 0, 1, . . . , l.

We can β-shadow it with some point x ∈ Ω, i.e., d(φix, x′i) = d(φix, xi+s) < βfor i = 0, 1, . . . , l. Then, φ−sx β-shadows (xi)s≤i≤s+l with the times synchronized

with the indices as required, since d(φi(φ−sx), xi) = d(φi−sx, x′i−s) < β for i =s, s+ 1, . . . , s+ l.

(4) Finally, suppose we have an infinite α-pseudo orbit (xi)−∞<i<∞.

For m ≥ 1, choose x(m) which β/2-shadows (xi)−m≤i≤m and let x be a limit

point of (x(m))m≥1 (recall that M is compact). Then, x β-shadows (xi)−∞<i<∞.Indeed, check it on a range, i.e., [−w,w] for an arbitrary number w. Pick a largem ≥ w so that d(φix, φi(x(m))) < β/2 for all i ∈ [−w,w]. Also d(φi(x(m)), xi) < β/2

for all i ∈ [−w,w], since x(m) β/2-shadows (xi)−m≤i≤m and so (xi)−w≤i≤w. Thus

d(φix, xi) < β for all i ∈ [−w,w].

19. Specification. February 19 (Notes by SS)

Corollary 19.1. Given β > 0, there is α > 0 such that if x ∈ Ω and d(φpx, x) < α, thenthere exists y ∈ Ω with φpy = y which β-shadows x, φx, . . . , φp−1x.

Proof. We may assume that β < ǫ/2, where ǫ is the expansive constant. Choose α as in thePseudo-Orbit Shadowing Theorem (Theorem 18.1). Then (xi)−∞<i<∞, where xi = φkx, i ≡k mod p, k ∈ 0, 1, . . . , p− 1, i.e.,

· · · , φp−1x, x, φx, · · · , φp−1x, x, φx, · · ·

is an α-pseudo orbit, since x, φx, . . . , φp−1x is an α-pseudo orbit and by hypothesis,

d(φ(φp−1x), x) < α.

So take y ∈ Ω which β-shadows it. Then φpy also β-shadows it and so for all j ∈ Z,d(φjy, φj(φpy)) ≤ d(φjy, xj) + d(xj , φ

j(φpy)) ≤ β + β < ǫ. By expansiveness, φpy = y.

We extend these pseudo-orbit acrobatics to specification — See DGS, p.193ff.Specification

Definition 19.1. Let X be a compact metric space and φ : X → X a homeomorphism.

(1) We say that the system (X,φ) has weak specification if given ǫ > 0, there existsK(ǫ) > 0 such that given x1, x2 ∈ X, given intervals I1 = [a1, b1], I2 = [a2, b2] ⊂ Z

with big enough gap (switching time) a2 − b1 ≥ K(ǫ), and given integer p > 0 bigenough that switching-back-time p− (b2 − a1) ≥ K(ǫ), there is a point x ∈ X withφpx = x such that d(φix, φix1) < ǫ for i ∈ I1 and d(φix, φix2) < ǫ for i ∈ I2.

(2) We say that (X,φ) has specification if we can do the same with any number k ≥ 1of orbit pieces : Given ǫ > 0, there exists K(ǫ) > 0 such that for any k ≥ 1, givenx1, . . . , xk ∈ X, given intervals Ij = [aj , bj ] ⊂ Z, j = 1, . . . , k with aj+1 − bj ≥ K(ǫ)


for j = 1, . . . , k − 1, and given an integer p > 0 such that p − (bk − a1) ≥ K(ǫ),there is a point y ∈ X with φpy = y such that d(φiy, φixj) < ǫ for all i ∈ Ij andj = 1, . . . , k.

Example 19.1 (Shift of finite type (SFT) and Higher Block Presentation). We will showthat every topologically mixing SFT has specification.

Let A = 0, 1, . . . , r− 1 be a finite alphabet. Then (AZ, σ) = Σr = the full r-shift, with Shift of Fi-nite Type(SFT)

the shift transformation σ defined by (σω)n = ωn+1 for n ∈ Z, ω = (ωn)n∈Z ∈ AZ.

Define d(ω, ζ) = 1/2k, where k = inf |m| : ωm 6= ζm for any ω, ζ ∈ AZ.A shift of finite type (SFT) determined by an r × r matrix B of 0’s and 1’s (transition

or incidence matrix) is ΣB = ω ∈ AZ : Bωnωn+1 = 1, for all n ∈ Z. Then ΣB is a closed

shift-invariant subset of AZ, and (ΣB , σ|ΣB) is a 1-step SFT.

The r×r matrix B corresponds to a finite directed graph on r vertices labeled 0, 1, . . . , r−1with directed edge from i to j if and only if Bij = 1. Thus ΣB = the set of all doubly infinitepaths on the associated graph, and Bij = 0 means there is no edge from the symbol i to thesymbol j, and in any · · ·ω−1ω0ω1 · · · ∈ ΣB, you never see the 2-block ij. More generally,we can give a finite list S of blocks in

0, 1, . . . , r − 1∗ = ∪k≥00, 1, . . . , r − 1k

and define

Σ(S) = ω ∈ AZ : no block in S appears (as a consecutive string) in ω.

If n = maximal length of blocks in S, then Σ(S) is an (n-1)-step SFT.Let ΣB be a 1-step SFT and fix m > 0. Define a new alphabet

Am =

m︷︸︸︷

A× · · · ×A = m-blocks of symbols on original alphabet

and define the m-th higher block code ζ(m) : ΣB → (Am)Z by

· · ·ω0ω1 · · ·ωm−1ωm · · ·ζ(m)

7−→ · · ·A0A1 · · ·

where Aj = ωjωj+1 . . . ωj+m−1 ∈ Am.

Then, ζ(m) : ΣB → ζ(m)(ΣB) is continuous, 1-1, onto and commutes with σ. So, ζ(m) :(ΣB , σ) → (ζ(m)(ΣB), σ) is a topological conjugacy. Generally, for any n,m ∈ N, if Σ(S) is

an n-step SFT, then ζ(m)(Σ(S)) is a higher-block presentation of Σ(S) and a 1-step SFTon the large alphabet Am.

For example, if A = 0, 1 and S = 101, then Σ(S) ∼= ζ(2)(Σ(S)), which is a 1-stepSFT on A2 = 00, 01, 10, 11. See Figure 40.

20. Specification in shifts of finite type. February 21 (Notes by SS)

Proposition 20.1. (1) A SFT ΣB is topologically ergodic if and only if B is irreducible,i.e., for all i, j ∈ 0, 1, . . . , r − 1, there exists n ≥ 1 such that (Bn)ij > 0, orequivalently, the graph is strongly connected, i.e., for any two vertices i, j, thereexists a path from i to j.

40 Karl Petersen

01

10

00 11

Figure 40. Higher block representation of Σ(101)

(2) A SFT ΣB is topologically mixing if and only if B is primitive, i.e., there existsn ≥ 1 such that Bn > 0, i.e., (Bn)ij > 0 for all i, j, or equivalently, the graph isstrongly connected and aperiodic.

Proposition 20.2. If (ΣB , σ) is a topologically mixing SFT, then it has specification.SFT andspecifica-tion Proof. Let m be such that (Bm)ij > 0 for all i, j (from topological mixing for cylinder sets

of length 1) i.e., we can get from any symbol i to any symbol j provided only that the pathhas length at least m. In order to see the idea, consider first a simple case, when ǫ = 1.Then, choose the switching time K(ǫ) = m.

Now suppose we are given k ≥ 1, x1, . . . , xk ∈ ΣB, intervals Ij = [aj , bj ] ⊂ Z, j = 1, . . . , k,with aj+1 − bj ≥ m, j = 1, . . . , k − 1, and an integer p ≥ m+ (bk − a1).

Since j1 = a2 − b1 + 1 > m, there exists a path u11 . . . u1j1

of length j1 from the symbol

u11 = x1b1 to the symbol u1j1 = x2a2 . Similarly, for l = 1, . . . , k, since jl = al+1 − bl + 1 > m

with ak+1 = p + a1, there exists a path ul1 . . . uljlof length jl from the symbol ul1 = xlbl to

the symbol uljl = xl+1al+1

, where xk+1ak+1

= x1a1 .Then define x∗ as follows:

x∗i =

xji for i ∈ Ij = [aj , bj ] and j = 1, . . . , k,

uli−bl+1 for bl ≤ i < al+1 and l = 1, . . . , k with ak+1 = p+ a1.

Now x∗i is defined for i = a1, a1 + 1, . . . , p+ a1 − 1.Let x∗ = · · · (x∗a1 · · · x

∗p+a1−1)(x

∗a1 · · · x

∗p+a1−1)(x

∗a1 · · · x

∗p+a1−1) · · · . This construction is

given schematically in Figure 41.Then x∗ ∈ ΣB , σ

px∗ = x∗, and for all i ∈ Ij , j = 1, . . . , k, d(σix∗, σixj) < 1/20 = 1, since

(σix∗)0 = x∗i = xji = (σixj)0. Note d(y, z) < 1 ⇔ y0 = z0 (otherwise d = 1).For arbitrary ǫ > 0, choose r such that 1/2r < ǫ and let K(ǫ) = m+ 2r.Now suppose we are given k ≥ 1, x1, . . . , xk ∈ ΣB, intervals Ij = [aj , bj ] ⊂ Z, j = 1, . . . , k,

with aj+1 − bj ≥ m+ 2r, j = 1, . . . , k − 1 and an integer p ≥ m+ 2r + (bk − a1).Since al+1−bl ≥ m+2r for l = 1, . . . , k, with ak+1 = p+a1, jl = (al+1−r)−(bl+r)+1 > m

for l = 1, . . . , k. So there exists a path ul1 . . . uljlof length jl from the symbol ul1 = xlbl+r to

the symbol uljl = xl+1al+1−r

, where xk+1ak+1−r

= x1a1−r.


Then define x∗ as follows:

x∗i =

xji for aj − r ≤ i ≤ bj + r and j = 1, . . . , k,

uli−bl−r+1 for bl + r ≤ i < al+1 − r, l = 1, . . . , k with ak+1 = p+ a1.

Now x∗i is defined for i = a1 − r, a1 − r + 1, . . . , p+ a1 − r − 1.Let x∗ = · · · (x∗a1−r · · · x

∗p+a1−r−1)(x

∗a1−r · · · x

∗p+a1−r−1) · · · .

Then x∗ ∈ ΣB , σpx∗ = x∗, and for all i ∈ Ij , j = 1, . . . , k, d(σix∗, σixj) < 1/2r < ǫ, since

for l ∈ [−r, r] and i ∈ [aj, bj ], aj − r ≤ i+ l ≤ bj + r and so (σix∗)l = x∗i+l = xji+l = (σixj)l,

i.e., σix∗ and σixj agree on the central (2r + 1)-block (Note that d(y, z) < 1/2r ⇔ y and zagree on the central (2r + 1)-block). Hence it has specification.

21. Specification in Axiom A systems. February 24 (Notes by KJ)

Last time we proved that a topologically mixing shift of finite type has specification. Specificationcont.Today we show that

Theorem 21.1. If (X,φ) is a topological dynamical system which is topologically mixing,expansive and has the pseudo-orbit shadowing property, then it has specification. Therefore,in an Axiom A system every elementary part of (Ω, φ) has specification.

Proof. (See DGS, pp 232 ff) The proof is similar to the proof of the corollary of the pseudo-orbit shadowing property: a tight enough pseudo-orbit which comes back to its beginning,has a periodic shadowing point. We do the same thing but for several orbit pieces at once.

Let β ≥ 0 and β < (expansive constant)/2. Let α be the pseudo-orbit tightness parameterthat produces β-shadowing. Now we need to find K(ǫ).

Let U = U1, U2, . . . , Ur be a cover of X by α-balls. Using topological mixing, chooseK large enough so that if n ≥ K then φ−nUi ∩ Uj 6= 0 for all i, j = 1, . . . , r. This allows usto get from one part of the space to another, providing we allow a time at least K for thetransition.

Turning to specification, suppose we are given the

Ii = [ai, bi], xi ∈ X, i = 1 . . . k,

with gaps ai − bi ≥ K, p − (bk − ai) ≥ K (ǫ = β).Define xk+1 = x1, ak+1 = a1 + p.Now we are going to make our pseudo-orbit. Define an α-pseudo orbit as follows:

zi = φjxj if aj ≤ i < bj for all j = 1, . . . , k.

any validpath

p+a1

a b2 2

same

symbolsas in x 2

same

symbolsas in x k

a bk k

a1

:x*

same

symbolsas in x 1

b1

any validpath

any validpath

Figure 41. Construction of x∗i for a1 ≤ i ≤ p+ a1 − 1

42 Karl Petersen

For j = 1, . . . , k find yj ∈ U(φbjxj) ∩ φ−aj+1+bj (U(φaj+1xj+1)), where U(x) is a member of

U to which x belongs. Since aj+1 − bj ≥ K, the intersection is nonempty because of theabove argument based on topological mixing.

Further define

zi = φi−bjyj if bj ≤ i < aj+1.

Thus the distance between φb1x1 and y1 is less than α, and so on:

d(φbj−1xj , yj) < α

for j = 1, 2, . . . k. We used strong mixing to get us from an α-neighborhood of one point toan α-neighborhood of another point with the orbit of one point.

At the end, we land within α of φa1x1. This gives an α-pseudo-orbit of length p. Repeat,to get an α-pseudo-orbit (zi)i∈Z by defining zi+p = zi for other i’s.

Now, by the pseudo-orbit shadowing property, there is an x ∈ X such that d(φix, zi) < βfor all i ∈ Z. But we also have

d(φix, φi+px) ≤ d(φix, zi) + d(zi, φi+px)

≤ d(φix, zi) + d(zi+p, φi+px)

≤ 2β

≤ expansive constant (for all i)

and so φpx = x. Of course the orbit pieces of x on the Ii β-shadow those of the xi.

Remark 21.1. Note that (Ω(φ), φ) may not be topologically mixing. The elementary partsare invariant sets, and the xi’s could have come from any of them.

Theorem 21.2 (DGS Chapter 21). Let (X,φ) be a topological dynamical system with weakspecification (for example, an elementary part of the nonwandering set of an Axiom Asystem). Then the following are true:

(1) The periodic points are dense (Proof: have a periodic point shadow one orbit pieceof one point).

(2) (X,φ) is topologically mixing (Proof: choose two orbit pieces, one in U and one inV , and have a point orbit shadow the two pieces).

(3) (X,φk), (X×Y, φ×ψ) (where (Y, ψ) has weak specification) and any factor of (X,φ)all have weak specification.

(4) Exercise: htop(X,φ) > 0.

(continued on the next day)

22. Consequences of specification. February 26 (notes by KJ)

Theorem 22.1 (cont.). Let (X,φ) be a topological dynamical system with weak specificationResults onspecifica-tion

(for example, an elementary part of the nonwandering set of an Axiom A system). Thenthe following are true:


(1) The periodic points are dense (Proof: have a periodic point shadow one orbit pieceof one point.)

(2) (X,φ) is topologically mixing (Proof: choose two orbit pieces, one in U and one inV , and have a point orbit shadow the two pieces).

(3) (X,φk), (X×Y, φ×ψ) (where (Y, ψ) has weak specification) and any factor of (X,φ)all have weak specification.

(4) (Bowen) htop(X,φ) > 0.(5) (Sigmund) In the compact metric space I (with respect to the weak∗ topology) of φ-

invariant Borel probability measures on X, the set E of ergodic invariant measures(which is equal to the set of extreme points of the compact convex set I) is residual(a dense Gδ).

Remark 22.1. This result is extremely different from uniquely ergodic systems —in systems with weak specification, there are many ergodic measures.

(6) (Sigmund – proof by Parthasarathy) The set of strongly mixing measures in I isfirst category in I. [Remember that strongly mixing means that for all A,B ∈B, µ(T−nA ∩ B) → µ(A)µ(B) as n → ∞. Page 199 of DGS has a reference toParthasaranthy (1961). He must have proved this using a more general situationthan specification.]

(7) Every µ ∈ I has a generic point: There is x ∈ X such that

1

n

n−1∑

k=0

f(φkx) →

∫

Xfdµ for all f ∈ C(X).

Equivalently, there are x’s for which the averages of the point masses 1n

∑n−1k=0 δφkx

converge to µ in the weak∗ topology.

Remark 22.2. Note that if µ is ergodic, it is not hard to find generic points. Givenf , the Ergodic Theorem gives a set of full µ-measure such that 1

n

∑f(φkx) →

∫

X fdµ. Then we could get a set of full µ measure that would be good for all f in acountable dense set in C(X). To get these points to work for an arbitrary g, givenǫ > 0, choose one of our good f from the countable dense set with |g − f | < ǫ

3 . Then∣∣∣∣

1

n

∑

g(φkx)−

∫

gdµ

∣∣∣∣≤

∣∣∣∣

1

n

∑

(g − f)(φkx)

∣∣∣∣

+

∣∣∣∣

1

n

∑

f(φkx)−

∫

fdµ

∣∣∣∣+

∣∣∣∣

∫

(f − g)dµ

∣∣∣∣

≤ǫ

3+ǫ

3+ǫ

3

if n is chosen so that∣∣ 1n

∑f(φkx)−

∫fdµ

∣∣ < ǫ

3 . This is a corollary of the theoremon p. 202 ff. of DGS, and tells us that we can find generic points for any measure,even the non-ergodic ones.

(8) (Sigmund) The set of points x ∈ X with maximum oscillation:

Vφ(x) = the set of limit points of

1

n

n−1∑

k=0

δφkx : n = 1, 2, . . .

= I,

44 Karl Petersen

is residual.

Remark 22.3. For points of maximum oscillation, not even statistical descriptionof the behavior of O(x) is possible. Thus according to statement (8) there is aresidual, that is to say, a topologically typical, set of points whose orbits are beyonddescription, in that we can make no statement (even statistically) of what is goingon. This is really bad chaos. With good chaos, you get an attractive measure thatstatistically describes long-term average behavior.

In systems with specification averages of point masses along a typical orbit shadowevery measure.

This property, in combination with the Kolmogorov (algorithmic) complexity ofthe orbit, is discussed in Homer White’s thesis.

For more on this theorem, see Chapter 22 in DGS.

Theorem 22.2 (Bowen). Let (X,φ) be expansive and have specification. For each n =1, 2, . . . let Pn = cardx ∈ X : φnx = x. Then

(1) htop(φ) = limn→∞1n log Pn

(2) (X,φ) has a unique measure of maximal entropy (hµ(φ) = htop(φ)), ((X,φ) is called

intrinsically ergodic) which is given by (among other ways) limn→∞1Pn

∑

φnx=x δxin the weak∗ topology.

In a toral automorphism, we have one basic set, which is topologically mixing, and Lebes-gue measure is the unique measure of maximal entropy. As discussed earlier, the periodicpoints are the points with rational coordinates. The convergence to Lebesgue measurein statement 2 of Theorem 22.2 should remind us of Riemann integration of continuousfunctions. The same argument works for Haar measure on compact abelian groups.

Note that Pn is finite for every n: if we had infinitely many fixed points, for example, wewould defeat expansiveness. DGS gives a bound for the number of periodic points.

Our interest in this theorem is mainly for the elementary parts of (Ω, φ).DGS calls the unique measure in Theorem 22.2 the ‘Bowen measure’ — but it is not the

same as the SRB measure.

22.1. More consequences of the pseudo-orbit shadowing property.

Theorem 22.3 (Anosov Closing Lemma). If (M,φ) is an Anosov system (the entire mani-AnosovClosingLemma

fold M is a hyperbolic set) then the periodic points are dense in Ω(φ). Consequently, (M,φ)is Axiom A.

This is a loose end left over from the beginning weeks of the class.

Proof. We review previous arguments in the pseudo-orbit shadowing property and othersto see where we used the density of periodic points in Ω(φ).

• Stable Manifold Theorem: This holds true on hyperbolic sets, hence on all of M .• Canonical coordinates [x, y] ∈W s

ǫ (x) ∩Wuǫ (y) ∩Ω. We used expansiveness. It took

effort to prove that [x, y] ∈ Ω, where we used “homoclinic pinball” and density ofperiodic points. But to prove this theorem we don’t require that [x, y] be in Ω, justin M .


• The proof of the pseudo-orbit shadowing property did not use that periodic pointsare dense in Ω, just hyperbolicity and canonical coordinates.

• The corollary on closing up periodic orbits is also fine, since it just uses expansivenessand hyperbolicity.

Now, let x ∈ Ω(φ) and let β > 0. . We are trying to show that there is z ∈ M which isperiodic and within β of x. Choose α as in Corollary 19.1 and let U = Bα/2(x). We knowthat there is y ∈ U such that φny ∈ U, since x is in the nonwandering set. Then there is az which β shadows y, φy, . . . φny and is periodic, hence is within β of x.

23. Anosov Closing Lemma. February 28 (Notes by KJ)

23.1. More consequences of the pseudo-orbit shadowing property.

Proposition 23.1. If (M,φ) is an Axiom A dynamical system, then there is a neighborhood Fundamentalneighbor-hood

U of Ω(φ) such that⋂

n∈Z

φnU = Ω(φ).

Proof. Take β < (expansive constant)/2, α as in the pseudo-orbit shadowing theorem, andγ < (α + (expansive constant))/2, and such that if d(x, y) < γ, then d(φx, φy) < α/2. LetU = y ∈M : d(y,Ω) < γ.

(Showing that ∩n∈ZφnU ⊂ Ω(φ)) Suppose that y ∈ ∩n∈Zφ

nU . This means that for eachn ∈ Z we can find xn ∈ Ω with d(φny, xn) < γ. Then the xn’s are an α-pseudo-orbit:

d(φxn, xn+1) ≤ d(φxn, φn+1y) + d(φn+1y, xn+1)

< α/2 + γ < α

Now find x ∈ Ω whose orbit β-shadows the xn’s. Then for every n ∈ Z,

d(φny, φnx) ≤ d(φny, xn) + d(xn, φnx)

≤ γ + β

≤ (expansive constant).

Therefore y = x ∈ Ω.The reverse inclusion is clear since Ω is φ-invariant.

Remark 23.1. Similarly, for each j there is a neighborhood Ej of Ωj such that ∩n∈ZφnEj =

Ωj. (In the proof above, make sure that α, β, γ are smaller than the fixed distance of theΩj’s from each other.)

The proposition implies that if something is not in Ω, it can’t stay in U all the time —at some time it gets outside of U .

Proposition 23.2. In a general (compact) topological dynamical system (X,φ) with non-wandering set Ω, for each x ∈ X,

d(φnx,Ω) → 0 as n→ ∞

d(φ−nx,Ω) → 0 as n→ ∞.

46 Karl Petersen

If (Ω, φ) is, as in the Axiom A case, the disjoint union of finitely many closed sets Ωj, thenfor each x ∈ X there are j(x), k(x) such that

d(φnx,Ωj(x)) → 0 and d(φ−nx,Ωk(x)) → 0

as n→ ∞.Thus if

W s(Ωj) = x ∈ X : d(φnx,Ωj) → 0

W u(Ωj) = x ∈ X : d(φ−nx,Ωj) → 0

as n→ ∞, then

X = ⊔jWs(Ωj) = ⊔jW

u(Ωj)

where ⊔ means disjoint union.

Proof. Let x ∈ X and let ω(x) = φnx : n ≥ 0′ where U ′ denotes the set of limit pointsof U . We will show that ω(x) ⊂ Ω(φ). (This will prove the first part of the Proposition: Ifthe distance didn’t go to 0, then there would be a neighborhood U of Ω and infinitely manyni’s with φ

nix 6∈ U . Then there would be a limit point of ω(x) outside of U , hence not inΩ.)

Let z ∈ ω(x), say φnix→ z, ni ≥ 0.We will show that z ∈ Ω, i.e. that every neighborhoodof z returns to itself in forward time. Let V be a neighborhood of z. Then find φnix, φnjx ∈V with ni < nj. Then φ

ni−njV ∩ V is not empty, since φnjx ∈ V and φnjx = φnj−niφnix ∈φnj−niV. Therefore, z ∈ Ω and the first part is proved.

For the second statement, use the limit set α(x) = φ−nx, n ≥ 0′.Now we want to show that each ω(x) is in one Ωj. For each j, define Uj, Vj to be

disjoint open neighborhoods of Ωj,∪k 6=jΩk respectively, so that Uj ∩ Vj = ∅. Suppose wehave infinitely many φnx ∈ Uj and Vj. Then we can find an increasing sequence ni with

φnix ∈ Uj and φni+1x ∈ Vj with φ

nix → z ∈ Ωj and φni+1x → w ∈ Vj ⊂ U cj But then also

w = φ(z), which is impossible by the invariance of Ωj.

24. More consequences of pseudo-orbit shadowing. Markov partitions.March 3 (Notes by LK)

We begin with one more consequence of pseudo-orbit shadowing.

Proposition 24.1. Let Ωj be a basic set of an Axiom A system (M,φ). Then

(1) W s(Ωj) =⋃

x∈Ωj

W s(x) and W u(Ωj) =⋃

x∈Ωj

W u(x).

(2) Given ǫ > 0, for each j there exists a neighborhood Uj of Ωj such that⋂

k≥0

φ−kUj ⊂

⋃

x∈Ωj

W sǫ (x) and

⋂

k≥0

φkUj ⊂⋃

x∈Ωj

W uǫ (x).

Proof.


x

W (y)

[x,y]

y W (x)s

u

ε

ε

Figure 42. A rectangle R

(1) That W s(Ωj) ⊃⋃

x∈Ωj

W s(x) is trivial. To show W s(Ωj) ⊂⋃

x∈Ωj

W s(x), we pick

a y ∈ W s(Ωj). Let ǫ > 0, let β = ǫ/2, let α be determined by the Pseudo-Orbit Shadowing Theorem for β, and let γ < α/2. Pick N large enough thatd(φny,Ωj) < γ for all n > N . This means we can find points xn ∈ Ωj withd(φny, xn) < γ for n > N . Since γ < α/2, (xn)n≥N is an α-pseudo-orbit in Ωj.Take x ∈ Ω that β-shadows (xn)n≥N . If β is small enough, we must have x ∈ Ωj,since its orbit β-shadows points of Ωj and the Ωj are closed and pairwise disjoint.Hence

φNy ∈W sǫ (φ

Nx) ⊂W s(φNx)

and thus

y ∈ φ−NW s(φNx) =W s(x).

The other direction follows similarly.(2) Using N = 0 and Uj = y ∈ M : d(y,Ωj) < γ in part (1), we showed that

⋂

k≥0

φ−kUj ⊂⋃

x∈Ωj

W sǫ (x).

24.1. Markov partitions. Fix a basic set Ωj of an Axiom A system (M,φ) and an ǫ > 0 Markovpartitionssmall enough for the Stable Manifold Theorem to hold. We define a rectangle as a set R ⊂ Ωj

which has small enough diameter that if x, y ∈ R then [x, y] is defined and [x, y] ∈ R. SeeFigure 42.

We say that R is proper if R is closed and R = intR, where the interior of R is relative toΩj. A Markov partition of Ωj is a finite family R1, . . . , Rm of proper rectangles such that:

(1) all Ri have small diameters (even compared to ǫ),(2) Ωj = R1 ∪ · · · ∪Rm,(3) intRi ∩ intRj = ∅ if i 6= j,

48 Karl Petersen

sx

iW (x,R )

R i

Figure 43. W s(x,Ri)

φR j R i

jW ( x, R )φ

W (x, R )φs

i

s

xφ

Figure 44. φW s(x,Ri)

(4) LetW s(x,R) =W sǫ (x)∩R andW u(x,R) =W u

ǫ (x)∩R. If x ∈ intRi and φx ∈ intRj ,then φW u(x,Ri) ⊃W u(φx,Rj) and φW

s(x,Ri) ⊂W s(φx,Rj). See Figures 43 and44.

Thus the image of each unstable section goes all the way across, in the unstable direction,any rectangle that it gets mapped to. We include an illustration in Figure 45 of rectanglesthat map badly and do not satisfy the last condition in the definition of a Markov partition.

25. Exercises on solenoid and Markov partitions. March 5 (Notes by LK)

Exercise 4.

(1) The solenoid. Let S1 = R/Z = [0, 1] with 0 ∼ 1. Let X = (S1)Z be the space of

all doubly-infinite sequences x = (xi)i∈Z where 0 ≤ xi < 1 for all i. We endow Xwith the product topology and let σ : X → X be the shift map. X also has theproduct group structure and σ is a continuous group automorphism. Let G = x ∈X : xn+1 = 2xn for all n.


R j

jW ( x, R )φs

xφ

W (x, R )s

φ i

Rφ i

R j

φ W (x, R )i

Rφ i

φ x

W ( x, R )φ j

u

u

Figure 45. Rectangles that map badly

Show that G is closed and σ-invariant. Also, show that (G,σ) is an automorphismof a compact abelian group and is topologically conjugate to the solenoid system(Λ, φ). Can you use this conjugacy to find the stable and unstable sets of points?

(2) Determine whether the partitions that we used to code the closed invariant setsin the horseshoe and solenoid constructions are Markov partitions. Recall that wefound the invariant sets as factors of symbolic systems starting with some partitionof the original space.

Notetaker’s remark: We also completed the definition of a Markov partition in today’slecture. For clarity, I have moved this discussion into the March 3 lecture notes.

26. Existence of Markov partitions. March 7 (Notes by LK)

We begin with a discussion of rectangles in shifts of finite type. Let ΣB ⊂ AZ, whereB is an r-by-r matrix with entries 0 or 1 and A = 0, 1, . . . , r − 1. We define canonicalcoordinates in the following manner: if ω, ξ ∈ ΣB are close enough that ω0 = ξ0 (i.e.d(ω, ξ) < 1) then we let [ω, ξ] = . . . , ξ−2, ξ−1, ξ0, ω1, ω2 . . .. We note that [ω, ξ] is in ΣBbecause the transitions are allowed since ω0 = ξ0. Further, this point is unique in ΣB. If

50 Karl Petersen

ρ = [ω, ξ] then ρ ∈W s1 (ω) ∩W

u1 (ξ) = ρ. That is,

d(φkρ, φkω) ≤ 1 for all k ≥ 0 and d(φkρ, φkξ) ≤ 1 for all k ≤ 0.

So a cylinder set [a] is a rectangle in this setting. That is, if ω, ξ,∈ [a], then [ω, ξ] ∈ [a]. Wenext prove the following warm-up observation about rectangles in Ωj.

Lemma 26.1. If, in a basic set Ωj of an Axiom A system (M,φ), R is a proper rectangleWarm-upLemma with small enough diameter compared to ǫ, then

(3) intΩjR =

⋃

x∈R

(intWuǫ (x)∩ΩW

u(x,R) ∩ intW sǫ (x)∩Ω

W s(x,R))

Figure 46 illustrates Lemma 26.1. This lemma can also be stated in terms of boundaries,as in Bowen:

∂ΩjR = ∂sR ∪ ∂uR

where

∂sR = x ∈ R : x /∈ intWuǫ (x)∩ΩW

u(x,R),

∂uR = x ∈ R : x /∈ intW sǫ (x)∩Ω

W s(x,R).

Proof. If x ∈ intΩjR, then W u(x,R) = R ∩ (W u

ǫ (x) ∩Ω) is a neighborhood of x in W uǫ ∩Ω

and thus x ∈ intWuǫ (x)∩ΩW

u(x,R). Similarly, x ∈ intW sǫ (x)∩Ω

W s(x,R).Conversely, suppose x is in the right-hand side of Equation 3. We want to show if

y ∈ Ωj is close enough to x, then y ∈ R. For y near x, we know [x, y] ∈ W sǫ (x) ∩ Ω

and [y, x] ∈ W uǫ (x) ∩ Ω. Further, [x, y] ∈ W s(x,R) and [y, x] ∈ W u(x,R) since canonical

coordinates are continuous functions of y, and x is in the (relative) interiors of these sets.Thus [x, y] and [y, x] are in R. Then [[y, x], [x, y]] = y is also in R.

Example 26.1 (Toral automorphisms have Markov partitions). For a construction in theMarkovpartitionsof toralautomor-phisms

2-dimensional case, see page 250 of Petersen’s book. We will outline the procedure here. LetA be a 2-by-2 integer matrix with determinant ±1. Then A has one real eigenvalue greaterthan 1, and one real eigenvalue less than one. The resulting eigenspaces have irrational

s

R

W (x,R)x

W (x,R)u

W (x)

W (x)

s

u

ε

ε Ω

Ω

R

R

R

R

u

u

s

s

Figure 46. An illustration of Lemma 26.1


Figure 47. One eigenspace

Figure 48. Both eigenspaces

Figure 49. Extending the eigenspaces to obtain a Markov partition

slope. We use these eigenspaces to make the partitions. To get rectangles, we make surethat we stop extending the eigenspace when we intersect the other eigenspace. We give anillustration of this procedure in Figures 47, 48, and 49 and encourage the reader to examinethe details in Petersen’s book.

Lind and Marcus explicitly find a Markov partition for the toral automorphism

A =

(1 11 0

)

We note that once you have a Markov system, you have a shift of finite type; sometimesit is easier to just work on this SFT. In fact, Lind and Marcus give an elementary definitionof a Markov partition in terms of a coding to a SFT. Denker, Grillenberger, and Sigmundgeneralize these concepts to topological dynamical systems on page 241 of their book.

52 Karl Petersen

Theorem 26.2 (Bowen). Let (M,φ) be an Axiom A diffeomorphism and Ωj a basic set.Markovpartitionsof Ωj

Then there are Markov partitions of Ωj into rectangles of arbitrarily small diameter.

Proof. Let ǫ be as in the Stable Manifold Theorem (Theorem 14.1). Take a positive β muchless than ǫ and less than 1

2 the expansive constant. Let α > 0 be as in the Pseudo-OrbitShadowing Theorem (Theorem 18.1) for β. Choose γ < α/2 such that d(x, y) < γ impliesthat d(φx, φy) < α/2.

Choose a finite γ-dense set P = p1, p2, . . . , ps in Ωj . Look at the symbolic space ofα-pseudo-orbits using P .

Σ(P ) = ω ∈ PZ : d(φωi, ωi+1) < α for all i ∈ Z.

This is actually a SFT. Give Σ(P ) the product topology. There is a map

θ : Σ(P ) → Ωj

defined byθ(ω) = the unique x in Ωj whose orbit β-shadows (ωi)i∈Z.

We need to check the existence and uniqueness of such an x. If ω ∈ Σ(P ), then the orbitof ω defines an α-pseudo-orbit and there exists a point x β-shadowing it by the Pseudo-OrbitShadowing Theorem. Uniqueness follows because if there is another such y which β-shadows(ωi), then d(φ

ix, φiy) < 2β for all i, but β is less then half the expansive constant. Hencex = y.

Clearly, θσ = σφ, so that the following diagram commutes:

Σ(P )σ

−−−−→ Σ(P )

θ

y

yθ

Ωjφ

−−−−→ Ωj

We next show that θ is onto. Given x ∈ Ωj, since P is γ-dense we can choose ωi ∈ Pwithin γ of φix for all i. Then θ(ω) = x because the orbit of x and the orbit of θ(ω) bothβ-shadow the (ωi)i∈Z.

27. Start the proof of existence of Markov partitions. March 17 (Notes byNP)

Editor’s Note: There are some overlaps in the notes concerning the next several weeks.(Several times the lecturer retraced his steps, and several notetakers expanded their write-ups for coherence.)

This week we will be continuing the proof of Bowen’s theorem proving the existence ofMarkov partitions for the nonwandering set of an Axiom A diffeomorphism (Theorem 26.2).

We prove that θ is continuous by contradiction—suppose it is not. Then there exists anη > 0 such that for all n = 1, 2, ... we can find points ω(n), ζ(n) ∈ Σ(P ) such that

ω(n)(i) = ζ(n)(i) for |i| ≤ n,

butd(θω(n), θζ(n)) ≥ η.

Figure 50 illustrates the central blocks of ω(n) and ζ(n).


ω(n)

ζ(n)-n

-n n

n

0

0

Figure 50. ω(n) and ζ(n) are the same on a central block.

If x(n) = θ(ω(n)) and y(n) = θ(ζ(n)), we can pass to a subsequence and assume x(n)→x

and y(n)→y, so that d(x, y) ≥ η.

Now x(n)β-shadows ω(n) and y(n)β-shadows ζ(n), and ω(n)(i) = ζ(n)(i) for |i| ≤ n, so

d(φix(n), φiy(n)) ≤ 2β for |i| ≤ n.

This implies that

d(φix, φiy) ≤ 2β < expansive constant

for all i, and this contradicts the expansiveness of φ.We have shown that θ is a continuous factor map from a shift of finite type onto a

basic set. Next we shall show that this mapping also maps certain rectangles in Σ(P ) ontorectangles in Ωj.

Consider the cylinder set [pi] = ω : ω0 = pi in Σ(P ). It is a rectangle since for any

ω, ζ ∈ [pi], [ω, ζ] = ..., ζ−2, ζ−1, ζ0, ω1, ω2, ...,

which is clearly in W s1 (ω) ∩W

u1 (ζ) ∩ [pi]. (Cylinder sets corresponding to longer blocks are

also rectangles in Σ(P )).Define Ti to be the image under θ of the rectangle [pi]; we will show it is also a rectangle

for all i = 1, 2, ..., r. These rectangles in Ωj are crucial to the construction of the Markovpartition of Ωj—refinements of them form the said partition.

Note that Ti is closed and the diameter of Ti is less than 2β, because any point y ∈ Ωjwhich beta-shadows a point in [pi] must be within β of pi, since this is the time-zero entryin the pseudo-orbit.

Let x, y ∈ Ti, and suppose that

ω = ..., ω−1, pi, p′i, ...

θ→ x , and

ζ = ..., ζ−1, pi, p′′i , ...

θ→ y.

We claim that

θ[ω, ζ] = [θω, θζ] = [x, y],

implying that [x, y] ∈ θ[pi] = Ti. We know that

[ω, ζ] = ..., ζ−1, ζ0, ω1, ...

= ..., ζ−1, pi, p′i, ...,

54 Karl Petersen

so

d(φmθ[ω, ζ], φmθω) ≤ 2β for m ≥ 0 , and

d(φmθ[ω, ζ], φmθζ) ≤ 2β for m ≤ 0.

Thus θ[ω, ζ] ∈ W s2β(x) ∩W

u2β(y) = [x, y]. We have ascertained that Ti is a rectangle in

Ωj.Now we will check the Markov mapping property for the Ti’s. The following is a proof

that if x ∈ Ti and φx ∈ Tk, then

φW s(x, Ti) ⊂W s(φx, Tk).

The proof for the unstable manifolds is analogous.Let y ∈W s(x, Ti) =W s

ǫ (x) ∩ Ti. Look upstairs in the shift space; does it follow that thepreimages of x and y under θ are identical to the right of zero? That is, can we show that

ω = ...ω−1pipk...θ→ x , and

ζ = ...ζ−1pipk, ...θ→ y?

Since y ∈W sǫ (x), y = [x, y] = θ[ω, ζ]. So we find that

φy = φ[x, y] = φ[θω, θζ] = θσ[ω, ζ]

= θ(...ζ−1ζ0ω1ω2...)

= θ(...ζ−1pipkω2...) ∈ Tk.

Noting that φy ∈ W sǫ (φx) because φW s

ǫ (x) ⊂ W sǫ (φx), we see that y ∈ W s(x, Ti) implies

that φy ∈ W s(φx, Tk), finishing the proof of the Markov mapping property for the stablemanifolds. Using an analogous argument for the unstable manifolds completes the proofthat the Ti’s have the Markov mapping property.

We have obtained a family of rectangles of small diameter which satisfy the Markovmapping property. However, this family may not be proper or essentially disjoint. We mustrefine the rectangles to obtain these properties so that a Markov partition is formed.

We will begin the next part of the proof in the next class period, by creating an essentiallydisjoint family of subrectangles of the Ti’s.

28. More of the proof. March 19 (Notes by NP)

Recall that Tk = θ[pk], k = 1, 2, ...r is a family of rectangles which satisfy the Markovmapping property. We will extract refinements of these rectangles which retain these pro-perties but which also are proper and essentially disjoint. For any m,k with Tm ∩ Tk 6= ∅,define four subsets of Tm as follows:

T 1m,k = x ∈ Tm : W u(x, Tm) ∩ Tk 6= ∅,W s(x, Tm) ∩ Tk 6= ∅

T 2m,k = x ∈ Tm : W u(x, Tm) ∩ Tk = ∅,W s(x, Tm) ∩ Tk 6= ∅

T 3m,k = x ∈ Tm : W u(x, Tm) ∩ Tk 6= ∅,W s(x, Tm) ∩ Tk = ∅

T 4m,k = x ∈ Tm :W u(x, Tm) ∩ Tk = ∅,W s(x, Tm) ∩ Tk = ∅.

Figure 51 illustrates these refinements.


m

kT

T

b = 1

TT

Tm,k4

m,kT3

m,k2 1

m,k

direction

directionunstable

stable

Figure 51. The refinements of Tm in terms of Tk.

We show that these T bm,k’s are rectangles; take x, y ∈ T bm,k, then z = [x, y] ∈ T bm,k since

W s(z, Tm) intersects Tk if and only if W s(x, Tm) does, and

W u(z, Tm) intersects Tk if and only if W u(y, Tm) does.

Then each T bm,k is also a rectangle, and, by the warm-up remark, so is each int(T bm,k) (if it

is not empty).Now we must avoid boundaries (which are probably topologically horrible); we will show

that the set of all points whose stable and unstable sets do not hit the correspondingboundaries of any neighboring rectangles is open and dense in Ωj. Since it will be shown

that the interiors of the T bm,k’s cover this open dense set, we can make a partition of theopen dense set. The closures of the rectangles forming this partition will be proper andessentially disjoint and will ultimately be our Markov partition.

Each Tk is closed, so ∂Tk is nowhere dense, hence⋃

k

intTk is open and dense in Ωj.

Discard any Tk for which intTk = ∅.

56 Karl Petersen

Define the ‘good set’ of points to be

G = x ∈⋃

k

intTk : x ∈ intTk, d(Tk, Tm) < 4β ⇒W sǫ (x) ∩ ∂

sTm = ∅,

W uǫ (x) ∩ ∂

uTm = ∅.

G is open, because ∂sTk and ∂uTk are closed, andW

sǫ (x) andW

uǫ (x) vary continuously with

x. We show that G is also dense in Ωj.Let k and m be such that d(Tm, Tk) < 4β, and define

Gsm,k = x ∈ intTk : W sǫ (x) ∩ ∂

sTm = ∅.

We will show that Gsm,k is dense in Tk. A similar proof holds for the sets Gum,k. Then, since

G =⋃

k

⋂

m

(Gsm,k ∩ Gum,k), a finite union of finite intersections, we find that G is open and

dense.To prove that Gsm,k is dense in Tk, note that if y ∈ intTm, then W

sǫ (y) ∩ ∂

sTm = ∅. We

prove this by contradiction, assuming that there is a z ∈W sǫ (y)∩∂

sTm. Since ∂sTm = z ∈

Tm : z /∈ intWuǫ (z)∩Ω(W

uǫ (z)∩Tm), we can find z′ /∈ Tm with z′ ∈W u

ǫ (z) and z′ arbitrarily

close to z. We can use canonical coordinates in Gsm,k since β is much smaller than ǫ, and we

see that the point y′ = [z′, y] is close to y. If it is close enough to y, which can be achievedby letting z′ be appropriately close to z, then it must be in intTm. But then, as can beseen in Figure 52, z′ = [y′, z] must be in Tm, which is impossible. So we see that any point

T

W (y)

y

z

y’

z’

W (z)

m

s

u

ε

ε

Figure 52. If y ∈ intTm, then Wsǫ (y) ∩ Tm = ∅.

in the interior of Tm cannot have stable or unstable sets intersecting the stable or unstableboundaries of Tm, respectively.

Given x ∈ Tk, find x1 ∈ intTk with x1 ≈ x. (This approximation can be chosen to bearbitrarily accurate). Fixing Tm within a distance 4β of Tk, we will move x1 slightly ifnecessary to find x2 ∈ Gsm,k with x2 ≈ x (again with arbitrary accuracy). If W s

ǫ (x1)∩Tm =

∅, then x1 ∈ Gsm,k and we do not need to adjust x1 at all. Otherwise, we will find an x2 ≈ x1


which does have this property, as follows. Suppose z ∈ W sǫ (x1) ∩ ∂

sTm. Find y ≈ z withy ∈ intTm, as seen in Figure 53. By the preceding paragraph, W s

ǫ (y) ∩ ∂sTm = ∅. Letx2 = [y, x1], so x2 ≈ x1 and x2 ∈ G

sm,k (if y is close enough to z).

Wsε (y)

εWs (x1

)Τ

Τ

Wε

u1(x )y

z

x

x2

1m

k

Figure 53. If W sǫ (x1) ∩ Tm 6= ∅, there is a point x2 arbitrarily close to x1

with W sǫ (x1) ∩ Tm = ∅.

This shows that any point x ∈ Tk can be approximated by elements of Gsm,k, makingthis set dense in Tk. Using an analogous argument to show that Gum,k is dense in Tk, it is

clear that the set Gsm,k ∩Gum,k is dense in Tk. In fact,

⋂

m

Gsm,k ∩Gum,k is a dense set. Since

Ωj =⋃

k

Tk, we see that the set

G =⋃

k

⋂

m

Gsm,k ∩Gum,k

is dense in Ωj .

29. Proof continued. March 24 (Notes by MA)

This week we continue the proof of the existence of Markov partitions for the nonwan-dering set Λ of an Axiom A diffeomorphism (M,φ) . Let us first summarize what has beendone so far.

Using the Stable Manifold Theorem (Theorem 7.2) followed by the Pseudo-Orbit Shado-wing Theorem (Theorem 18.1) we obtained an SFT Σ (P ) which maps onto Ωc, a basic set,through a continuous factor map θ. In Σ (P ) we have cylinder sets [pk], which are rectangles,and map to rectangles Tk in Ωc. These cover Ωc and satisfy the Markov Mapping Property.However this family of rectangles need not be proper nor need they be essentially disjoint.The purpose of this week’s notes is to construct a Markov partition by properly cutting upthe rectangles Tk into subrectangles.

58 Karl Petersen

29.1. A candidate for a Markov partition. Suppose two rectangles Tm and Tk intersect.We cut Tm into smaller rectangles T bm,k as follows:

T 1m,k = x ∈ Tm : W u (x, Tm) ∩ Tk = ∅,W s (x, Tm) ∩ Tk = ∅ ,

T 2m,k = x ∈ Tm : W u (x, Tm) ∩ Tk = ∅,W s (x, Tm) ∩ Tk 6= ∅ ,

T 3m,k = x ∈ Tm : W u (x, Tm) ∩ Tk 6= ∅,W s (x, Tm) ∩ Tk = ∅ ,

T 4m,k = x ∈ Tm : W u (x, Tm) ∩ Tk 6= ∅,W s (x, Tm) ∩ Tk 6= ∅ .

In this manner we obtain at most 4 subrectangles, for some may be empty. Successivelycutting in this manner we obtain a new set of rectangles whose only intersections are alongboundaries. We can ignore these boundaries by restricting ourselves to a ’good set’ Gdefined as follows:

G = x ∈ Ωc :Wsǫ (x) ∩ ∂

sTk = ∅,W uǫ (x) ∩ ∂uTk = ∅

for all k such that Tk hits some Tn to which x belongs.

Last week we showed that G was open and dense in Ωc. We now establish some otherproperties of G.

First we show that G ⊂ ∪(

T bm,k

). If x ∈ G, then x ∈ Tm for some m but x /∈ ∂Tm so

x ∈ T m. By the definition of G, W s

ǫ (x) ∩ ∂sTk = ∅ and W u

ǫ (x) ∩ ∂uTk = ∅ for all k suchthat Tk hits Tm. Now if W u

ǫ (x, Tm) hits Tk then it must hit T k . Similarly if it misses then

it must hit (T ck). This puts x in the interior of some T bm,k.

Next we show that each G∩(

T bm,k

)is an open rectangle. Clearly each is open. Suppose

x, y ∈ G∩(

T bm,k

). Then [x, y] ∈ T bm,k since T bm,k is a rectangle. So [x, y] ∈W s

ǫ (x)∩Wuǫ (y)

which misses boundaries, so [x, y] ∈(

T bm,k

). We must still show that [x, y] ∈ G. Suppose

otherwise. Then we can assume, without loss of generality, that W sǫ ([x, y]) ∩ ∂

sTn 6= ∅ forsome Tn which hits Tm. But W

sǫ ([x, y]) = W s

ǫ (x) so W sǫ (x) ∩ ∂

sTn 6= ∅ contradicting atx ∈ G.

Finally, since the sets

G ∩(

T bm,k

)

partition G, we will use these sets as our Markov

partition. Relabel these open rectangles as R1, ..., Rm. They are pairwise disjoint by defi-nition, and from the beginning of the construction (using cylinder sets) we see that theserectangles can be constructed with arbitrarily small diameter. Since G is open and dense inΩc, we see that the closures Ri will cover Ωc. We claim that these closures form a Markovpartition for Ωc.

To show that these closures form a Markov partition for Ωb, we need only show that therectangles are proper, essentially disjoint, and have the Markov Mapping Property (MMP).

Since Ri is open and Riis the largest open set lying in Ri, Ri ⊂ Ri

. From this is

follows that Ri ⊂ Riso

Ri ⊂ Ri⊂ Ri = Ri

and hence Ri= Ri. Therefore the rectangles are proper.


To see that the rectangles Ri are essentially disjoint, note that for i 6= j, Ri⊂ Ri ⊂ Rcj

since Ri ⊂ Rcj and Rcj is closed. Similarly Rj⊂ Rj ⊂

(

Ri)c

since by the first step,

Rj ⊂(

Ri)c

and(

Ri)c

is closed . So Ri∩Rj

= ∅.

It remains to be shown that the rectangles Ri satisfy the Markov Mapping property.Since the boundaries may cause problems in the proof, we show that if MMP holds forx ∈ G then MMP holds for all x ∈ Ωc.

Suppose MMP holds on G. Let x ∈ Riand φx ∈ Rj

. Take x′ ∈ Ri

∩ φ−1

(

Rj)

∩ G

very close enough to x. We must show that φW u(x,Ri

)⊃W u

(φx,Rj

)and φW s

(x,Ri

)⊂

W s(φx,Rj

). We will prove the inclusion for the stable part since the unstable part will

follow similarly.Note that

W s(x,Ri

)=

[x, y] : y ∈W s

(x′, Ri

)

(this does not depend on x′ being in the good set G) since if z ∈ W s(x,Ri

)then z =

[x, [x′, z]]. The reverse inclusion is clear.Since MMP holds on G (i. e. φW s

(x′, Ri

)⊂W s

(φx′, Rj

)) we have

φ[x, y] : y ∈W s

(x′, Ri

)=

[φx, φy] : y ∈W s

(x′, Ri

)

⊂[φx, z] : z ∈W s

(φx′, Rj

).

Thus

φW s(x,Ri

)⊂

[φx, z] : z ∈W s

(φx′, Rj

).

Finally, from an earlier argument, we have that[φx, z] : z ∈W s

(φx′, Rj

)=W s

(φx,Rj

)

and so we have the inclusion (for the stable part).To complete the proof of existence we now need only show that MMP holds on the good

set G.Suppose x ∈ Ri

∩ φ−1

(

Rj)

⊂ G. We need to show that φW s(x,Ri

)⊂ W s

(φx,Rj

).

In order to do this, we will show that G∩ φ−1G is dense in W s(x,Ri

), and so we will then

only need to show that φ(W s

(x,Ri

)∩G ∩ φ−1G

)⊂W s

(φx,Rj

).

Note that G ∩ φ−1G is dense in Ωc (both G and φ−1G are residual). Suppose t ∈W s

(x,Ri

). Take z ∈ G ∩ φ−1G ∩ Ri very close to t. Now z may not be in W s

(x,Ri

),

but w = [x, z] is. By continuity w is close to t. Here we need to remember that the initialrectangles Ri (without closure) were rectangles in G so that w = [x, z] ∈ Ri ⊂ G. Further,if φx, φz ∈ G, then φw = [φx, φz] ∈ G by the definition of G. (Note: x ∈ Ri ∩ φ

−1Rj , so

φx ∈ Rj ⊂ G.) Therefore, G ∩ φ−1G is dense in W s(x,Ri

).

To show that φ(W s

(x,Ri

)∩G ∩ φ−1G

)⊂ W s

(φx,Rj

), note that φW s

ǫ (x) ⊂ W sǫ (φx),

so we only need to show φ(W s

(x,Ri

)∩G ∩ φ−1G

)⊂ Rj . Take y ∈W s

(x,Ri

)∩G∩φ−1G.

We must show that φx and φy lie in the same rectangles T bm,k, for then φy ∈ Rj.

If x ∈ Tk ∩ φ−1Tm, then y ∈ W sǫ (x) ∩ Ri ⊂ W s

ǫ (x) ∩ Tk = W s (x, Tk) . But since therectangles Tk have MMP we have φW s (x, Tk) ⊂ W s (φx, Tm) and so φy ∈ W s (φx, Tm) ⊂

60 Karl Petersen

Tm. Similarly, if y ∈ Tk ∩ φ−1Tm then φx ∈ Tm. Thus φx and φy lie in the same rectangles

Tr.We now need to show that φx and φy lie in the same rectangles T bm,k.

Suppose φx, φy ∈ Tm and Tm ∩ Tk 6= ∅. We have already seen that W s (φx, Tm) =W s (φy, Tm) so we only need check

W u (φx, Tm) ∩ Tk 6= ∅ if and only if W u (φy, Tm) ∩ Tk 6= ∅.

Assume φz ∈ W u (φx, Tm) ∩ Tk for some z. Then by MMP, whenever x ∈ Ti (and hencealso y ∈ Ti), φz ∈ φW u (x, Ti) and so z ∈ W u (x, Ti) ⊂ Ti. Let z

′′ = [z, y] . Then φz′′ =[φz, φy] ∈W s

ǫ (φz) ∩Wuǫ (φy) and φz′′ ∈ Tm since Tm is a rectangle. Once we get φz′′ ∈ Tk

we will have the desired result W u (φy, Tm) ∩ Tk =W uǫ (φy) ∩ Tm ∩ Tk 6= ∅.

Since x and y lie in the same Riwe have that x and y lie in the same rectangle Ti

as above (different index i) so z′′ = [z, y] ∈ W sǫ (z) ∩ Ti = W s (z, Ti). This gives φz′′ ∈

φW s (z, Ti) ⊂W s (φz, Tk) by the MMP for the Ti, since φz ∈ Tk; therefore φz′′ ∈ Tk.

30. Near the end of the proof. March 31 (Notes by PS)

Final Details in Markov ProofTo finish the proof on the existence of Markov partitions, we need to show that φz′′ ∈ Tm.

What we are in the process of showing is:

x ∈ intRi ∩ φ−1(intRi) ∩G⇒ φ(W s(x,Ri)) ⊂W s(φ(x), Rj)

We took

y ∈W s(x,Ri) ∩G ∩ φ−1(G)

and we are showing that

φ(y) ∈W s(φ(x), Rj).

In fact, such y are dense in W s(x,Ri). We showed that φ(x) and φ(y) are always in thesame Tm. We are trying to show that φ(x) and φ(y) are always in the same T bm,k. Recall

that the interiors of T bm,k partition G into int(Rν).We assume that

W u(φ(x), Tk) ∩ Tm 6= ∅

and try to show that

W u(φ(y), Tk) ∩ Tm 6= ∅,

because T bm,k is defined by these intersection.We took z such that

φ(z) ∈W u(φ(x), Tk) ∩ Tm ⊂ Tk ∩ Tm

for k 6= m. Put z′′ = [z, y]. Then

φz′′ = [φ(z), φ(y)] ∈W sǫ (φ(z)) ∩W

sǫ (φ(y))


Tk

W

T

φ(

m

φ(

u

y) x)

φ(z)

φ(y)

Tk

W

Tm

φ(

u

x)

z / /

Thus, φ(z′′) ∈ Tk because φ(z), φ(y) ∈ Tk.Now, φ(z′′) ∈W u(φ(z), Tk). We need only find some φ(z′′) ∈ Tm. Note that

φ(z) ∈W u(φ(x), Tk) ⊂ φ(W u(x, Ti))

for some i by the Markov Mapping property. Pulling back through φ−1, z ∈ Ti and z ∈ Ttfor some t 6= i, where the inverse images of Ti and Tt meet Tk and Tm respectively.

The pseudo-orbits would appear as follows:

· · · pipk · · ·

and

· · · ptpm · · ·

Now,

z ∈W u(x, Ti) ∩ Tt.

Therefore

W u(y, Ti) 6= ∅

because x, y ∈ T bi,t. Take

62 Karl Petersen

z′ ∈W u(y, Ti) ∩ Tt.

Then

z′′ = [z, y] = [z, z′]

since W u(y) =W u(z′). But [z, z′] ∈W s(z, Tt), since both z, z′ ∈ Tt, which is a rectangle.Therefore:

φ(z′′) ∈ φ(W s(z, Tt)) ⊂W s(φ(z), Tm) ⊂ Tm

where φ(W s(z, Tt)) ⊂W s(φ(z), Tm) follows from the Markov mapping property.It is conceivable that we might have a problem with our subshift. In particular, can we

ensure t 6= i ? If not, we simply extend our original alphabet of pi so that there are enoughpoints making this possible.

We may thus assume that P contains enough points so that whenever ω ∈ Σ(p), thenthere is a ω ∈ Σ(p), with ω(0) 6= ω(0) but θ(ω) = θ(ω). We can ensure this by “splitting”each point of P into two very nearby points.

31. End of the proof. April 2 (Notes by PS)

We have another lingering question in the Markov Proof: How can we know φ(z′′) ∈ Tm?According to Bowen (p. 82), you might have q0

′ = pt = ps, which causes problems withthe statement “Now z ∈W u(x, Ts) ∩ Tt so that x, y are in the same T ns,t”, since T

ns,t is only

defined for s 6= t. If s = t, T ns,t = Ts = Tt.

Tk

Tm

Tns,t

We know that φ(z) ∈W u(φ(x), Tk)∩ Tm and φ(x), φ(y) ∈ the same Tk. We showed thatz ∈W u(x, Ti) ⊂ Ti. Setting z

′′ = [z, y] gives

φ(z′′) = [φ(x), φ(y)] ∈ Tk ∩Wsǫ (φ(z)) ∩W

uǫ (φ(y)) =W s(φ(z), Tk) ∩W

uǫ (φ(y)) ⊂ Tk

Now, x ∈ Ti ⇒ y ∈ Ti. Therefore,

z′′ = [z, y] ∈W sǫ (z) ∩ Ti =W s(z, Ti).

So,


φ(z′′) ∈ φ(W s(z, Ti)) ⊂W s(φ(z), any T for which φ(z) ∈ T )

(the containment holds by the Markov property—for example, φ(W s(z, Ti) ⊂W s(φ(z), Tm)).

Theorem 31.1. Given R = R1, . . . , Rr is a Markov partition for Ωb into proper closedessentially disjoint rectangles and

Ai,j =

1 if intRi ∩ φ−1(intRj) 6= ∅

0 otherwise

If ω ∈ ΣA ⊂ 1, . . . , rZ , we can define a factor map π : (ΣA, σ) → (Ωb, φ) by

π(ω) =⋂

j∈Z

φ−j(Rωj

)

which is one-to-one over the residual set:

Y = Ω \⋂

j∈Zφ−j

(∪ri=1 (∂Ri)

)

Thus, the dynamical systems are “almost” the same topologically. There are only coun-table many such systems, in a sense: finitely many Ri and finitely many choices for thematrix A, so there are countably many over all.

Lemma 31.2. Let ǫ be small enough so that canonical coordinates are well-defined. Let

C ⊂W uǫ (x) ∩ Ωb

andD ⊂W s

ǫ (x) ∩Ωb.

Then [C,D] is a proper rectangle if and only if intC = C and intD = D (with respect tothe relative topologies of W u

ǫ (x) ∩ Ωb and Wsǫ (x) ∩ Ωb )

Proof. First note that [C,D] is closed and a rectangle. To see that it is closed, if [cn, dn] ∈[C,D], then by taking subsequences, assume cn → c ∈ C and dn → d ∈ D. By the continuityof [, ] we have [cn, dn] → [c, d] ∈ [C,D].

To see that it is a rectangle, if [c1, d1] and [c2, d2] are in [C,D], then [c1, d1] ∈ W s(c1)and [c2, d2] ∈W u(d2) so that [[c1, d1], [c2, d2]] = [c1, d2] ∈ [C,D].

Step 1: Assume intC = C and intD = D. Let [u, v] ∈ [C,D] and try to find [c, d] ∈

int[C,D] with [c, d] near [u, v]. If we can do this, then [C,D] ⊂ int[C,D] and is thus aproper rectangle.

Now, pick c near u and d near v with c ∈ intC and d ∈ intD.Claim: [u, v] ≈ [c, d] because of the continuity of [, ].Claim: [c, d] ∈ int[C,D].Recall the earlier remark that had as a consequence that if R is a rectangle, then intR is

a rectangle as well. Thus

intΩbR =

⋃

y∈R(int(W s

ǫ (y)∩R)W s(y,Ω) ∩ int(Wu

ǫ (y)∩R)Wu(y,Ω)

But y = [c, d] is in the right-hand side of the above equation. For example, [c, d] ∈int(W s

ǫ (y)∩Ω)Ws(y,R) because d ∈ intD.

64 Karl Petersen

Wuε (x)

Wsε(x)

C

D

[u,v]

[c,d]

v

u

If y′ ∈ W s(y,R) is sufficiently near y, then d′ = [d, y′] ≈ d will still be in D, sinced ∈ intD. And then y′ = [c, d′] ∈ [C,D].

Step 2: Conversely, assume [C,D] is proper. Given c ∈ C, d ∈ D, we want to findu ∈ intC, v ∈ intD with u ≈ c, v ≈ d.

Find y ∈ int[C,D] so that y ∈ intW s(y, [C,D]) ∩ intW u(y, [C,D]), with y near to [c, d].Let x = [c, d]. Then

u = [y, x] ≈ [[c, d], x] = c

and

[x, y] ≈ [x, [c, d]] = d.

Claim: [y, x] ∈ intC. Let y = [p, q] with p ∈ C and q ∈ D, so that [y, x] = [[p, q], x] =p ∈ C. Moving [p, q] slightly still leaves us in [C,D], so moving p slightly and still leaves usin C.

32. More end of the proof. April 4 (Notes by PS)

Question: Are rectangles connected?

Wsε (x)

R

ε (y)Wu

y

x [x,y]

Take a rectangle R ⊂ Ωb(x) ⊂ Ω. Note that Ω is often very ugly, and its relativetopology can be very bad. Intersection with Ω can ruin connectivity. Also, rectanglestypically have terrible boundaries. W s

ǫ (x) can hit R repeatedly. For example, for hyperbolicautomorphisms of the 3-torus, boundaries of rectangles in the Markov partition contain norectifiable arcs.


x Wεs

(x)

33. Back to coding

Definition 33.1. Let S and R be closed rectangles with S ⊂ R. Then S is called au-subrectangle of R if

(1) S 6= ∅.(2) S is proper (i.e. S = intΩS )(3) If s ∈ S, then W u(s, S) = W u(s,R). (That is, S extends all the way across R in

the unstable direction.)

u(x)W

x

RS

The key to getting nonempty intersections is to map u-subrectangles into u-subrectanglesusing the Markov property.

Let R = R1, . . . , Rn be a Markov partition of Ωb of proper closed rectangles withintRi ∩ intRj 6= ∅ for i 6= j.

Lemma 33.1. If S is a u-subrectangle of Ri and intRi ∩ φ−1(intRj) 6= ∅ (i.e. Ai,j = 1 ),

then φ(S) ∩Rj is a u- subrectangle of Rj.

(R j )φ

φ(S)

φ(S) (R j )φ

R j

66 Karl Petersen

This is the key to showing that for ω ∈ ΣA, then π(ω) = ∩j∈Zφ−j

(Rωj

)6= ∅. Note we

have nested sets ∩n−nφ−j

(Rωj

)⊃ ∩n+1

−n−1φ−j

(Rωj

)⊃ . . .

Bowen uses this to induct on ∩nj=1φn−j

(Rωj

)= Rωn ∩ φ

(∏n−1j=1 φ

n−1−j(Rωj

))

where∏n−1j=1 φ

n−1−j(Rωj

)plays the role of the u-subrectangle S. The above Lemma makes this

induction work properly.Note: so far we haven’t used the fact that the rectangle is proper. The rectangles

must be proper because the Markov mapping property is only known to hold for x ∈intRi ∩ φ

−1(intRj), so that we know thatφ(W s(x,Ri)) ⊂W s(φ(x), Rj) andφ(W u(x,Ri)) ⊃W u(φ(x), Rj)Caution: ∂Ri can (a) be horrible, (b) can intersect W u

ǫ (x) and Wsǫ (x) in an ugly manner

and (c) map under φ in an awful way.

Proof of Lemma 33.1. Letx ∈ intRi ∩ φ

−1(intRj).

Let D =W s(x,Ri) ∩ S.a) First we show that D is relatively proper (i.e. D = intW s

ǫ (x)∩ΩD and D 6= ∅.

R iS

D

x

Wsε (x)

Note S =⋃

y∈DWu(y,Ri) implies D 6= ∅.

x

s

To see D ⊃ intW sǫ (x)∩Ω

D:

y ∈ D ⊂ S


implies

W u(y,Ri) =W u(y,Rj) ⊂ S.

To see D ⊂ intW sǫ (x)∩Ω

D:If s ∈ S, then [x, s] ∈ D because it is in each of W s

ǫ (x), Ri and W uǫ (s) and thus in

W u(s,Ri) =W u(s, S) ⊂ S.Furthermore, S = [W u(x,Ri),D]:

(1) S ⊃ [W u(x,Ri),D] is clear.(2) If s ∈ S, write s = [[s, x], [x, s]], where [s, x] ∈W u(x,Ri) and [x, s] ∈ D.

Since S is proper, D is relatively proper via the preceding lemma ([C,D] is proper if andonly if C and D are relatively proper.

b) Next show that φS ∩Rj 6= ∅:We claim that

φS ∩Rj =⋃

y∈D

φ(W u(y,Ri)) ∩Rj =⋃

y∈D

W u(φ(y), Rj).

To check this: if y ∈ D ⊂W s(x,Ri) then (since ax ∈ intRi)

φ(y) ∈ φ(W s(x,Ri)) ⊂W s(φ(x), Ri) ⊂ Rj

by the Markov Mapping property.If

y ∈ intW sǫ (x)∩Ω

D

then

φ(W u(y,Ri)) ⊃W u(φ(y), Rj)

Thus

φ(W u(y,Ri)) ∩Rj ⊃W u(φ(y), Rj)

Take y ∈ intRi and φ(y) ∈ intRj. Since such y’s are dense in D, we get

φ(S) ∩Rj =⋃

y∈D

φ(W u(y,Ri)) ∩Rj =⋃

y∈D

W u(φy,Rj)

which is dense because D is proper. Now y ∈ intD implies y ∈ intRi, so that

φ(S) ∩Rj =⋃

y′∈φ(D)

W u(y′, Rj) 6= ∅

since φ(D) 6= ∅.This is proper because

⋃

y′∈φ(D)

W u(y′, Rj) = [W u(φ(x), Rj), φ(D)]

as in part (a). Thus φ(S)∩Rj is proper. We need only check a few more things to completethe Lemma.

68 Karl Petersen

34. Obtaining symbolic dynamics. April 7 (Notes by RJ)

The lemma stated last time comes from Bowen’s (3.14).After the end of the proof of the Theorem on Existence of Markov partitions, add:

Remark 34.1. The proof shows that if intRi ∩φ−1intRj 6= ∅, then for all x′ ∈ Ri ∩φ

−1Rj ,

φW s(x′, Ri) ⊂W s(φx′, Rj)

and

φW u(x′, Ri) ⊃W u(φx′, Rj).

(We used a point x ∈ intRi ∩φ−1 intRj ∩G to prove the tough claim. This did not need

the Ri to be proper.) As long as rectangle interiors overlap, our statements are okay; thisjustifies the stronger statements of the mapping property made by Mane and Bowen. Forexample, if x′ ∈ Ri ∩ φ

−1Rj, then

φW s(x′, Ri) = [φx′, φy] : y ∈W s(x,Ri)

⊂ [φx′, z] : z ∈W s(φx,Rj)

= W s(φx′, Rj).

Figure 54. The argument shows that as long as we’re in the interior, thewhole thing maps correctly but boundary points might have problems.

Lemma 34.1. If S is a u-subrectangle of Ri and intRi ∩ φ−1 intRj 6= ∅, then φS ∩ Rj is

an u-subrectangle of Rj .

Proof. Let x ∈ Ri ∩ φ−1Rj.

(1) S =⋃

y∈DWu(y,Ri) = [W u(x,Ri),D], where D =W s(x,Ri) ∩ S.

This is so because if s ∈ S, then [x, s] ∈ D, and s = [[s, x], [x, s]]. The reverseinclusion is easy. Thus D is nonempty and relatively (in W s

ǫ (x) ∩ Ω) proper, sinceS is.

(2) φS ∩Rj =⋃

y∈D φWu(y,Ri) ∩Rj =

⋃

y∈DWu(φy,Rj).

The reason for this is that if y ∈ D ⊂W s(x,Ri), then

φy ∈ φW s(x,Ri) ⊂W s(φx,Rj) ⊂ Rj ,


by the preceding remark. Also, by applying the remark to y,

φW u(y,Ri) ⊃W u(φy,Rj),

hence

φW u(y,Ri) ∩Rj =W u(φy,Rj).

So, as in (1), φS ∩Rj = [W u(φx,Rj), φD].W u(φx,Rj) is relatively proper, by the preceding lemma, because Rj is proper

and

Rj = [W u(φx,Rj),Ws(φx,Rj)].

(As in the previous calculation, s = [[s, φx], [φx, s]].)

(3) φ : W sǫ (x) ∩ Ω →W s

ǫ (φx,Ω) with open image.Let y ∈ W s

ǫ (x) ∩ Ω. If φz is close enough to φy, then φkz is ǫ-close to φkx forall k ≥ 0. For z is within ǫ of x if the neighborhood of φy in which φz lies issmall enough. And φkz is within ǫ of φkx for k ≥ 1 because z ∈ W s

ǫ (x) and soφz ∈W s

ǫ (φx). Therefore, φD is relatively proper (in W sǫ ∩Ω). And hence, φS ∩Rj

is proper.We note further that φS∩Rj is nonempty because the representation in (2) shows

that φS ∩Rj ⊃ 0 6= ∅.

(4) Now we check the u-subrectangle condition: if y ∈ φS ∩ Rj , then W u(y,Rj) =W u(y, φS ∩ Rj). It is clear that W u(y,Rj) ⊃ W u(y, φS ∩ Rj). It remains then toshow W u(y,Rj) ⊂ φS ∩Rj . If y ∈ φS ∩Rj , then (by (2)) y ∈ W u(y′, Rj) for somey′ ∈ φDj , so

W u(y,Rj) =W u(y′, Rj) ⊂ φS ∩Rj.

(This is okay since the diameter of Ri is very small compared to ǫ.)

The point of this machinery is to be able to prove the theorem on Symbolic Dynamics.

Proof. (For ω ∈ ΣA, πω =⋂

j∈Z φ−jRωj

is a factor map ΣA → Ωb which is one-to-one over

the residual set Y = Ωb \⋃

j∈Z φ−j(

⋃ri=1 ∂Ri).)

We show first that for each n = 1, 2, . . .,⋂nj=−n φ

−jRωj6= ∅.

Let a1 . . . an be an allowed word in ΣA (i.e., Aaiai+1 = 1 for all i = 1, 2, . . . n − 1). Notethat for n > 1

n⋂

j=1

φn−jRaj = φ(

n−1⋂

j=1

φn−1−jRaj ) ∩Ran ,

and Ra1 is proper and nonempty. Using induction, assume that S =⋂n−1j=1 φ

n−1−jRaj is a

(proper and nonempty) u-subrectangle of Ran−1 . What about φS ∩ Ran? Since intRai ∩φ−1 intRai+1 6= ∅ for all i and Aan−1an = 1, then φs ∩ Ran is a (nonempty and proper)

u-subrectangle of Ran . Thus⋂nj=1 φ

−jRaj 6= ∅.

Note: Just to get this intersection nonempty, the “proper stuff” was not used.

70 Karl Petersen

35. Symbolic dynamics. Entropy. April 9 (Notes by RJ)

Let R = R1, R2, . . . , Rr be a Markov partition of Ωb.

(1) We showed that if a1a2 . . . an is an allowed word in ΣA (i.e. intRaj ∩ φ−1Rai+1 6= ∅

for all i), then⋂nj=1 φ

−jRaj 6= ∅, by induction using the first lemma.Consequently, if ω ∈ ΣA, then for all n ≥ 0,

n⋂

j=−n

φn−jRωj6= ∅,

and hencen⋂

j=−n

φ−jRωj= φ−n

2n+1⋂

j=1

φ(2n+1)−jRωj−n−1 6= ∅.

Remark 35.1. This did not use the stuff in the lemma about “proper.” But, wedid get that not only is

⋂nj=−n φ

−jRωj6= ∅, but also it’s proper (by induction, using

the lemma), hence its interior is nonempty, so for all n,

n⋂

j=−n

intφ−jRωj6= ∅.

Using compactness,

πω =

∞⋂

j=−∞

φ−jRωj6= ∅ for all ω ∈ ΣA.

(2) πω is just one point. For if x, y ∈ πω, then φjx, φjy are in the same Rωjfor

all j ∈ Z. Hence d(φjx, φjy) ≤ expansive constant for all j, hence x = y. Thusπ : ΣA → Ωb is defined as a map.

(3) πσ = φπ. This is obvious.

(4) π is continuous, by the same proof that showed θ : Σ(p) → Ωb is continuous. Thatproof only used the fact that if ξj = ωj for |j| ≤ n, then x = Πξ and y = Πωhave d(φjx, φjy) small (< 2β) for |j| ≤ n. (If π weren’t continuous, we could findsequences agreeing on arbitrarily long blocks whose images remained at least somedistance apart. Take a limit point of this sequence, etc.)

(5) The set Y = Ωb\⋃∞

−∞ φ−j(⋃rk=1 ∂Rk) is residual because the Rk’s are proper, hence

have nowhere dense boundaries. We show that π maps ΣA one-to-one onto Y . Eachpoint y ∈ Y has a uniquely determined itinerary ω ∈ ΣA: ωj is the unique k suchthat φjy ∈ intRk. (The orbit O(y) never hits any ∂Rk, and intRk ∩ intRk′ = ∅ fork 6= k′.) So for any y, there is one and only one ω ∈ ΣA such that πω = y.

(6) π(ΣA) = Ωb, since Y is dense in Ωb and π(ΣA) is closed.

Corollary 35.1. σ : ΣA → ΣA is topologically transitive. If (Ωb, φ) is topologically mixing,then so is (ΣA, σ).


Proof. (Ωb, φ) is known to be topologically transitive and is topologically mixing if it hasonly one elementary part. This is the proof that requires the “proper” machinery. Takecylinder sets [a1, . . . , ar] = C; [b1, . . . , br] = D ⊂ ΣA. (Recall that a typical basic open setis ω ∈ ΣA : ωi = ai for |i| ≤ r.) Then πC ⊃

⋂ri=−r φ

−r intRai , a nonempty open set C0

in Ωb. (Similarly, for πD ⊃ D0.) Say φnC0 ∩D0 6= ∅. (There exist such n 6= 0. If (Ωb, φ) istopologically mixing, then this happens for |n| ≥ N(C0,D0).) Then, since π is onto,

∅ 6= π−1(φnC0 ∩D0) = σnπ−1C0 ∩ π−1D0

⊂ σnC ∩D.

That proves it.

Remark 35.2. We can also use this “proper machinery” to show that π : ΣA → Ωb is atmost r2 to 1 (where r is the size of the alphabet, the number of rectangles in the Markovpartition).

35.1. Entropy, pressure, equilibrium states, Gibbs states. The first half of Bowenis a good reference for the upcoming material. One may also wish to see Petersen’s bookas a reference on entropy; and Walters is a good resource for pressure and stuff.

Let X be a compact metric space, and let φ be a homeomorphism on X. Then (X,φ) isa topological dynamical system.

We consider these definitions of the topological entropy, htop(X,φ) = htop(φ) = h(φ) = h.

(1) Open-cover definition (Adler-Konheim-McAndrew)

Let U be an open cover of X and denote by N(U) the minimum possible numberof elements in any subcover of U . Let H(U) = logN(U).

Now,

htop(U , φ) = limn→∞

1

nH(U ∨ φ−1U ∨ · · · ∨ φ−n+1U)

(this exists by sub-additivity). The topological entropy is then

htop(X,φ) = supUhtop(U , φ).

This definition was motivated by the definition of entropy for measures (in turndrawing from Shannon’s definitions of entropy and capacity in information theory).If µ is a φ-invariant measure on X, then for a finite, measurable partition α =A1, . . . , Ar of X, define:

H(α) = −r∑

i=1

µ(Ai) log µ(Ai),

hµ(α, φ) = limn→∞

1

nH(α ∨ φ−1α ∨ · · · ∨ φ−n+1α),

and

hµ(φ) = supαhµ(α, φ).

Here α ∨ φ−1α ∨ · · · ∨ φ−n+1α represents the common refinement of the partitionsinvolved.

72 Karl Petersen

36. Entropy, pressure, Gibbs measures. April 11 (Notes by RJ)

Again, we discuss a topological dynamical system (X,φ). We can consider a family ofmetrics on (X,φ) given by

Dn(x, y) = sup0≤k≤n−1

d(φkx, φky).

36.1. Bowen’s definitions of topological entropy. We introduce two other definitionsfor htop(X,φ), involving the notions of n, ǫ-separated and n, ǫ-spanning sets.

Definition 36.1. For ǫ > 0, n = 1, 2, 3, . . ., a set A ⊂ X is called n, ǫ-separated if for all x,y ∈ A, x 6= y, there exists an integer k with 0 ≤ k ≤ n− 1 such that d(φkx, φky) ≥ ǫ.

Definition 36.2. For ǫ > 0, n = 1, 2, 3, . . ., a set B ⊂ X is called n, ǫ-spanning if everypoint of X is within ǫ of φkx, for some x ∈ B, 0 ≤ k ≤ n− 1.

Then we have

htop(X,φ) = limǫց0

↑ lim supn→∞

1

n(log supcardA : A is n, ǫ-separated)

and

htop(X,φ) = limǫց0

↑ lim supn→∞

1

n(log infcardB : B is n, ǫ-spanning).

Note that supcardA : A is n, ǫ-separated is equal to the maximum possible number ofdisjoint open ǫ/2-balls with respect to Dn (i.e., how many such balls can be packed into X)and that infcardB : B is n, ǫ-spanning is the minimum possible number of ǫ-balls withrespect to Dn that cover X.

Theorem 36.1. (Dinaburg, Goodman, and Goodwyn):VariationalPrinciplefor entropy

htop(X,φ) = supµ∈M(X,φ)

hµ(X,φ).

Recall that M(X,φ) is the set of φ-invariant Borel probability measures on X. (SeeDGS, p.138; KEP, p.239.) M(X,φ) is compact and metrizable in the weak⋆- topology.

Any measure that achieves this supremum is called maximal. In general, there may notbe any such maximal measures, but in an expansive system, they always exist. (See DGS,p.139, Walters, p.224.) Also, more generally, maximal measures exist in any system forwhich µ → hµ(φ) is upper semi-continuous (USC). If the supremum is attained, it mightnot be unique; then we say we have a phase transition.

If there is only one maximal measure, then the system is called intrinsically ergodic.A continuous function V : X → R is called a potential function. Let SnV (x) =

∑n−1k=0 V (φkx) for n ≥ 1. The pressure of V and φ is defined to be

P (V, φ) = limǫց0

↑ lim supn→∞

1

n(log sup

∑

x∈A

eSnV (x) : A is n, ǫ-separated),

or equivalently,

P (V, φ) = limǫց0

↑ lim supn→∞

1

n(log inf

∑

x∈B

eSnV (x) : B is n, ǫ-spanning).


The relationship between these definitions and those for topological entropy can be seen bynoting that cardA =

∑

x∈A 1 =∑

x∈A e0. Thus htop(φ) = P (0, φ).

Theorem 36.2. (Ruelle, Walters) VariationalPrinciplefor pres-sure

P (V, φ) = supµ∈M(X,φ)

hµ(φ) +

∫

XV dµ.

(See Bowen, Walters.)〈V 〉 represents the expected, or average, value of the energy. (There may be a negative

thrown in from the physics.) In this way, P (V, φ) = sup(−“free energy”). So the pressureis related to minimizing the free energy.

The difference between entropy and average energy tells us how much “free energy” thereis in a system. And, somehow, natural systems want to minimize this quantity.

Any measure that achieves this supremum is called an equilibrium state. Again, if (X,φ)is expansive (or µ→ hµ(φ) is USC), then equilibrium states exist.

If there is more than one equilibrium state, then we say we have a “phase transition.”Nature gives us many good illustrations of phase transitions. For example, consider spring-time in the Arctic. As the earth gets a little warmer, the solid ice begins melting intothe water. Yet, simultaneously, it is still cold enough that some water continues to freeze.Similarly, consider a large chunk of magnetic material. At room temperature, the moleculesin this material all point the same way and are magnetized. However, if you heat it up, thematerial will completely demagnetize. The atoms will act in a Bernoulli, completely chaoticstate—pointing up, down, or in any random direction. Somewhere between solid ice andflowing water, between a powerful magnet and a chunk of dead rock—when a system justdoesn’t seem to quite know what to do—we have a phase transition.

36.2. Gibbs Measures. Gibbs measures were brought into ergodic theory by Sinai around1967. Previous significant work had been done by Dobrushin and several others going backto Gibbs, finding physically significant measures as descriptions of equilibrium states ofsystems with interacting components.

Let ΣA be a (two-sided) shift of finite type (SFT) on 1, . . . , rZ. For a continuousfunction V : ΣA → R, define the n-range variation of V to be

varnV = sup|V (x)− V (y)| : x, y ∈ ΣA, xk = yk for |k| ≤ n.

How fast does the dependence of the function on coordinates near the center decay? Onecondition for fairly rapid decay is the following. We say that V has summable variation if∑∞

n=0 varnV <∞.

Definition 36.3. A measure µ on ΣA is called a Gibbs measure with potential function Vif there exist constants c > 0 and P such that

c ≤µ[x0, . . . , xn]

exp−nP + SnV (x)≤

1

c

for all x ∈ ΣA and all n ≥ 0.

74 Karl Petersen

Thus the idea of Gibbs measure is a generalization of that of Markov measure. A Markovmeasure µ with transition matrix (Mij) is a Gibbs measure with potential function V (x) =logMx0x1 and pressure P = 0. This is because

eSnV (x) ≍Mx0x1Mx1x2 . . .Mxn−2xn−1 ≍ µ[x0x1 . . . xn−1].

37. Motivations from physics. April 14 (Notes by SS)

Definition 37.1. Let ΣA = topologically mixing SFT ⊂ 1, . . . , rZ. Then a (shift-invariant) probability measure µ on ΣA is called a Gibbs measure with potential functionV ∈ C(ΣA) and pressure P ∈ R if there is a constant c > 0 such that

1

c≤

µ[x0x1 · · · xn−1]

exp−nP + SnV (x)≤ c

for all x ∈ ΣA and all n ≥ 1, where SnV (x) =∑n−1

k=0 V (σkx).

Remarks 37.1.

(1) A Markov measure on ΣA, say 1-step with stochastic transition matrix (Mij), is aGibbs measure with potential function V (x) = log(Mx0x1) and pressure P = 0 :

µ[x0 · · · xn−1] = px0Mx0x1Mx1x2 · · ·Mxn−2xn−1

and

SnV (x) =∑n−1

k=0 log(Mxkxk+1) = log(Mx0x1 · · ·Mxn−1xn)

so that

exp−nP + SnV (x) =Mx0x1 · · ·Mxn−1xn ≍ µ[x0 · · · xn−1].

So, a Gibbs measure is a nice generalization of a Markov measure.

(2) For a Gibbs measure µ,

µ[x0x1 · · · xn−1]

µ[y0y1 · · · yn−1]≍ exp[SnV (x)− SnV (y)].

If (x, y) ∈ RA = Gibbs equivalence relation (which means, there exists K such thatxk = yk for all k with |k| ≥ K), then (if V depends on just a few coordinates, or hasits dependence on far-out coordinates fall off quickly, say

∑∞n=0 varnV <∞), there

exists the limit ρµ(x, y), called the Radon-Nikodym derivative of the equivalencerelation RA:

ρµ(x, y) = limn→∞

µ[x−n · · · xn]

µ[y−n · · · yn]= exp

∞∑

k=−∞

[V (σkx)− V (σky)]

︸︷︷︸

a finite sum

where [x−n · · · xn] is a configuration = block or word.

Idea from statistical mechanics : Fix a finite region F and suppose the coordinatesof x are fixed outside of F . Then, the relative probabilities of the different possible


configurations within F are supposed to be given in a Gibbsian way, determined bythe energies of the configurations :

P (C1)

P (C2)∼eE(C1)

eE(C2).

See J. Gibbs, Y. Sinai, D. Ruelle, R. Dobrushin, or C. Preston’s “Gibbs states oncountable sets”.

(3) Suppose that we have a physical system that can be in situations 1, · · · , Q.For example, the phase space X (e.g., X = R6N = (position, momentum)-space

for an ideal gas of N particles, or constant-energy surface in R6N ) is decomposedinto many cells with associated (system) energies E1, E2, . . . , EQ and probabilitiespi = µ(Ei), i = 1, . . . , Q. By the state of the system, we mean the probabilitymeasure µ, which describes the probabilities of the observable (macro)configurationsof the system.

How do we choose the pi so as to minimize the Helmholtz free energy :

1

kT

Q∑

i=1

piEi

︸︷︷︸

〈E〉

−hµ

where k = Boltzmann’s constant, T = temperature and hµ =∑Q

i=1 pi log pi ?There is reason to believe that nature prefers such states, i.e., choices of the pi.

We might call this the Physical Variational Principle.If the total energy 〈E〉 is fixed, then this is the same as maximizing entropy hµ.In general, our system might be in contact with a much larger one (say a heat

source or sink), and so its expected energy might be allowed to change.From a small calculation, the best µ has pi ∼ e−βEi (the same β = 1/kT for all

i).For fixed energy, maximizing entropy yields the Maxwell-Boltzmann velocity dis-

tribution, in which

pi =e−Ei/kT

∑

j e−Ej/kT

.

The denominator Z =∑

j e−Ej/kT , a normalizing constant, is the partition function

(or Zustandssumme). ( See Bowen, p.4, p.34 and Walters, p.217, p.227.)

38. Maximizing entropy or free energy. April 16 (Notes by SS)

Example 38.1. Let X = R6N (3 coordinates of position, momentum for each particle) inan ideal gas of N particles. Then cut (macro)phase (position-momentum) space R6 intocells with associated energies E1, E2, . . . , Er. Then a probability measure µ onX determinespi = µ(Ei) and a macro-configuration ~n = (n1, n2, . . . , nr) gives the numbers ni of particlesin the various cells.

Maybe Ei =12mVi

2, if there is only kinetic energy; or there may be potential energy (due

to position), or interaction energies among the particles. If Ei =12mVi

2, then pi ∼ e−Vi2,

76 Karl Petersen

and the Physical Variational Principle yields the Maxwell-Boltzmann velocity distributionfor a hard-sphere gas:

The best pi’s are

e−Ei/kT

∑

j e−Ej/kT

.

Another approach seeks to maximize the relative probability W of a state µ. Note thathere we are talking about the probability of occurrence of some probability measure thatdescribes probabilities of observable configurations of the system. Associated with µ arethe expected values of the occupation number ni of the different macro-configurations,with associated energies Ei. So we ask how to choose the ni so as to maximize W , or,equivalently, log W . If W is the relative probability of a state ~n, then W ∼ the numberof micro-configurations that produce the macro-configuration ~n, so that, considering allmicro-configurations to be equiprobable,

W =N1!N2! · · ·Nr!

N !.

In the fixed energy case, if pi = Ni/N , then by Stirling’s Formula maximizing W is thesame as maximizing the thermodynamic entropy S = k log W , and this is the same asmaximizing hµ = −

∑pi log pi.

By the way, what is temperature? It’s a subtle quantity: if U =∑niEi is the internal

energy of the system, then the temperature T = ∂U/∂S.

39. Existence and uniqueness of equilibrium states. April 18 (Notes by SS)

Exercise 5. Recall that an attractor (in a topological dynamical system (X,φ)) is a clo-sed φ-invariant set Λ for which there is a neighborhood U of Λ such that φU ⊂ U and∩n≥0φ

nU = Λ.

(1) Does it change the definition to require the existence of such an open U?(2) Does it change the definition to require that there exists such an open U with

φU ⊂ U?We call an attractor tight if for any η > 0, we can find such a U (Λ ⊂ open

V ⊂ U, φU ⊂ U,∩n≥0φnU = Λ) with U ⊂ y ∈ X : d(y,Λ) < η.

(3) Is every attractor tight?(4) What if Λ = Ωb = a basic set in an Axiom A system?

Example 39.1. Suppose that our system consists of particles of types 1, · · · , r locatedat the points of the integer (1-dimensional) lattice in R. A configuration of the system is a

point x ∈ 1, · · · , rZ, i.e., at each point of Z, we have a symbol from 1, · · · , r :

micro-configuration x = · · · x−2x−1x0x1x2 · · ·

Suppose that we have a function U of one coordinate (U(x) = U(x0)) which gives the(potential) energy due to particle x0 being at 0, i.e.,

presence of symbol x0 at 0 contributes energy U(x) = U(x0);presence of symbol xk at k contributes energy U(σkx) = U(xk).Also suppose that there is an interaction energy between each pair of particles due to the

presence of xk at k and xn at n,


I(n − k;xk, xn).

Then cut X = ΣA into cells [x−m · · · xm] and estimate the energy involved with a cell :

Em[x−m · · · xm] =m∑

k=−m

[U(σkx) +∑

−m≤k<n≤m

I(n− k;xk, xn)],

or taking account also the interactions of the xk,−m ≤ k ≤ m with symbols outside therange [−m,m],

Em[x−m · · · xm] =m∑

k=−m

[U(σkx) +1

2

∑

j 6=k

I(k − j;xj , xk)]

︸︷︷︸

−V (σkx)

.

According to the Physical Variational Principle, good probabilities to try to assign to thecylinder sets [x−m · · · xm] ⊂ ΣA are

µm[x−m · · · xm] ∼e−Em[x−m···xm]/kT

∑

B=[y−m···ym] e−Em(B)/kT

.

Thinking about the definition of pressure shows that

Z =∑

B=[y−m···ym]

eS2m+1V (σ−my) ∼ e(2m+1)P ,

therefore

µm[x−m · · · xm] ∼eS2m+1V (σ−mx)

e(2m+1)P (V,φ),

and henceµm[x−m · · · xm]

exp−(2m+ 1)P + S2m+1V (σ−mx)≍ 1.

We hope that these measures µm converge as m → ∞ to a Gibbs measure on ΣA whichis an equilibrium state for V (sometimes they don’t exist or are not unique).

Theorem 39.1. (Sinai, Ruelle, Bowen) If V : ΣA → R has summable variation (e.g.,is Holder continuous), then there is a unique equilibrium state µV on ΣA with potentialfunction V , and µV is also the unique Gibbs measure with potential function V .

To prove this, we follow the idea of Walters, Ruelle’s operator theorem and g-measures,Trans. A.M.S. 214 (1975), 375-387.

First, we can reduce to Σ+A = 1-sided SFT ⊂ 1, · · · , rN with the same allowed blocks,

because we have the following :

Proposition 39.2. If V : ΣA → R has summable variation, then there are V1 with sum-mable variation and continuous u : ΣA → R such that V = V1 + u− uσ (i.e., V and V1 arecohomologous) and V1(x) only depends on x0x1 · · · , i.e.,

xk = yk for all k ≥ 0 implies that V1(x) = V1(y).

(And then V and V1 have the same Gibbs states.)

78 Karl Petersen

Let g ∈ C(Σ+A) and g > 0 with

∑

y∈σ−1x g(y) = 1 for all x ∈ Σ+A.

Then define L = Lg : C(Σ+A) → C(Σ+

A) by

Lf(x) =∑

y∈σ−1x

g(y)f(y) forf ∈ C(Σ+A),

and let L∗ : C(Σ+A)

∗ → C(Σ+A)

∗ be the dual operator on measures i.e.,

L∗µ(f) =

∫

fd(L∗µ) =

∫

Lfdµ = µ(Lf).

Then, a measure µ on Σ+A is called a g-measure if L∗µ = µ.

Theorem 39.3. (Keane, Inv. Math. 16 (1972), 309-324) Let Σ+A be a 1-sided topologically

mixing SFT and g ∈ C(Σ+A) with

∑∞n=1 varn log g < ∞. Then for all f ∈ C(Σ+

A), Lnf

converges uniformly to a constant µ(f). This defines a measure µ on Σ+A, which is the

unique g-measure.

Theorem 39.4. (Ledrappier, ZW 30 (1974), 185-202) Let g ∈ C(Σ+A) be such that g > 0

and∑

y∈σ−1x g(y) = 1 for all x ∈ Σ+A, and let µ ∈ C(Σ+

A)∗. Denote by B the Borel σ-algebra

of Σ+A. Then the following are equivalent:

(1) µ is a g-measure on Σ+A,

(2) µ is σ-invariant and for all f ∈ L1(µ),

Eµ(f | σ−1B)(x) =∑

z∈σ−1σx

g(z)f(z) for µ-a.e. x,

(3) µ is σ-invariant and an equilibrium state for V = log g.

Note that using f = χx0=a in (2) the right-hand side is just g(a, x1, x2, · · · ) and so g isa sort of Radon-Nikodym derivative of µ:

g(x) = limAցx

µ(A)

µ(σA)= lim

n→∞

µ[x0x1 · · · xn]

µ[x1 · · · xn]for µ-a.e. x.

40. Ruelle’s Operator Perron-Frobenius Theorem, g-measures. April 21,1997 (Notes by KJ)

40.1. From last time. . . We have Σ+A, a one-sided mixing shift of finite type with transi-

tion matrix A.A g-function is defined to be a function g ∈ C(Σ+

A) such that g > 0 and Σy∈σ−1xg(y) = 1.

Define the operator L = Lg : C(Σ+A) → C(Σ+

A) by

Lf(x) = Σy∈σ−1xg(y)f(y).

The operator L has an adjoint L∗ : C(Σ+A)

∗ → C(Σ+A)

∗. We say µ is a g-measure if L∗µ = µ.

Theorem 40.1 (Keane. Inv Math, 1972). Let Σ+A and g be as above and assume that

Σ∞n=0 varn(log g) <∞.

Then for every f ∈ C(Σ+A), L

nf converges uniformly to a constant µ(f) and µ is the uniqueg-measure.


Theorem 40.2 (Ledrappier. ZW, 1974). Let Σ+A and g be as above. Then the following

are equivalent:

(1) µ is a g-measure on Σ+A.

(2) µ is σ-invariant and for every f ∈ L1(µ) Eµ(f |σ−1B)(x) = Σz∈σ−1σxg(z)f(z) almost

everywhere dµ.(3) µ is σ-invariant and an equilibrium state for V = log g.

Remark 40.1. For a g-measure µ, P (log g, σ) = hµ(σ) +∫log g dµ = 0.

Remark 40.2. L(f σ) = f .

40.2. Infinite-dimensional extension of the Perron-Frobenius Theorem for non-

negative matrices. Recall the Perron-Frobenius Theorem for nonnegative matrices (seeParry and Tuncel or Lind and Marcus). In all that follows, we will assume that the shiftof finite type is aperiodic. The theorem says that there exists a unique largest positiveeigenvalue λ with positive left and right eigenvectors r and l. The entries in the n’th powerof the matrix are asymptotic to λn, and in fact are between two constant multiples of λn.

We can normalize that matrix by replacing the nonzero entries with ri/λrj in the i, jentry. From this we get a stochastic matrix.

If you start out with a 0, 1 matrix for a shift of finite type, then the stochastic matrix givesyou the measure of maximal entropy. If your function is of two coordinates, say eV (x1,x2),then the stochastic matrix gives the equilibrium measure for that V .

Theorem 40.3 (Ruelle’s operator Perron-Frobenius Theorem). Let Σ+A be a topologically

mixing shift of finite type. Let V ∈ C(Σ+A) with Σ∞

n=0 varn V < ∞. Define L = LV :

C(Σ+A) → C(Σ+

A) by

LV f(x) =∑

y∈σ−1x

eV (y)f(y).

(Note that this L = Lg in the previous notation, with g = eV (x), but this is probably nota g-function, since it probably does not sum to one.) Then there are λ > 0, the “positiveeigenvalue,” ρ ∈ C(Σ+

A) with ρ > 0, the “left eigenvector,” and ν ∈ C(Σ+A)

∗ a positive

measure on Σ+A, the “right eigenvector,” such that

Lρ = λρ,L∗ν = λν,Lnf

λn→ ν(f)ρ

for all f ∈ C(Σ+A) uniformly on Σ+

A as n→ ∞.

Ingredients of proof (see Walters, Trans. AMS 214 (1985), 375-387, for details): (1) Usethe Schauder-Tychonoff Theorem (continuous self-maps of compact convex sets inlocally convex spaces have fixed points) to get ν as fixed point of the operatorν → L∗ν/(L∗ν)I where I represents the constant function. In order for the Schauder-Tychonoff Theorem to apply, use the Borel measures as the convex set.

Take λ = L∗ν(I). This method gives both the measure and the eigenvalue.(2) Use a similar method to get ρ.(3) The hard part is to get asymptoticity. For this, use the previous theorem about

g-measures, taking

g =eV ρ

λρ(σ)

80 Karl Petersen

(in analogy to the finite-dimensional plan.) Then use the unique g-measure µg(f) =ν(fρ) and its convergence properties from Keane and Ledrappier.

As a direct application, we obtain most of the following theorem. For the rest, see Bowen,pp. 21-27, and Walters, reference above.

Theorem 40.4 (Existence of Gibbs measures and equilibrium states). Let Σ+A be a one-

sided topologically mixing SFT, V ∈ C(Σ+A) with Σ∞

n=1 varn V <∞. Then there is a uniqueequilibrium state µV for V . It is the unique Gibbs state for V and is given by µV (f) = ν(fρ)with ν, ρ as given in the Ruelle Theorem. µV is the unique g-measure for g = eV ρ/λρ(σ).Also (Σ+

A, σ, µV ) is exact, hence strongly mixing, and its natural extension is Bernoulli(since the time-zero partition is weakly Bernoulli).

Corollary 40.5. Similarly on (ΣA, σ).

This follows from a trick developed by Sinai to replace an arbitrary V by one that onlydepends on positive coordinates:

Lemma 40.6. If V : ΣA → R has summable variation, then there is u ∈ C(ΣA) such thatW = V +u−u σ has the property that if xk = yk for all k ≥ 0 then W (x) =W (y). (ThenV and W have the same Gibbs states, justifying our working on Σ+

A)

There remain three major theorems to present:

Theorem 1: Holder-continuous potential functions on basic sets of Axiom A systemshave unique equilibrium states.

Theorem 2: In a C2 Axiom A system, almost every point of the manifold M is at-tracted in forward time to some basic set (a “tight attractor”).

Theorem 3: The asymptotic statistics of the orbit of Lebesgue-a.e. point of M aredescribed by a certain equilibrium measure (the SRB measure) on the attractorwhich it approaches.

Due to lack of time, we will have to skip many details of their proofs.Ruelle proved the following theorem for Anosov systems, and Bowen proved it for Axiom

A systems.

Theorem 40.7 (Theorem 1—Ruelle, Bowen). Let Ωb be a basic set in an Axiom A system(M,ϕ). Let V : Ωb → R be Holder continuous (|V (x)−V (y)| ≤ c d(x, y)η for some c, η ≥ 0for all x, y ∈ Ωb). Then there is a unique equilibrium state µV for V . µV is ergodic.In addition, if (Ωb, ϕ) is topologically mixing, then µV is Bernoulli ( i.e., (Ωb, ϕ, µV ) ismeasure-theoretically isomorphic to a two-sided Bernoulli shift).

Proof. We use the fact that we have great symbolic dynamics using the map π, which isone-to-one on a residual set.

We show next time that V1 = V π has summable variation on ΣA. We then need to getthis function to depend on only half of the coordinates (or more precisely, replace it by acohomologous one with this property).

This will give us µV1 on ΣA, which is the unique equilibrium state for V1. We then useπ to push µV1 down: let µV = µV1π

−1. µV1 is fully supported: it gives positive measure toopen sets.


Noting that the factor map π is actually an isomorphism of the systems (ΣA.σ, µV1) and(Ωb, ϕ, µV ), the assertions will follow fairly easily.

(Continued on next day)

41. Symbolic dynamics yields existence of equilibrium states on basic sets.April 23, 1997 (Notes by KJ)

41.1. Sketch of proof of the existence of equilibrium states on basic sets. Wetreat the case when (Ωb, φ) is topologically mixing; for the non-mixing case consider Ωb =X1 ∪ ... ∪Xm and replace φ by φm.

Take a Markov partition R = R1, . . . , Rr into sets of small diameter, and use it to getthe coding π : ΣA → Ωb. Let V1 = V π : ΣA → R. Then V Holder implies that V1 hassummable variation, as follows: Suppose we have ω, ξ ∈ ΣA with ωk = ξk for |k| ≤ n. If

x = πω, and

y = πξ,

then φjx, φjy are in the same Ri for |j| ≤ n. Therefore d(x, y) has to be kind of small,much smaller than the maximum diameter. It is exponentially small in n (by Hirsch-Pugh),i.e., there exists a ∈ (0, 1) such that d(x, y) ≤ an, for all n. Hence |V1(ω) − V1(ξ)| =|V (x)− V (y)| ≤ c(an)ρ, and varn V1 ≤ c(aρ)n is summable in n.

By the previous theorem on the existence of equilibrium and Gibbs states on a shift offinite type (Theorem 40.4), there exists a unique equilibrium (Gibbs) state µV1 for V1 onΣA. Define µV on Ωb by µV (E) = µV1(π

−1E). Since π is a factor mapping it commutes withthe shift σ and φ. Since Ωb is topologically mixing, µV is a φ invariant Borel probabilitymeasure on Ωb and is measure-theoretically mixing.

We have a factor mapping

π : (ΣA, σ, µV1) → (Ωb, φ, µv)

between measure-preserving systems. We claim that π is actually an isomorphism. To showthis, we check that the set where π is not 1-1 has measure zero.

For

x 6∈⋃

j∈Z

φjr⋃

i=1

(∂sRi ∪ ∂uRi),

π−1x is exactly one point. Define ∂sR = ∪ri=1∂sRi, and define ∂uR similarly. We claim

that µV1(π−1∂sR) = µV1(π

−1∂uR) = 0, hence π is 1-1 on the set of full measure

ΣA\ ∪j∈Z φjπ−1(∂sR∪ ∂uR).

We show here that µV1(π−1∂sR) has measure zero; the other case is similar.

Let Ds = π−1∂sR, which is a proper closed subset of ΣA because ∂sR ⊂ Ωb, the bounda-ries are closed, and π is continuous. Note that σDs ⊂ Ds since φ∂

sR ⊂ ∂sR by an earlierfact, section 3.15, p. 84 in Bowen. This forces it to have measure zero: let D = ∩n≥0σ

nDs.Then D is a proper, closed, σ-invariant set. This forces µV1(D) = 0 or 1 since µV1 is ergodic.ButDc is open and not empty since µV1 is a Gibbs measure and hence fully supported on ΣA,

82 Karl Petersen

so µV1(Dc) > 0. Therefore µV1(D) = 0 and µV1(Ds) = 0 (since µV1(D) = limµV1σ

n(Ds)).This gives us a measure-theoretical isomorphism, which tells us what we need.

Now we check that µV is an equilibrium state for V :

P (V, φ) ≥ hµV (φ) +

∫

Ωb

V dµV (by Variational Principle)

= hµV1 (σ) +

∫

ΣA

V1dµV1 (by isomorphism)

= P (V1, σ) (since µV1 is an equilibrium state for V1)

≥ P (V, φ).

The last inequality follows from the fact that (Ωb, φ) is a factor of (ΣA, σ) and pressure, likeentropy, cannot increase under factor mappings (see Bowen, Prop 2.13, p. 55). Therefore,P (V, φ) = hµV (φ) +

∫

ΩbV dµV , which means that µV is an equilibrium state for V .

Now we check for uniqueness. Suppose that µ on Ωb is also an equilibrium state for V .Lift µ to ν, a σ-invariant measure on ΣA, using Lemma 4.1. Then

hνσ +

∫

ΣA

V1dν ≥ hµφ+

∫

Ωb

V dµ (since the integrals are equal)

= P (V, φ) (since µ is an equilibrium state)

= P (V1, σ) (from above).

Therefore ν is an equilibrium state for V1, hence ν = µV1 , so µ = νπ−1 = µV by ourdefinition.

We summarize the idea of the argument: Good symbolic dynamics presents a residualinvariant set as a tight topological factor of an SFT, so a Gibbs or equilibrium state on ΣAgives a Gibbs or equilibrium state on Ωb.

41.2. Theorems 2 and 3. We now move on to sketching the proofs of the remainingTheorems 2 and 3:

Lebesgue-almost every point of M is attracted to one of the basic sets, and the statisticsof Lebesgue-almost every orbit are given by SRB measure, which is an equilibrium state.

The measure we want comes from applying Theorem 1 to a good V .Assume (M,φ) is Axiom A and φ is a C2 Axiom A diffeomorphism. It is known that

certain parts of the following do not hold in the C1 case. We have that the basic set Ωb ishyperbolic, so for p ∈ Ωb,

Dpφ : Eup → Euφp

where Eup is the expanding subspace. Define Ju(p) to be the Jacobian of this map, thedeterminant of Dpφ in local coordinates (recall that M is an oriented Riemannian manifoldwith an adapted metric).

Recall that∫

A Jφω =∫

φA ω, where ω is the volume form, and Jφ, the Jacobian, is the

local volume distortion by φ. We use the Jacobian above to define the special potential


function Vu(p) = − log |Ju(p)|. Note that Vu ≤ 0, since Dpφ expands on Eup . Then V − u isHolder on Ωb (by the Hirsch-Pugh theorem), and therefore by the preceding theorem thereexists a unique equilibrium state for it:

µVu = µSRB,

the Sinai-Ruelle Bowen measure on Ωb. (This measure is actually independent of the choiceof metric on M , since changing the metric will replace Vu by a function cohomologous toVu + a constant.)

42. Finding attractors in Axiom A systems. April 24 (Notes by LK)

We begin with an Axiom A system (M,φ), a basic set Ωb ⊂ Ω(φ) ⊂M , and we constructa special equilibrium state to go with this system. We let

Dpφ : Eup → Euφp

and define the Jacobian of this map Ju(p) = g(p) = local volume distortion factor. We usethe potential function

Vu(x) = − log Ju(x)

for x ∈ Ωb. Since the map is expanding on this unstable space, we have that Ju(x) ≥ 1 andhence Vu(x) ≤ 0.

Recall that we defined the metric

Dn(x, y) = sup0≤k≤n−1

d(φkx, φky).

We will be working with balls with respect to this metric and finding the measure of theseballs with respect to m, the manifold measure given by the volume form, and µSRB = µVu ,the SRB-measure. The following lemma lets this process begin.

Lemma 42.1. (Volume Lemma) For small ǫ > 0, there exists cǫ > 1 such that

1

cǫ≤m(BDn(x, ǫ))

exp[SnVu(x)]≤ cǫ,

for all x ∈ Ωb and all n ≥ 1.

The key ingredient to the proof of the Volume Lemma is the chain rule. We multiply theJacobian along an orbit of a point, and then compare that to the Lebesgue measure of theset that is within ǫ of x and stays within distance ǫ of the orbit of x.

We see hints of the connections between entropy (and pressure), volume growth, dimen-sion, and Lyapunov exponents. There are many formulas relating these concepts, such asthose of Young, Newhouse, and Yomdin.

We use the Volume Lemma to get the following proposition.

Proposition 42.2. (1) On (Ωb, φ), we have

P (Vu, φ) = limn→∞

1

nlogm(

⋃

x∈Ωb

BDn(x, ǫ)) ≤ 0.

(2) If m(⋃

x∈Ωb

W sǫ (x)) > 0, then P (Vu, φ) = 0 (and hence hµSRB

(φ) = −

∫

Ωb

VudµSRB).

84 Karl Petersen

Proof. (Proof of 1) If A is a maximal n, δ-separated set in Ωb, then each BDn(x, ǫ), forx ∈ Ωb, is contained in some BDn(y, ǫ+δ) for some y ∈ A (by the definition of n, δ-separatedand the definition of Dn). Thus, from the Volume Lemma,

m(⋃

x∈Ωb

BDn(x, ǫ)) ≤ cǫ+δ∑

y∈A

exp(SnVu(y)).

There is also a reverse inequality that shows that these two things are comparable. Wetake the limit as n → ∞ of 1

n times the log of both sides. The right-hand side gives thepressure, and the left-hand side gives the desired formula.

(Proof of 2) Let us use the abbreviated notation

⋃

x∈Ωb

W sǫ (x) =W s

ǫ (Ωb);

then

W sǫ (Ωb) ⊂

⋃

z∈Ωb

BDn(z, ǫ)

(since W sǫ (x) ⊂ BDn(x, ǫ)). If m(W s

ǫ (Ω)) = η > 0, then for all n ≥ 0,

m(⋃

z∈Ωb

BDn(z, ǫ)) ≥ η > 0,

so (from (1)),

P (Vu, φ) ≥ lim1

nlog η = 0.

Remark 42.1. Whenever P (V, φ) = 0 and µ is the equilibrium state for V (µ = µV ),we have such a formula for hµ(φ) = −

∫V dµV . In particular, we always have an integral

formula for entropy for g-measures.

Example 42.1. A Markov measure is an equilibrium state with P = 0, hence

hµ(σ) = −

∫

logMx0x1dµ(x) = −∑

i,j

piMij logMij .

Recall that Ωb is an attractor if there exists a neighborhood U of Ωb such that φU ⊂ Uand

⋂

n≥0 φnU = Ωb. We say that Ωb is a tight attractor if for all η > 0, we can find such

U ⊂ Bη(Ωb) = y ∈M : d(y,Ωb) < η.

We give an illustration of a possible non-tight attractor. Let Λ be an attractor lying inthe plane L with some dense orbit. Suppose there are an attracting fixed point (AFP) andrepelling fixed point (RFP) outside of L. Suppose there is a point y which is close to RFPwhose orbit gets close to Λ but then heads towards AFP. Let p = φk(y) be the point closeto Λ. We illustrate this in Figure 55.

Take a neighborhood B of Λ that also includes the one point p, but none of the rest ofthe orbit of y. Let U = B ∪ φB ∪ φ2B . . .. Then φU ⊂ U , and U contains points at asignificant distance from Λ. We illustrate B and φB in Figure 56.


.

. . . . ....

.

.

.. .y

p

L

Λ

RFP AFP

p.φ

Figure 55. Attractor Λ

.

. . . . ....

.

.

.. .y

p

L

Λ

RFP AFP

p.φ

B

.

. . . . ....

.

.

.. .y

p

L

Λ

RFP AFP

p.φ

φ B

Figure 56. B and φB

Then⋂

n≥0 φnU = Λ. If we have infinitely many such “outside orbits” swooping close

to Λ, then we cannot have a tight attractor. We will see that the Ωb’s that are not tightattractors have Lebesgue measure 0.

Proposition 42.3. Ωb is a tight attractor if and only if W sǫ (Ωb) is a neighborhood of Ωb.

86 Karl Petersen

Proof. Recall that a neighborhood of Ωb is a set which contains an open set that containsΩb. Suppose that W s

ǫ (Ωb) is a neighborhood of Ωb. Let η > 0, and choose k so thatλkǫ < η (recall λ ∈ (0, 1) from the Stable Manifold Theorem), and let U = φkW s

ǫ (Ωb).Since W s

ǫ (Ωb) ⊂ Bǫ(Ωb), and also

φW sǫ (x) ⊂W s

ǫ (φx),

we have φU ⊂ U . Also, Ωb =⋂

n≥0 φnU , because points of W s

ǫ (Ωb) have their forwarditerates approach Ωb exponentially fast.

Conversely, look at the proof of Proposition 24.1. Set η to be the γ found in this proof.Find

U ⊂ y ∈M : d(y,Ωb) < η = Bη(Ωb)

with

φU ⊂ U and Ωb =⋂

n≥0

φnU.

Then for all n ≥ 0, φnU ⊂ U ⊂ Bη(Ωb), so U ⊂ φ−nBη(Ωb). Thus

Ωb ⊂ U ⊂⋂

n≥0

φ−nBη(Ωb) ⊂W sǫ (Ωb)

by Proposition 24.1. Therefore, W sǫ (Ωb) is a neighborhood of Ωb.

We would like to determine which of the Ωb’s are tight attractors. To answer this, wewill need two other technical lemmas.

Lemma 42.4. (1) If there exists an x ∈ Ωb such that W uǫ (x) ⊂ Ωb, then Ωb is a tight

attractor.(2) If Ωb is not a tight attractor, then there exists γ > 0 such that for all x ∈ Ωb, there

exists y ∈W uǫ (x) with d(y,Ωb) > γ.

Proof. 1 ⇒ 2 is easy by compactness of Ωb. To prove 1, we use that

Ux =⋃

y∈Wuǫ (x)

W sǫ (y)

is a neighborhood of x in M . We try to find a neighborhood Bδ(Ωb) ⊂W sǫ (Ωb) and use the

previous proposition. We use the fundamental neighborhood U of the nonwandering set forwhich

⋂

j∈Z φjU = Ω(φ). Take a periodic point p in Ux and look at Xp =W u(p) ∩ Ω. Using

this machinery, we are able to produce a neighborhood of Ω that is in W sǫ (Ωb).

Lemma 42.5. For small ǫ, δ > 0, there exists a constant c(ǫ, δ) such that if x ∈ Ωb andy ∈ BDn(x, ǫ), then,

m(BDn(y, ǫ) ≥ c(ǫ, δ)m(BDn (x, ǫ)).

Theorem 42.6. Let Ωb be a basic set of a C2 Axiom A system (M,φ). The following areequivalent:

(1) Ωb is a tight attractor.(2) m(W s(Ωb)) > 0.(3) P (Vu, φ|Ωb

) = 0.


Proof. 1 ⇒ 2: Recall

W s(Ωb) =⋃

x∈Ωb

W s(x) =⋃

x∈Ωb

⋃

n≥0

φ−nW sǫ (φ

nx).

If Ωb is a tight attractor, then W sǫ (Ωb) is a neighborhood of Ωb and hence has positive

Lebesgue measure (since m is positive on open sets). Hence m(W s(Ωb)) > 0 becauseW s(Ωb) ⊃W s

ǫ (Ωb).2 ⇒ 3: Use statement 2 of Proposition 42.2, which said that

m(⋃

x∈Ωb

W sǫ (x)) > 0 ⇒ P (Vu, φ) = 0.

We have

W s(Ωb) =⋃

n≥0

φ−n(⋃

x∈Ωb

W sǫ (φ

nx)) =⋃

n≥0

φ−n(⋃

x∈Ωb

W sǫ (x)) =

⋃

n≥0

φ−nW sǫ (Ωb).

By nonsingularity, if m(W sǫ (Ωb)) = 0 then m(W s(Ωb)) = 0; therefore m(W s

ǫ (Ωb)) > 0 andhence P (Vu, φ|Ωb

) = 0.3 ⇒ 1: If Ωb is not a tight attractor, then find γ as in Lemma 42.4 such that for all

x ∈ Ωb, there exists y ∈ W uǫ (x) such that d(y,Ωb) > γ. In fact, we will choose x’s from

some n, ǫ-separated set in Ωb and find the corresponding y(x, n)’s with those γ’s.From Lemma 42.5 and Proposition 42.2, we can show

P (Vu, φ) = limn→∞

1

nlogm(

⋃

z∈Ωb

BDn(z, ǫ)) < 0.

This contradicts our assumption that P (Vu, φ) = 0, and hence Ωb is a tight attractor.

The following corollary summarizes the global dynamics of the system (M,φ).

Corollary 42.7 (Theorem 2). For m-a.e. p ∈M , there is b(p) such that d(φnp,Ωb(p)) → 0as n → ∞; that is, the orbit of almost every point of M approaches, in forward time, abasic set which is a tight attractor.

Proof. Recall from Proposition 23.2

M =r⋃

b=1

W s(Ωb),

and this equals, up to a set of m-measure 0,

⋃

Ωb tight attr.

W s(Ωb).

88 Karl Petersen

43. Attracting measures. April 28 (Notes by KS)

We begin by saying a little more about Lemma 42.4 from last time:

Lemma (Lemma 42.4). If there exists an x ∈ Ωb with W uǫ (x) ⊂ Ωb then Ωb is a tight

attractor.

Sketch of proof: Recall that to show Ωb is a tight attractor, we need to show that W sǫ (Ωb)

is a neighborhood of Ωb.Take x0 ∈ Ωb such that W u

ǫ (x0) ⊂ Ωb and look at the set

Ux0 =⋃

y∈Wuǫ (x0)

W sǫ (y).

This is a neighborhood of x0 in M (see Hirsch-Pugh).Let p ∈ Ux0 be a periodic point (recall that periodic points are dense) and let m be such

that φm(p) = p. For small β > 0, we have W uβ (p) ⊂ Ux0 (and actually W u

β (p) ⊂ Ωb—let

z ∈ W uβ (p), then z ∈ W s

ǫ (y) for some y, and φj(z) is close to Ω for all j ∈ Z. Then if ǫ

and β are small enough, by our results on Fundamental Neighborhoods (Proposition 23.1),z ∈ Ω and hence z ∈ Ωb).

Now,

W u(p) = ∪k≥0φkmW u

β (p) ⊂ Ωb,

since Ωb is φ-invariant, so Xp =W u(p) ∩Ω ⊂ Ωb. Also, for some N ,

Ωb = Xp ∪ φXp ∪ · · · ∪ φNXp.

If x ∈ Y = ∪Nk=0φkW u(p), then W u

ǫ (x) ⊂ Ωb (since d(φ−nz, φ−nx) ≤ ǫ for all n ≥ 0 and

d(φ−nx, φ−nφkp) → 0 for some k = 0, . . . , N imply z ∈W u(φkp) = φkW u(p) ⊂ Ωb).As above, we deduce that Ux ⊂ W s

ǫ (Ωb), and thus ∪x∈Y Ux ⊂ W sǫ (Ωb). Even though Y

is dense in Ωb, this is not yet quite enough to show that W sǫ (Ωb) is a neighborhood of Ωb.

However, because of the Hirsch-Pugh result, according to which W uǫ (x) and W s

ǫ (y) varycontinuously with x and y, one can find a single δ > 0 such that Bδ(x) ⊂ Ux for each x inY .

Then Ωb ⊂ ∪x∈YBδ(x) ⊂ W sǫ (Ωb), so that W s

ǫ (Ωb) is a neighborhood of Ωb, and henceΩb is a tight attractor.

We now move to the Sinai-Ruelle-Bowen Theorem (Theorem 3 in our list). This theoremtells about the asymptotic statistics of the orbits. (Note that we already know somethingfrom the Corollary 42.7 (Theorem 2).)

Theorem 43.1 (Theorem 3). Let Ωb be a basic set in a C2 Axiom A system (M,φ). Assumethat Ωb is a tight attractor. Then for m (Lebesgue measure) a.e. x ∈ W s(Ωb) (and hencein the basin of Ωb), for each f ∈ C(Ωb),

(4)1

n

n−1∑

k=0

f(φkx) −→

∫

Ωb

fdµSRB.

Proof. Part 1. Main idea: Show that m and µSRB have the same asymptotics for the setsBDn(p, ǫ) = y ∈M : d(φkp, φky) < ǫ for all k = 0, 1, . . . , n− 1.


We know that

m(BDn(p, ǫ)) ≍ exp(SnVu(p)),

by the Volume Lemma, and also,

µSRB(BDn(p, ǫ)) ≍ exp(SnVu(p)),

since µSRB is a Gibbs measure (in the topologically mixing case—without topological mix-ing, it is still true. See Bowen, Section 4.4).

So for x ∈ Ωb (or W s(Ωb)), we have by the Ergodic Theorem (recall that µSRB is anergodic measure):

1

n

n−1∑

k=0

f(φkx) −→

∫

Ωb

fdµSRB

for µSRB-a.e. x.We would like to show that this convergence holds for a.e. x with respect to m. We can

do this by using the measure asymptotics to estimate the m-measure of the “bad” set ofx ∈W s(Ωb) in terms of the µSRB-measure.

Part 2. Show that the theorem holds for just one f ∈ C(Ωb)—we do it for each f in acountable dense set in C(Ωb), take the union of all the resulting “bad” sets of m-measure 0,then use standard approximation arguments to extend to all f ∈ C(Ωb).

Part 3. Let f ∈ C(Ωb) and let δ > 0. Set

Cn(δ) = p ∈M :∣∣Anf(p)− f

∣∣ > δ,

where

Anf(p) =1

n

n−1∑

k=0

f(φkp) and f =

∫

Ωb

fdµSRB.

Let Eδ = ∩m≥1 ∪n≥m Cn(δ) (note that the intersection is decreasing). The set Eδ is theset of p which are δ-bad for infinitely many n. We know that µSRB(Eδ) = 0, and we wantto show that m(Eδ ∩W

s(Ωb)) = 0 as well.Choose ǫ > 0 small enough so that |f(x)− f(y)| < δ when d(x, y) < ǫ (using uniform

continuity), and small enough that the requirements for canonical coordinates, etc., aremet.

Part 4. Fix m > 0 and define Rm, Rm+1, . . . inductively by letting Rm be a maximalfamily of disjoint Dm-balls of radius ǫ in Ωb ∩ Cm(2δ) and Rm+1 be a maximal family ofdisjoint Dm+1-balls of radius ǫ in Ωb ∩ Cm+1(2δ) which are also disjoint from the balls inRm. Let R

′n ⊂ Ωb ∩Cn(2δ) denote the set of centers of the balls in Rn.

Then

(5) W sǫ (Ωb) ∩ ∪∞

n=mCn(3δ) ⊂ ∪∞n=m ∪x∈R′

nBDn(x, 2ǫ).

(If y is an element of the left-hand side of (5), then y ∈ W sǫ (z) for some z ∈ Ωb. But

since y ∈ Cn(3δ) for some n ≥ m, and |Anf(z)−Anf(y)| < δ, we have z ∈ Cn(2δ). Bymaximality of Rn, for some k between m and n, and some x ∈ R′

k, BDn(z, ǫ)∩BDk(x, ǫ) 6=

∅. Thus

y ∈ BDn(z, ǫ) ⊂ BDk(z, ǫ) ⊂ BDk

(x, 2ǫ),

which is what we needed to show.)

90 Karl Petersen

Therefore, by the Volume Lemma,

(6) m(W sǫ (Ωb) ∩ ∪∞

n=mCn(3δ)) ≤∞∑

n=m

∑

x∈R′

n

C2ǫ exp(SnVu(x)).

We now use the µSRB-measure to estimate the right-hand side of (6) from above.Part 5. Let Um = ∪∞

n=m ∪x∈R′

nBDn(x, ǫ). Since x ∈ R′

n ⊂ Cn(2δ), each BDn(x, ǫ) ⊂Cn(δ), so Um ⊂ ∪∞

n=mCn(δ). Therefore, by the Ergodic Theorem,

0 = µSRB(Eδ) = limm→∞

µSRB (∪∞n=mCn(δ)) ,

so µSRB(Um) → 0.Part 6. There exist positive constants bǫ and C such that

µSRB(Um) ≥ bǫ

∞∑

n=m

∑

x∈R′

n

exp(SnVu(x))

≥ Cm(W sǫ (Ωb) ∩ ∪∞

n=mCn(3δ)).

In the topologically mixing case, the first inequality holds using the Gibbs condition onµSRB . If (M,φ) is not topologically mixing, then the inequality still holds (see Bowen, 4.4).The second inequality holds by Part 4.

Therefore, as m→ ∞, we havem(W sǫ (Ωb)∩∪

∞n=mCn(3δ)) → 0 and som(W s

ǫ (Ωb)∩E3δ) =0.

Part 7. Let δ′ > 3δ and n ≥ 0. Then the set Eδ′ = lim∣∣Anf − f

∣∣ ≥ δ′ is φ-invariant, so

Eδ′∩Ws(Ωb) = Eδ′∩∪n≥0φ

−nW sǫ (Ωb). Now, for each n, since φ preserves sets of m-measure

0, andEδ′ ∩ φ

−nW sǫ (Ωb) ⊂ φ−n(E3δ ∩W

sǫ (Ωb)),

we have m(Eδ′ ∩ φ−nW s

ǫ (Ωb)) = 0. Thus, m(Eδ′ ∩Ws(Ωb)) = 0.

Part 8. Finally, the set of x for which (4) does not hold is equal to ∪∞n=1E1/n. Since

m(E1/n ∩Ws(Ωb)) = 0, we conclude that the convergence holds for m-a.e. x in W s(Ωb).

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