Special ergodic theorems and dynamical large deviations

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IOP PUBLISHING NONLINEARITY

Nonlinearity 25 (2012) 3189–3196 doi:10.1088/0951-7715/25/11/3189

Special ergodic theorems and dynamical largedeviations

Victor Kleptsyn1, Dmitry Ryzhov2 and Stanislav Minkov3

1 CNRS, Institute of Mathematical Research of Rennes, IRMAR, UMR 6625 du CNRS2 Chebyshev Laboratory, Department of Mathematics and Mechanics, St.-Petersburg StateUniversity, Russia3 Moscow State University, Faculty of Mechanics and Mathematics, GSP-1, Leninskie Gory,Moscow, 119991, Russia

E-mail: victor.kleptsyn@univ-rennes1.fr

Received 9 February 2012Published 15 October 2012Online at stacks.iop.org/Non/25/3189

Recommended by L Bunimovich

AbstractLet f : M → M be a self-map of a compact Riemannian manifold M , admittinga global SRB measure µ. For a continuous test function ϕ: M → R anda constant α > 0, consider the set Kϕ,α of the initial points for which theBirkhoff time averages of the function ϕ differ from its µ–space average by atleast α. As the measure µ is a global SRB one, the set Kϕ,α should have zeroLebesgue measure.

The special ergodic theorem, whenever it holds, claims that, moreover, thisset has a Hausdorff dimension less than the dimension of M . We prove that forLipschitz maps, the special ergodic theorem follows from the dynamical largedeviations principle. We also define and prove analogous result for flows.

Applying the theorems of Young and of Araujo and Pacifico, we concludethat the special ergodic theorem holds for transitive hyperbolic attractorsof C2-diffeomorphisms, as well as for some other known classes of maps(including the one of partially hyperbolic non-uniformly expanding maps) andflows.

Mathematics Subject Classification: 37D20, 37A25, 37A50

1. Introduction

1.1. Basic concepts

Let f : M → M be a self-map of a compact Riemannian manifold M to itself. The famousBirkhoff ergodic theorem states that if µ is an ergodic invariant measure for f , then for any

0951-7715/12/113189+08$33.00 © 2012 IOP Publishing Ltd & London Mathematical Society Printed in the UK & the USA 3189

3190 V Kleptsyn et al

continuous function ϕ ∈ C(M) and for µ-almost every x ∈ M the time averages ϕn of thefunction ϕ at the point x converge to the space average ϕ:

limn→∞ ϕn(x) = ϕ, where ϕn(x) := 1

n

n−1∑k=0

ϕ ◦ f k(x), ϕ :=∫

M

ϕ dµ. (1)

The conclusion of this theorem has one large drawback. It is usually much more natural(in particular, from the ‘physical’ point of view) to take an initial point x, which is generic inthe sense of the Lebesgue measure. But if the measure µ is supported on some ‘thin’ set (forinstance, on a point), one cannot deduce anything for initial points outside this set.

This leads to another famous notion, the Sinai–Ruelle–Bowen measure (or SRB measurefor short):

Definition 1. An invariant measure µ is called an (observable) SRB measure for f : M → M

if there is a set U ⊃ M of positive Lebesgue measure such that for every x ∈ U and for everycontinuous function ϕ on M one has

limn→∞ ϕn(x) → ϕ. (2)

An SRB measure is called global if one can take U = M .

Not all the systems possess SRB measures. The first example of a system with no SRBmeasures, the heteroclinic attractor (see [5, 14]), was given by Bowen himself. However,under certain assumptions, e.g. hyperbolicity of the system, one can guarantee its existence;we refer the reader to [17] and to [4, section 14] for the survey of this subject.

But even the existence of an SRB measure sometimes turns out to be insufficient. In somesituations (see [6–9), it is necessary not only to conclude that the set of ‘bad’ initial conditionshas zero Lebesgue measure, but also to state the same for its images under Holder (but notnecessarily absolutely continuous!) conjugacy between two dynamical systems.

One of the ways of handling this difficulty bases on the analysis of Hausdorff dimensionof this set. It leads to the concept of the special ergodic theorem, proposed by Yu Ilyashenko,to which this paper is devoted. We will now introduce some auxiliary notations, and presentthis concept in definition 2.

Taking a continuous test function ϕ ∈ C(M) and any α � 0, we define the set of (ϕ, α)-nontypical points as

Kϕ,α := {x ∈ X: lim

n→∞ |ϕn(x) − ϕ| > α}.

In other words, Kϕ,α is the set of points for which the statement ‘time averages convergeto the space one’ is violated ‘in an essential way’.

By definition, if µ is a global SRB measure, then Leb(Kϕ,0) = 0. Thereinafter we assumethat µ actually is a global SRB measure. Otherwise we can restrict all our considerations toits (invariant) basin of attraction U ⊂ M , that is, the set of all points for which (2) holds.

Definition 2. Say that for (f, µ) the special ergodic theorem holds, if for any continuousfunction ϕ ∈ C(M) and any α > 0 the Hausdorff dimension of the set Kϕ,α is strictly less thanthe dimension of the phase space:

∀ϕ ∈ C(M), α > 0 dimH Kϕ,α < dim M. (3)

Remark 1. Formally speaking, to state the property (3) one doesn’t need to claim the existenceof an SRB measure. Though, this property itself immediately implies that µ actually is anSRB measure. Indeed, the set Kϕ,0 of points for which the equality (2) between time and spaceaverages does not hold is exhausted by a countable number of sets Kϕ,1/n of zero Lebesguemeasure.

Special ergodic theorems and dynamical large deviations 3191

Having introduced this notion, Ilyashenko [6] proved that the special ergodic theoremholds for the circle doubling map; later, Saltykov [13] also proved it for the linear Anosovmap on the 2-torus. Analogous questions were also considered in the work of Gurevich andTempelman [3], where the Hausdorff dimensions of level sets of Birkhoff averages for finitespin lattice systems were evaluated.

The aim of this paper is to prove the special ergodic theorem for a larger class ofmaps. Namely, we notice and prove that the special ergodic theorem can be obtained as acorollary of another property of the system, the dynamical large deviations principle. Thisimplies that the special ergodic theorem holds, for instance, for all the transitive hyperbolicC2–diffeomorphisms, for partially hyperbolic non-uniformly expanding C1-diffeomorphisms,and for some other examples.

Moreover, in the last part of the paper we present an example of C∞-smoothdiffeomorphism of a 2-sphere such that it admits a global SRB measure but for which the specialergodic theorem doesn’t hold. One can also see [11] for different geometric constructions thatalso lead to such an example.

1.2. Large deviations principles

The dynamical large deviations principle is a dynamical counterpart of the large deviationsprinciple in the probability theory. The latter states that under some assumptions on thedistribution of i.i.d. random variables ξj , for any α > 0 the probability of a large deviationP(|(ξ1 + · · · + ξn)/n − Eξ | > α) decreases exponentially in n as n goes to infinity, see, forexample, [15].

Definition 3. Suppose that µ is an invariant measure for a self-map f : M → M , andm be an arbitrary measure on M . Say, that for (f, µ) the dynamical large deviationsprinciple for reference measure m holds, if for any ϕ ∈ C(M), α > 0 there exist constants4

C = C(α, ϕ), h = h(α, ϕ) > 0 such that

∀n ∈ N m(Kϕ,α,n) � C(α, ϕ)e−nh(α,ϕ), (4)

where

Kϕ,α,n := {x ∈ M: |ϕn(x) − ϕ| � α}.Usually, for m in (4) stands either for the Lebesgue measure or the invariant measure µ

itself. In this paper we will use the case m = Leb (otherwise our arguments would not work).Under this assumption, µ in definition 3 is a global SRB measure due to the Borel–Cantellilemma type arguments.

The dynamical large deviations principle with respect to the invariant measure was widelystudied (see, for example, [10, 11]). Unfortunately, we cannot use any of these numerous resultsdirectly, unless the invariant measure is absolutely continuous with respect to the Lebesgue one.

1.3. Theorem and applications

Theorem 1 (Main result). Let f : M → M be a Lipschitz self-map of compact Riemannianmanifold M and µ be a global SRB measure. If the dynamical large deviations principle withLebesgue reference measure holds for (f, µ), then for this map the special ergodic theoremalso holds.

4 For a fixed function ϕ, the function h = h(α) is called the ‘rate function’ or ‘Cramer function’.

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As will be shown in the next section, the proof of theorem 1 is indeed rather simple.However, the conclusion itself seems very interesting to us—especially due to the possibilitiesfor applying it to the study of the skew products.

There are numerous applications of the theorem.

Definition 4. A C1-map f : M → M is non-uniformly expanding, if

lim supn→+∞

1

n

n∑j=0

log ||Df (f j (x))−1|| < 0

for Lebesgue almost all x. A C1-map f : M → M is partially hyperbolic non-uniformlyexpanding, if the tangent bundle of M decomposes into direct sum of two continuoussubbundles such that one of them is uniformly contracting, and the other is non-uniformlyexpanding.

Corollary 1. The special ergodic theorem holds:

(i) for transitive hyperbolic attractors of C2-diffeomorphisms; in particular, for transitiveC2 Anosov diffeomorphisms;

(ii) for the stochastic maps from real quadratic family f (x) = x2 + c;(iii) for typical maps from general C2 unimodal families;(iv) for partially hyperbolic non-uniformly expanding C1-diffeomorphisms;(v) for Lorenz-like maps.

All the statements follow from theorem 1 and the large deviations principles for thesemaps. The principle for the first item was proven in [16], see theorem 2 (2) and item (ii) in thediscussion after it. For other items, see [2], section 2.

1.4. Flows

All the above concepts and statements can be defined for flows as well as for maps. Namely,let {ft } be a flow on a compact Riemannian manifold M with an invariant measure µ. For anycontinuous function ϕ ∈ C(M) and any x ∈ M one defines its T -time average ϕT over thetime T > 0 as

ϕT (x) := 1

T

∫ T

t=0ϕ ◦ ft (x) dt. (5)

In the same way as for the maps, given α � 0, we define the set of (ϕ, α)-nontypicalpoints:

Kϕ,α :={x ∈ X: lim

T →∞|ϕT (x) − ϕ| > α

},

where, as above, ϕ = ∫M

ϕ dµ denotes a space average of the function ϕ. We will also use thenotation

Kϕ,α,T := {x ∈ M: |ϕT (x) − ϕ| � α}.Definition 5. Say that for the flow F with an invariant measure µ the special ergodic theoremholds, if for any continuous function ϕ ∈ C(M) and any α > 0 the Hausdorff dimension ofthe set Kϕ,α is strictly less than the dimension of the phase space:

∀ϕ ∈ C(M), α > 0 dimH Kϕ,α < dim M. (6)

We also say that the dynamical large deviations principle with respect to the reference measurem holds for (F, µ), if there exist constants C = C(α, ϕ), h = h(α, ϕ) > 0 such that

∀T > 0 m(Kϕ,α,T ) � C(α, ϕ)e−nh(α,ϕ). (7)

Special ergodic theorems and dynamical large deviations 3193

In the same way as for the maps, we have the following

Theorem 2. Let µ be an invariant measure for a flow (a continuous action of R) {ft } on acompact Riemannian manifold M . Suppose that f1 : M → M is a Lipschitz self-map ofM . If the dynamical large deviations principle with the Lebesgue reference measure holds for({ft }, µ), then the special ergodic theorem also holds for this flow.

Corollary 2. The special ergodic theorem holds for Anosov flows, as well as for the Lorenzflow and, moreover, for geometric Lorenz flows.

The statements follow from the large deviations principles for these flows. For Anosovflows, this principle was proven in [16] (see theorem 2 (2) and item (ii) in the discussion afterit). For precise definitions of (geometric) Lorenz flows and for large deviations principles forthese flows see [1].

2. Proofs

2.1. Proof of the main theorem

Let f : M → M be Lipschitz map on a d-dimensional compact manifold M with a global SRBmeasure µ supported on it, for which the dynamical large deviations principle holds. Chooseand fix ϕ ∈ C(M), α > 0, and denote L := max(Lip(f ), 2).

We are going to prove that

dimH Kϕ,α � d0 := d − h(α/2, ϕ)

ln L< d.

That is, for each d ′ > d0 we want to find a cover of Kϕ,α with arbitrarily small d ′-dimensionalvolume. Thus we deduce that d ′-dimensional volume of Kϕ,α is zero, and hence dimH Kϕ,α �d ′ for all d ′ > d0.

To construct such a cover, we first choose (and fix) δ = δ(ϕ, α) > 0 such that

dist(x, y) < δ ⇒ |ϕ(x) − ϕ(y)| < α/2.

Then we have the following lemma:

Lemma 1. If x ∈ Kϕ,α,n, then B(x, r) ⊂ Kϕ,α/2,n, where

r = r(n) := L−nδ (8)

and B(x, r) is a ball centered at x with the radius r .

Proof. Let y ∈ B(x, r). Then dist(x, y) < L−nδ and consequently

dist(f k(x), f k(y)) < δ for k = 0, 1, . . . n.

Therefore, by the choice of δ, we have

|ϕ(f k(x)) − ϕ(f k(y))| < α/2 for k = 0, 1, . . . n,

which implies (by averaging over k) that |ϕn(x) − ϕn(y)| < α/2. Hence

|ϕn(y) − ϕ| � |ϕ − ϕn(x)| − |ϕn(x) − ϕn(y)| � α − α/2 = α/2. �Now, for any n, let r(n) be the same as in 8. Choose a maximal r(n)

2 -separated set Sn inM . The union of r(n)

2 -neighbourhoods of elements of Sn is a cover of M , and in particular,of Kϕ,α,n. Remove from this cover the neighbourhoods that do not intersect Kϕ,α,n: let

S ′n := {x ∈ Sn: Br(n)/2(x) ∩ Kϕ,α,n �= ∅}

3194 V Kleptsyn et al

and

Cn :={B

(x,

r(n)

2

)| x ∈ S ′

n

}.

By construction, Cn is still a cover of Kϕ,α,n. Also, noticing that any point of Kϕ,α belongsto an infinite number of Kϕ,α,n, we see that for any N ∈ N the union of Cn for n � N is a coverof Kϕ,α .

Let us estimate the d ′-dimensional volumes of these covers. First, estimate the cardinalityof Cn: note that radius r(n)/4 balls centred at the elements of Sn are pairwise disjoint. Hence,the volume of the union

Vn :=⋃x∈S ′

n

Br(n)/4(x)

is at least card(Cn) · c · r(n)d , where the constant c > 0 depends only on the geometry of themanifold M .

On the other hand, by lemma 1, Vn ⊂ Kϕ,α/2,n. Hence, due to the assumed large deviationsprinciple, Leb(Vn) � Ce−h(α/2,ϕ)n. Thus,

card(Cn) · c · r(n)d � Ce−h(α/2,ϕ)n

and hence

card(Cn) � C ′Lnde−h(α/2,ϕ)n = C ′Lnd0 .

Therefore we see that the d ′-dimensional volume of the union of covers Vn for n � N canbe estimated as

∞∑n=N

card(Cn) · r(n)d′ � const ·

∞∑n=N

L−n(d ′−d0)

This series converges, as d ′ > d0 by the choice of d ′. Thus, choosing N sufficientlylarge, we obtain covers of Kϕ,α with arbitrarily small d ′-dimensional volumes, and hencedimH Kϕ,α � d ′ for any d ′ > d0. This implies that

dimH Kϕ,α � d0,

which concludes the proof of theorem 1.

2.2. Proof of theorem 2

Suppose x ∈ Kϕ,α for the flow F . It means that there exists a sequence tn growing to ∞ suchthat

|ϕtn(x) − ϕ| > α.

Note that

|ϕtn(x) − ϕ[tn](x)| <maxy∈M |ϕ(y)|

[tn],

where [·] stands for an integer part of a value. Then for all sufficiently large n

|ϕ[tn](x) − ϕ| > α/2.

Hence x ∈ Kϕ,α/2 for the time 1 map f1 of a flow F . This set has a Hausdorff dimension lessthan the dimension of M due to theorem 1.

Special ergodic theorems and dynamical large deviations 3195

3. An example of a diffeomorphism for which the special ergodic theorem doesn’t hold

In this part of the paper we present an example of a C∞-smooth diffeomorphism of 2-spheresuch that it admits a global SRB measure but for which the special ergodic theorem doesn’thold. This example drastically simplifies the example that was constructed by the third authorin [12].

Consider the Cantor set K0, obtained from the interval I = [0, 1] by the standard infiniteprocedure of consecutive deleting the middle intervals of relative length 1/n. Namely, on thestep number n we take all the intervals that were obtained as a result of the previous steps anddelete their central parts of relative length 1/n. The set K0 has zero Lebesgue 1-measure andHausdorff dimension 1 (see [12], lemma 1).

Let K be a subset of S1 obtained from K0 by gluing the endpoints of an interval I . Thereexists a C∞ smooth function f : S1 → R with the following properties:

(a) f (t) = 0 if t ∈ K

(b) f (t) > 0 if t /∈ K .

We also use the C∞-diffeomorphism g of the segment [0,1] with two fixed points: 0 is anattractor and 1 is the parabolic repeller. All points of interval (0, 1) attract to 0. We also claimthat all the derivatives of the function g(r) − r at the point r = 1 are zero:

[g(r) − r](n)r=1 = 0 ∀n ∈ N. (9)

Now we define the function F of the unit disc D with the center Q by formula in standardpolar coordinates (r, t):

F(r, t) = ((1 − f (t)) · r + f (t) · g(r), t),

F (Q) = Q. F is the smooth diffeomorphism of D with invariant radii. Any point with angularcoordinate t ∈ K is fixed by F , and any point with angular coordinate t ∈ S1 \ K tends toQ. By the Fubini theorem, the Lebesgue-a.e. point of D tends to its center Q. Hence F hasa global SRB-measure µ that is a δ-measure sitting at the point Q. But the set L of the fixedpoints of F has the Hausdorff dimension 2 (see [12], lemma 2). Note also that L ∩ D1 alsohas the Hausdorff dimension 2, where D1 = {(r, t) ∈ D | r � 1/2}.

Let us choose a smooth test-function φ on D such that φ(Q) = 0, and φ(X) � 1 ifX ∈ D1. Then the µ-space average of φ (that is, φ(Q)) is 0. Therefore, the set Kφ,1 containsthe intersection L ∩ D1. Hence dimH (Kφ,1) = 2, and the map F doesn’t satisfy the specialergodic theorem.

To achieve such an example on a closed manifold S2, one should pass to the one-pointcompactification of D. The map F descends correctly to this compactification because theboundary of D consists of the fixed points for F , and is C∞ smooth because of the propertyequation (9). Thus we constructed a diffeomorphism of S2 that admits a global SRB-measure,but for which the special ergodic theorem doesn’t hold.

Acknowledgments

The authors are very grateful to Yu Ilyashenko, M Blank, A Bufetov, A Gorodetsky, Ya Pesinand P Saltykov for fruitful discussions. The first two authors would also like to thankthe organizers of the conference ‘Geometric and Algebraic Structures in Mathematics’for bringing them together. All three authors were partially supported by RFBR grant10-01-00739-a and joint RFBR/CNRS grant 10-01-93115-CNRS-a. The research of thesecond author was supported by the Chebyshev Laboratory (Department of Mathematics

3196 V Kleptsyn et al

and Mechanics, St-Petersburg State University) under the Russian Federation governmentgrant 11.G34.31.0026.

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