The Smooth Realization Problem: Area Preserving Di ...smooth observables. 2. The proof of the Main theorem is based on the work of Katok (1979) but requires some new ideas. I believe

The Smooth Realization Problem:Area Preserving Diffeomorphisms with

Polynomial Decay of Correlations

Yakov PesinPennsylvania State University

A joint work with Farruh Shahidi and Samuel Senti

August 3, 2020

Yakov Pesin Pennsylvania State UniversityThe Smooth Realization Problem: Area Preserving Diffeomorphisms with Polynomial Decay of Correlations

The Smooth Realization Problem

The smooth realization problem in dynamics asks whether there isa diffeomorphism of a compact smooth manifold, which has aprescribed collection of ergodic properties with respect to a naturalinvariant measure such as the Riemannian volume (or a smoothmeasure). Another interesting measure to consider is the MME.

A yet more interesting but substantially more difficult version ofthe smooth realization problem is to construct a volume preservingdiffeomorphism with prescribed ergodic properties on any givensmooth compact manifold. In other words, does the topology ofthe manifold can provide an obstruction for realizing a givenergodic property.


Some History

Starting with the basic ergodic property – ergodicity – Anosov andKatok (1970) constructed an example of a volume preservingergodic C∞ diffeomorphism.

Katok (1979) gave an example of an area preserving C∞ map withnon-zero Lyapunov exponents on any surface which is Bernoulli.

Brin, Feldman, and Katok (1981) extended this result byconstructing a volume preserving C∞ diffeomorphism, which isBernoulli, on any Riemannian manifold of dimension ≥ 5.

In their example the map has all but one non-zero Lyapunovexponents. Dolgopyat and Pesin (2002) constructed a volumepreserving C∞ Bernoulli diffeomorphism with all non-zeroLyapunov exponents on any Riemannian manifold of dimension≥ 2.


It is natural to ask if a compact smooth manifold admits a volumepreserving Bernoulli diffeomorphism with non-zero Lyapunovexponents that enjoys other important statistical properties such asexponential or polynomial decay of correlations (that is rate ofmixing), the Central Limit Theorem, and the Large Deviationsproperty; all three with respect to a natural class of observables,e.g., functions which are Holder continuous.


The Correlations Function

Let X be a measurable space and T : X → X a measurableinvertible transformation preserving a measure µ.Let H1 and H2 be two classes of real-valued functions on X calledobservables. For h1 ∈ H1 and h2 ∈ H2 define the correlationfunction

Corn(h1, h2) :=

∫h1(T n(x))h2(x) dµ−

∫h1(x) dµ

∫h2(x) dµ.


Decay of Correlations

1. T has polynomial decay of correlations (more precisely, admitspolynomial upper bound on correlations) w.r.t classes H1 and H2 ifthere exists γ1 > 0 s.t. for any h1 ∈ H1, h2 ∈ H2, and any n > 0,

|Corn(h1, h2)| ≤ Cn−γ1 ,

where C = C (h1, h2) > 0 is a constant.2. T admits a polynomial lower bound on correlations w.r.t classesH1 and H2 of observables if there exists γ2 > 0 s.t. for anyh1 ∈ H1, h2 ∈ H2, and any n > 0,

|Corn(h1, h2)| ≥ C ′n−γ2 ,

where C ′ = C ′(h1, h2) > 0 is a constant.3. T has exponential decay of correlations w.r.t classes H1 and H2

if there exists γ3 > 0 s.t. for any h1 ∈ H1, h2 ∈ H2, and any n > 0,

|Corn(h1, h2)| ≤ C ′′e−γ3n,

where C ′′ = C ′′(h1, h2) > 0 is a constant.Yakov Pesin Pennsylvania State University

The Smooth Realization Problem: Area Preserving Diffeomorphisms with Polynomial Decay of Correlations

Main Theorem

There are numbers γ2 > γ1 > 0 and β > 0 such that any compactsmooth connected and oriented surface M admits an areapreserving C 1+β diffeomorphism f = fM satisfying:

1 f has the Bernoulli property.

2 f has non-zero Lyapunov exponents almost everywhere withrespect to area m.

3 f admits a polynomial upper bound on correlations withrespect to the exponent γ1 and the class C ρ of all Holdercontinuous functions on M.

4 There is a nested sequence of subsets {Mj} that exhaust Msuch that f admits a polynomial lower bound on correlationswith respect to the exponent γ2 and the class C ρ of all Holdercontinuous functions h for which there is k = k(h) such thatsupp(h) ⊂ Mk .


Comments

1. In the two dimensional case Liverani and Martens (2005)constructed an example of an area preserving C∞ diffeomorphismof the 2-torus with non-zero Lyapunov exponents which haspolynomial decay of correlations with respect to the class ofsmooth observables.

2. The proof of the Main theorem is based on the work of Katok(1979) but requires some new ideas. I believe that modifying ourargument and using the approach of Brin, Feldman, and Katok(1981), and of Dolgopyat and P. one can extend the Main theoremto manifolds of dimension ≥ 2.


3. One can show that the map f in the Main theorem satisfies theCentral Limit Theorem and has the Large Deviation property.Moreover, f has a unique MME which is Bernoulli, has non-zeroLyapunov exponents almost everywhere, exponential decay ofcorrelations, satisfies the Central Limit Theorem and has the LargeDeviation property.

4. In view of our Main theorem it is interesting to know whetherany smooth compact Riemannian manifold admits a volumepreserving diffeomorphism with exponential decay of correlationsand whether it admits a diffeomorphism with a unique MME withrespect to which it has polynomial decay of correlations. Bothquestions seem to be quite difficult.


The Katok Map

Consider the automorphism of the two-dimensional torus T 2 givenby the matrix A := ( 5 8

8 13 ). It has four fixed points x1 = (0, 0),x2 = ( 1

2 , 0), x3 = (0, 12 ), and x4 = ( 1

2 ,12 ). For i = 1, 2, 3, 4 let

D ir = {(s1, s2) : s1

2 + s22 ≤ r2} be the disk of radius r centered at

xi and Dr =⋃4

i=1 Dir . Here (s1, s2) is the coordinate system

obtained from the eigendirections of A and originated at xi . Letλ > 1 be the largest eigenvalue of A.Choose 0 < α < 1 and a slow-down function ψ : [0, 1] 7→ [0, 1]satisfying:

1 ψ(u) = 1 for u ≥ r0 and some 0 < r0 < 1;

2 ψ′(u) > 0 for every 0 < u < r0;

3 ψ(u) = (u/r0)α for 0 ≤ u ≤ r02 .


There exists an area preserving C 2+κ, κ = 2α1−α , diffeomorphism

fT 2 of T 2 such that

1 It is topologically conjugate to A.

2 It has two continuous, uniformly transverse, invariantone-dimensional stable E s(x) and unstable Eu(x)distributions; for a.e. x ∈ T 2 the Lyapunov exponent alongthese distributions are negative and respectively, positive; theLyapunov exponents at the fixed points xi are zero;

3 It has two continuous, uniformly transverse, invariantone-dimensional foliations with smooth leaves called stableW s and unstable W u foliations.

4 It is isomorphic to a Bernoulli diffeomorphism.

5 In D ir , i = 1, 2, 3, 4, the map fT 2 is – up to a smooth

coordinate change – the time-1 map of the flow given by

s1 = s1ψ(s12 + s2

2) log λ, s2 = −s2ψ(s12 + s2

2) log λ.

The map fT 2 is called the Katok map.Yakov Pesin Pennsylvania State University


Young tower for the Katok map

Since the map fT 2 is topologically conjugate to the hyperbolicautomorphism A of the torus, it possesses a Markov partition ofarbitrary small diameter. Let P be an element of the partitionwhich is far away from the neighborhood Dr of the four fixedpoints. Let also τ = τ(x) be the first return time of x to P.Consider the set P ⊂ P consisting of points x for which τ(x) <∞.Since fT 2 preserves area, P has full measure in P. LetF (x) = f τ(x)(x) be the first return time map. Thus we obtain atower with base P, hight τ , and the base map F .


In fact, this tower is a Young tower (P., Senti, Zhang, 2019):

1 for each n > 0 the level set {x ∈ P : τ(x) = n} is a disjoinunion of s-sets Pn,i , i = 1, . . . , k(n) called partition elements;for x ∈ Pn,i , the full length stable curve through x lies in Pn,i ;f nT 2(Pn,i ) is a u-set;

2 k(n) ≤ C exp(hn) where C > 0 and h > 0 are constants andh < htop(fT 2) = log λ;

3 the induced map F has the bounded distortion property withrespect to the partition elements;

4 the inducing time is integrable:

S =∞∑n=1

m({x ∈ P : τ(x) = n}) =∞∑n=1

k(n)∑i=1

m(Pn,i ) <∞.

Our goal now is to obtain polynomial lower and upper bounds onthe rate of decay of the tale in the above sum, i.e.,

Sn = m({x ∈ P : τ(x) > n}).


Lower Bound of the Tale

Lemma

There is C1 > 0 such that for any 0 < α < 14

m({x ∈ P : τ(x) > n}) > C1n−(γ−1),

where γ = 12α + 5 · 2α−3 + 1

24 (note that γ > 2).

To see this write

m({x ∈ P :τ(x) > n}) =∞∑

k=n+1

m({x ∈ P : τ(x) = k})

=∞∑

k=n+1

∑Pn,i :τ(Pn,i )=k

m(Pn,k) >∞∑

k=n+1

m(Pn,`),

where for large n, Pn,` is chosen such that it travels some timeoutside Dr , then enters Dr only once and travels there for a longtime and finally travels some time before entering P.


Claim 1. There is Q > 0 such that for any N > 0 one can find ans-set Pn,` with n > N and numbers 0 < m1 < m2 satisfyingm1 < Q, n −m2 < Q, f k(Pn,`) ∩ Dr = ∅ for 0 ≤ k < m1 orm2 < k ≤ n and f k(Pn,`) ∩ Dr 6= ∅ for m1 ≤ k ≤ m2.

To show this it suffices to find Q > 0 such that for sufficientlylarge n there is an admissible word of length n of the formPW1PiW2P where the words Wj are of length l(Wj) < Q,j = 1, 2, and do not contain any of the symbols P or Pk (theelement of the Markov partition containing the fixed point xk fork = 1, 2, 3, 4), and the word Pi consists of the symbol Pi which isrepeated n − 2− l(W1)− l(W2) times.

Claim 2. There is C2 > 0 such that m(Pn,`) > C2n−γ .

The proof requires some technically involved estimates on thesolutions of differential equations

s1 = s1ψ(s12 + s2

2) log λ, s2 = −s2ψ(s12 + s2

2) log λ,

which describe the dynamics in the slow-down domains.


Upper Bound of the Tale

Lemma

There is C3 > 0 such that for any 19 < α < 1

4

m({x ∈ P : τ(x) > n}) < C3n−(γ′−1),

where γ′ = 12α + 1

2α+4 (note that γ > γ′ > 2).

To prove this given an s-set Pn,i with τ(Pn,i ) = n, choose anynumbers k = k(Pn,i ), p = p(Pn,i ), and two finite collections ofnumbers {km ≥ 0}m=1,...,p and {lm ≥ 0}l=0,...,p such that

1 k1 + k2 + · · ·+ kp = k and l1 + l2 + · · ·+ lp+1 = n − k ;

2 the trajectory of the set Pn,i under f jT 2 , 0 ≤ j ≤ n,

consecutively spends lm-times outside Dr and km-times insideDr .


Given 0 < p < k < n, consider the collections

Sk,n,p = {Pn,i : τ(Pn,i ) = n, k = k(Pn,i ), p = p(Pn,i )}.

Note that

m({x ∈ P : τ(x) = n}) ≤n∑

k=1

k∑p=1

maxPn,i∈Sk,n,p

{m(Pn,i )}Card Sk,n,p.

Claim 1. There are 0 < h < htop(f ), ε0 > 0, and C4 > 0 such thatε0 < htop(f )− h and

Card Sk,n,p ≤ C4p−2e(h+ε0)(n−k).

Claim 2. There exists ε0 > 0 such that for any Pn,i ∈ Sk,n,p,

m(Pn,i ) ≤ C5k−γ′e(− log λ+ε0)(n−k),

where C5 > 0 is a constant.Yakov Pesin Pennsylvania State University


Carrying the Katok Map to a Surface

Theorem

There are maps

ϕ1 : T 2 → S2, ϕ2 : S2 → D2, ϕ3 : D2 → M

such that the maps

fS2 = ϕ1 ◦ fT 2 ◦ ϕ−11 : S2 → S2,

fD2 = ϕ2 ◦ fS2 ◦ ϕ−12 : D2 → D2

are area preserving C 2+κ diffeomorphisms, fD2 is identity on theboundary of the disk and the map

fM = ϕ3 ◦ fD2 ◦ ϕ−11 : M → M

is area preserving and of class C 1+β for some β > 0.


1 ϕ1 is a double branched covering, is one-to-one on eachbranch, and C∞ everywhere except at the points xi ,i = 1, 2, 3, 4 where it branches;

2 ϕ2 unfolds S2 onto the unit disk D2 and is C∞;

3 ϕ1 and ϕ2 preserve area, i.e., (ϕ1)∗mT 2 = mS2 and(ϕ2)∗mS2 = mD2 .

The maps ϕ1 and ϕ2 were constructed by Katok. However,constructing the map ϕ3 in our case is quite difficult, since wehave to deal with finite regularity of the map fD2 .


Infinite vs. Finite Differentiability and Flatness

If the slow-down function ψ is C∞, then so is the map fD2 .Moreover, if the function ψ−1 is infinitely flat at zero, then fD2 isinfinitely flat at the boundary of the disk. In this case aconstruction of the map ϕ3 was given by Katok.In our case the slow-down function ψ is polynomial and as a result,the map fD2 is only of class C 2+κ and is finitely flat at theboundary: there is a sequence of open domains Vn ⊂ D2 s.t.

Vn ⊂ V n ⊂ Vn+1, ∪n≥1Vn = D2, Vn−1 ⊂ fD2(Vn) ⊂ Vn+1

and for every 0 < β < 2 + κ,

‖fD2 − Id‖C1+β(Vn+1\Vn−1) ≤ (rn−1)2+κ−β,

where rn = dist(Vn, ∂D2). This requires us to develop a specific

construction of the map ϕ3 which guarantees that the map fM isan area preserving diffeomorphism of class C 1+β for some β > 0.


The Embedding Theorem

Given a smooth compact connected oriented surface M, for any19 < α < 1

4 there exist β = β(α), 0 < β < 2 + κ, and a continuous

map ϕ3 : D2 → M such that

1 the restriction ϕ3|int D2 is a diffeomorphic embedding;

2 ϕ3(D2) = M;

3 ϕ3 preserves area; more precisely, (ϕ3)∗mD2 = mM where mM

is the area in M; moreover, mM(M \ ϕ3(int D2)) = 0;

4 the map fM := ϕ3 ◦ fD2 ◦ ϕ−13 is a C 1+β area preserving

diffeomorphism of the surface.


Completing the Proof of the Main theorem

The map fT 2 is a Young diffeomorphism and has a collection ofs-subsets Pn,i as well as the return time τ . DefinePn,i ,M := ϕ3(ϕ2(ϕ1(Pn,i ))) and the return time τM(x) = τ(y)where x = ϕ3(ϕ2(ϕ1(y))), y ∈ T 2. This represents fM as a Youngdiffeomorphism.By the above two lemmas

C1

nγ−1< m({x : τ(x) > n}) < C2

nγ′−1.


To establish lower and upper bounds on the decay of correlationswe need the following result by Gouezel, Sarig, Shahidi andZelerowicz.

Theorem. Assume that (M,m, f ) is a Young diffeomorphism forwhich the greatest common denominator of numbers {τi},gcd{τi} = 1 and for which m(τ > n) = O( 1

nν ) for some ν > 1.Assume also that for some C > 0 and all x , y ∈ Pn,i ,

d(f j(x), f j(y)) ≤ C max{d(x , y), d(f τi (x), f τi (y))}.

Then for any σ > 0 and h1, h2 ∈ C ρ(M):


1 Corn(h1, h2) = O( 1nν−1 ).

2 There exists a nested sequence of sets M1 ⊂ M2 · · · ⊂ M suchthat if h1, h2 are supported in Mk for some k > 0 then

Corn(h1, h2) =∞∑

n>N

m({x : τ(x) > N})∫Mh1 dm

∫Mh2 dm+rν(n),

where rν(n) = O(Rν(n)) and

Rν(n) =

1nν if ν > 2,log nn2 if ν = 2,

1n2ν−2 if 1 < ν < 2.


Documents

The Smooth Realization Problem: Area Preserving Di ...smooth observables. 2. The proof of the Main theorem is based on the work of Katok (1979) but requires some new ideas. I believe