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# On equilibrium states for partially hyperbolic horsehoes

I. Rios Instituto de Matemática e Estatística, Universidade Federal Fluminense  and  J. Siqueira Centro de Matemática da Universidade do Porto
###### Abstract.

We prove existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Hölder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced in [Díaz,L., Horita,V., Rios, I., Sambarino, M., Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes, Ergodic Theory and Dynamical Systems, 2009]. For the original three dimensional system, we consider potentials with small variation, constant on local stable manifolds, obtaining existence and uniqueness of equilibrium states.

## 1. Introduction

The Thermodynamic Formalism, inspired by Statistical Mechanics, is a part of Ergodic Theory that studies the behavior of invariant measures with regular Jacobians called equilibrium states. We formally define equilibrium states with respect to a potential as follows.

###### Definition 1.1.

Consider a continuous map on a compact metric space . We say that a -invariant probability measure is an equilibrium state for w.r.t. a potential if it satisfies

 hμ(G)+∫ϕdμ=supη{hη(G)+∫ϕdη},

where the supremum is taken over all -invariant probability measures.

The theory was introduced by Sinai in the 70s, and since then it has been extensively studied by many authors who have made great advances in the area. In , the author initiated the study of equilibrium states for Anosov diffeomorphisms which would be supplemented by Bowen and Ruelle. See ,  and . They considered a system that consists of an uniformly expanding map and a Hölder continuous potential. For that system they proved that it has an unique equilibrium state. Sinai, Ruelle and Bowen’s strategy was to use a finite Markov partition to build a conjugacy with a Markov shift of finite type. With some adjustments the proof extends to the case of uniformly hyperbolic maps.

In the context of non-uniformly hyperbolic (in dimension bigger than one), we mention the important contributions made by Sarig (see  and ), Buzzi and Sarig , who studied countable Markov shifts, Buzzi, Paccaut and Schmitt , who studied piecewise expanding maps, Oliveira and Viana , and Varandas and Viana , who considered non-uniformly expanding maps, among others.

More specifically, in , the authors proved existence and uniqueness of equilibrium states for a family of non-uniformly expanding maps associated with a class of Hölder continuous potentials with small variation. In their work, they considered finite and not generating Markov partitions. Some rectangles are uniformly expanding and others could have some contractive behavior. They used the notion of hyperbolic times introduced by Alves, Bonatti and Viana in  to construct special Gibbs-type measures.

Leplaideur, Oliveira and Rios  studied the existence and properties of equilibrium states, for a family of partially hyperbolic horseshoes introduced by Diaz et alii in . The three dimensional horseshoes were proved to be simultaneously at the boundary of the set of uniformly hyperbolic systems and the set of persistently non hyperbolic ones (see ). In , the authors studied the ergodic measures with support in the limit set, showing that there is a spectral gap on the set of central Lyapunov exponents. As a consequence, there is a phase transition for the potential , where is the Jacobian of the system restricted to the (one dimensional) central direction and . In fact, for a special value , there are two mutually singular equilibrium states. For , the equilibrium state is unique, and it is a Dirac measure. They proved the existence of equilibrium states for every (for every continuous potential, in fact), but the question of uniqueness for was left open. In , the authors explored geometric and ergodic properties of a class of horseshoes with similar dynamical properties.

Still in the context of the partially hyperbolic horseshoes introduced in , Arbieto and Prudente  proved that, for Hölder continuous potentials depending only on the unstable direction (thus constant on the center-stable one), the associated equilibrium state is unique.

Here we consider the same family of three dimensional horseshoes, taking the inverse map. We start by defining a two dimensional abstract map, obtained by projecting the system on two center-unstable leaves (center-stable leaves for the original map). We prove existence and uniqueness of equilibrium states, associated to Hölder continuous potentials with small variation, where, by small, we mean smaller than a fixed constant explicitly given. This part of the work is strongly inspired in the work of Oliveira and Viana , but there are many extra difficulties to circumvent in our case: the degree of the map is not constant, and the number of expanding rectangles of the Markov partition is too small.

Back to the three dimensional problem, we address the question of uniqueness of equilibrium states for a special class of Hölder continuous potentials, that depend only on the center-unstable direction (complementing the direction considered in ), and whose variation has the same bound as before.

This paper is organized as follows. In section 2, we introduce the family of horseshoes constructed in the paper  and the projected maps . We also state the main results of our work. The first main result (Theorem A) stablishes the class of potentials we deal with, and ensures the uniqueness of equilibrium states for the family . Theorem B has intrinsic interest, and will be used for the proof of Theorem A. It guarantees the existence of an eigenfunction and eigenmeasure for the Ruelle-Perron-Frobenius transfer operator and its dual, respectively, both associated with the same eigenvalue that is the spectral radius of the operator. Theorem C gives an answer to the three dimensional problem for a family of potentials varying only on the center-unstable direction.

In section 3 we prove Theorem B. We establish the existence of a reference measure and prove the abundance of hyperbolic times for almost every point with respect to it. We find an eigenfunction for the Transfer Operator and state some of its properties. In sections 4, and 5 we prove the existence and uniqueness of the equilibrium state, which is a non-lacunary Gibbs measure. Finally, in section 6, we prove Theorem C. We use the equilibrium state for the map to construct an equilibrium state supported on the three dimensional horseshoe, and prove the uniqueness.

## 2. Main results

Consider and let

 ~R0=[0,1]×[0,1]×[0,1/6]and~R1=[0,1]×[0,1]×[5/6,1].

Define

 F0(x,y,z):=F|~R0(x,y,z)=(ρx,f(y),βz),

where , and

We also define

 F1(x,y,z)=F|~R1(x,y,z)=(34−ρx,σ(1−y),β1(z−56)),

where e .

For , we define

 (1) F(X)={F0(X)ifX∈~R0F1(X)ifX∈~R1.

Points outside will be mapped injectively outside . Notice that depends on the parameters and .

In , Díaz et al showed that the non-wandering set for is partially hyperbolic, with one dimensional central direction, parallel to the -axis. The vertical direction is expanding and the horizontal direction, parallel to the -axis is contracting. The -axis direction has both behaviours.

Now we define the map , which is related to the projection of in two center-stable planes. We consider an abstract space consisting by three rectangles , and defined as follows:

 R1=[0,ρ]×[0,1]×{0},R2=[3/4−ρ,3/4]×[0,σ]×{0},R3=[0,ρ]×[2,3]×{5/6}.

The rectangles are inside two planes called and as in figure 3. Let , and let be defined by . See their graphs in figure 4.

Let . We define a map by its restrictions to each rectangle as follows:

 G1(x,y,z) = (αx,g0(y),0), G2(x,y,z) = (α(3/4−x),g1(y),5/6), G3(x,y,z) = (αx,g0(y),0).

The behaviour of the map is similar on and . In these rectangles, the vertical direction is expanding. In the horizontal direction sends the points from the right side to the left, except for the extreme points whose coordinates are fixed. In the rectangle both directions are expanding.

We consider the maximal invariant set of the union of the rectangles:

 Λ:=⋂n∈NG−n(3⋃i=1Ri),

and from now on we call the restriction .

Let be the matrix:

 A=⎛⎜⎝110001110⎞⎟⎠.

and consider the finite type subshift with transition matrix :

 ΣA={Θ=(θ0θ1θ2⋯)∈{1,2,3}N|Aθiθi+1=1}.

According to this the following transitions are allowed:

 1→1,2 2→3 3→1,2.

The map is semi-conjugated to the shift on . This result is an adaptation of a result in . [See Lemma 5.2]

Note that the points belonging to the rectangles 1 and 2 have two pre-images while points in the rectangle 3 have just one pre-image.

In figure 7, the steps in the generation of the set are depicted. Note that is not a Cantor set: it contains infinitely many lines segments. It is worth to evidence that is not conjugated to the shift . The entire segment is associated to a unique sequence on : the constant sequence .

The topological entropy of the subshift is given by:

 htop(σ)=log(1+√52).

By the semiconjugacy we have . Let . We are ready to formulate the first main result of this paper.

###### Theorem A.

Let be as above and be a Hölder continuous potential satisfying:

 (2) supϕ−infϕ

Then there exists a unique equilibrium state for the system with respect to the potential .

We point out that our class of potentials includes the zero potential, constant potentials and potentials similar to the ones studied in  with close to zero.

### 2.1. Ruelle-Perron-Fröbenius Operator

Let be the set of real continuous functions on . For a Hölder continuous potential we define an operator , called the Ruelle-Perron-Fröbenius operator which associates for each a function by:

 (3) Lϕψ(x)=∑y∈G−1(x)eϕ(y)ψ(y).

For each we consider the Birkhoff sum . Note that, iterating the operator we obtain, for each ,

 (4) Lnϕψ(x)=∑y∈G−n(x)eSnϕ(y)ψ(y).

We also consider the dual operator of , that satisfies

 ∫ψ dL∗ϕη=∫Lϕ(ψ) dη,

for every and every . Now we can stablish the second main theorem of this paper. Consider as in the statement of the Theorem A.

###### Theorem B.

Let be the spectral radius of the transfer operator . There exists a probability measure and there exists a positive function wich satisfies:

 L∗ϕν=λνeLϕh=λh.
###### Theorem C.

Let be the three dimensional partially hyperbolic horseshoe defined in the begin of this section. Let be a Hölder continuous potential with . Assume that does not depend on the -coordinate in each set and . Then there exists a unique equilibrium state for the system with respect to the potential .

## 3. Proof of Theorem B

We consider the cone of continuous non-negative functions on . Then is a normal cone which gives a partial order on . Since is a positive and bounded operator, it follows from Mazur’s Theorem [see ] that:

###### Proposition 3.1.

The spectral radius of , , is an eigenvalue for the dual operator associated an eigenmeasure .

We can stablish a lower bound to the eigenvalue of .

###### Lemma 3.1.

The spectral radius of , , sastisfies

 λ≥einfϕ+logω,where ω=(1+√52).
###### Proof.

By definition Therefore:

 ∥Lnϕ1(x)∥=∑y∈G−n(x)eSnϕ(y)≥#{G−n(x)}⋅eninfϕ.

Then, we have

 λ≥limsup[#{G−n(x)}⋅eninfϕ]1n=einfϕelogω=einfϕ+logω

since . ∎

### 3.1. Hyperbolic times

Given and symbols we define the cylinder set of length by:

 Rn(a0,⋯,an−1)={x∈Λ|x∈Ra0,G(x)∈Ra1,⋯,Gn−1(x)∈Ran−1}.

When there is no risk of confusion, we denote to refer to .

For each and let be the set of admissible words of length such that the proportion of symbols and are at least . That is

 #{0≤j≤n−1|aj=1,3}>γn.

We can associate words in with orbits, or more precisely, with cylinders sets. The points in these cylinders spend a big fraction of their orbits in the "non-hyperbolic" part of the system. We will study the exponential growth of the cardinality of these sets . We can estimate the cardinality by counting the number of admissible words of length whithin at most symbols .

By definition, to compose a word with symbols we have to follow the conditions below:

• a symbol two is followed by a three.

• a symbol is followed by or .

• a symbol is followed by or .

It means that once we have distributed the symbols the other symbols, except possibly for the first one, are already determined. Following this rules the number of words of length with symbols is given by If the first symbol in the word is not a we have two choices: or . According to this we have the follow estimate:

 (5) #I(γ,n)≤(1−γ)n∑k=02(n−kk).
###### Lemma 3.2.

For each , and we have:

 (n−kk) ≤ ek√2πk⋅e112k+1⋅(nk−1)k.
###### Proof.

In fact,

 (n−kk) = (n−k)!k!(n−2k)=(n−k)(n−k−1)⋯(n−k−(k+1))k! ≤ (n−k)kk!.

By Stirling’s Formula, see , we have:

Then

 (n−kk) ≤ (n−k)k√2πk(ke)ke112k+1 = ek√2πk⋅e112k+1⋅(nk−1)k,

which finishes the proof of the lemma. ∎

###### Lemma 3.3.

For large enough and we have:

 (n−kk)<(n−(k+1)k+1)for all k≤[αn].
###### Proof.

In fact, we have that the inequality

 (n−kk)=(n−k)!k!⋅(n−2k)!<(n−(k+1))!(k+1)!(n−2(k+1))!= = (n−(k+1)k+1)

holds, if and only if:

 (n−k)⋅(k+1)(n−2k)⋅(n−2k−1)<1.

If we have , then the previous inequality is equivalent to the following:

 −5k2+(5n−3)k+(−n2+2n)<0.

Which is always true. Then it is enough to verify conditions to give us . If we make we have:

 n2−4kn≥n2−4αn2−n>4α2n2+2αn≥4k2+2k,

where the first and the last inequalities always hold.To get the second inequality, it is enough to prove that:

 n2(−4α−4α2+1)−n(1−2α)>0.

Taking , the coefficient of is positive, and the inequality is true for large enough. ∎

Let and consider . Then

 #I(γ,n)≤2[αn](n−[αn][αn]).

Applying the lemma 3.2 we obtain:

 #I(γ,n) ≤ αn⋅eαn√2παn⋅e112αn+1⋅(nαn−1)αn ≤ αn⋅eαn√2παn⋅e112αn+1⋅[(1−αα)α]n.

Note that, if we make , we have and hence converges to one. Then we can establish the following proposition.

###### Proposition 3.2.

If then there exists which satisfies:

 #I(γ,n)≤eβnfor n large enough.
###### Proof.

We consider

 f(α,n) := 1nln⎛⎝αneαn√2παne112αn+1(1−αα)αn⎞⎠= = 1n(lnα+lnn2−112αn+1−ln(2π)2)+ + α+αln(1−α)−αlnα.

Taking the limit in we have

 h(α):=limn→∞f(α,n)=α+αln(1−α)−αlnα=α+αln(1−αα).

Since and , there exists with since . We have that converges to uniformly on any compact subset of . Then, for sufficiently large,

Thus and, therefore, there exists satisfying the lemma. ∎

Now we define a key tool in this paper: the concept of hyperbolic times, which was introduced by Alves in  and extensively studied in  by Alves, Bonatti and Viana.

###### Definition 3.1.

We say that is a (G,c)-hyperbolic time for , for some if

 n−1∏j=k∥∥DG(Gj(x))−1∥∥−1≥e2c(n−k),0≤k≤n−1.

If satisfies

 (6) σ−(1−γ)⋅(1e)γ>1,

there is such that

 (7) σ−(1−γ)⋅(1e)γ>e4c>1.

For that it is enough to consider .

From now on we will fix as in equation 7 and we will omit and in the notation, referring to a hyperbolic time, instead of -hyperbolic time.

In our setting we can prove that good admissible words are associated to cylinder sets with positive proportion of hyperbolic times. For this, we use the well known Pliss Lemma (see ).

###### Proposition 3.3.

There exists such that for every and every admissible word there is and numbers such that is hyperbolic for for all .

###### Proof.

Given , consider . Since we have in each rectangle the following:

 ∥∥DG(x)−1∥∥ = σif x∈R2 ∥∥DG(x)−1∥∥ ≤ eif x∈R1 ou R3,

we can estimate by

 bi≥⎧⎨⎩log(1e)=−1ifi=1,3log(σ−1)ifi=2.

Defining where , and since belongs to the complement of , we have

 a1+⋯+an−1 = n−1∑t=0bit≥(1−γ)nlog(σ−1)+γn(−1) > 4cn

where the last inequality follows from the choice of (see equation 7).

Let . Then for every . We consider and and then by the Pliss Lemma, there exist with and integer numbers for which the folllowing inequality holds:

 ak+⋯+ani≥2c(ni−k)  ∀ 0≤k≤ni−1  i=1,⋯,l.

Therefore for each we have:

 log⎛⎝ni−1∏j=k∥∥DG(Gj(x)−1)∥∥−1⎞⎠ = ni−1∑j=klog∥∥DG(Gj(x)−1)∥∥−1 ≥ ni−1∑j=kaj≥2c(ni−k).

Thus, is a hyperbolic time for if . ∎

###### Remark 3.1.

An useful property of hyperbolic times was given in . Roughly they proved that, if is a hyperbolic time for there exists a neighborhood of the -th iterate of for which one every inverse branch is exponentially contracting. Moreover, the radius of the neighborhood depends only on and on the map.

We will use this property to prove a type of distortion condition for the Birkhoff sums. For the precise statement and the proof see .

###### Proposition 3.4.

There exists a constant such that if belong to a hyperbolic cylinder , the following holds:

 |Snϕ(x)−Snϕ(y)|≤Kd(x,y)δ≤K0.
###### Proof.

Assume is Hölder continuous with constants . Then we have:

 |Snϕ(x)−Snϕ(y)| = |n∑j=1ϕ(Gn−j(x))−ϕ(Gn−j(y))| ≤ n∑j=1|ϕ(Gn−j(x))−ϕ(Gn−j(y))| ≤ Cn∑j=1d(Gn−j(x),Gn−j(y))δ.

Also, since and belong to the same hyperbolic cylinder, according to the Remark 3.1 for each we have:

 d(Gn−j(x),Gn−j(y))δ≤e−jc.

Then, there is a constant which depends only on and which satisfies:

 |Snϕ(x)−Snϕ(y)|≤K0.

### 3.2. Jacobians and conformal measures

###### Definition 3.2.

Let be a probability measure on . We say that a function is a Jacobian of with respect to if it satisfies:

 η(G(A))=∫AJηGdη,

for all mensurable set such that is injective.

In other words, given the measure , we consider by:

 G∗η(B)=η(G(B)) for all measurable subsetBof Λ.

The Radon-Nikodym theorem ensures that the Jacobian is essentially unique. Further, it coincides with the Radon-Nikodym derivative of the measure with respect to , . We will get an useful property regarding the Jacobian with respect to an iterated of .

###### Proposition 3.5.

If the Jacobian of with respect to , , exists, then for each the map also has a Jacobian and it is given by:

 JμGn=n−1∏j=0JμG∘Gj.
###### Proof.

For each function we define the support of by .

For topological spaces, one can establish another definition of Jacobian which is equivalent to the one given in the definition 3.2. We say that is a Jacobian of with respect to , if for any open set wich satisfying that is injective, and for all such that , the following holds:

 ∫G(U)ψ dμ=∫U(ψ∘G)JμG dμ.

Using this definition we can prove the proposition. We treat the case since the general case follows by induction. Given an open set suppose that is injective and is such that . We have to show that . In fact, we have:

 ∫G2(U)ψ dμ = ∫G(U)(ψ∘G)JμG dμ = ∫U[[(ψ∘G)⋅JμG]∘G]⋅JμG dμ = ∫U[(ψ∘G2)⋅(JμG∘G)]⋅JμG dμ.

In the next lemma we compute the Jacobian of the dual operator eigenmeasure .

###### Proposition 3.6.

If the spectral radio of , , is an eigenvalue associated with an eigenmeasure then we have:

 JνG=λe−ϕ.

(In this case, is said to be a conformal measure.)

###### Proof.

Given which satisfies injective, we have to show that . We have:

 ∫Aλe−ϕ dν = ∫e−ϕ⋅χA dλν = ∫e−ϕ⋅χA dL∗ϕν = ∫Lϕ(e−ϕ⋅χA) dν.

By the injectivity of in we get:

 Lϕ(e−ϕ⋅χA)(x) = = ∑y∈G−1(x)χA(y)=χG(A)(x)
 ∫Aλe−ϕdν=∫χG(A)(x)dν=ν(G(A)),

wich finishes the proof. ∎

Thus, combining Proposition 3.6 and Proposition 3.5 we obtain:

###### Lemma 3.4.

There exists satisfying:

 K−11≤JνGn(x)JνGn(y)≤K1,

for all , where is a hyperbolic cylinder.

###### Proof.

Given , the Proposition 3.5 implies that we can calculate the Jacobian of with respect to using the Jacobian of with respect to . Then, according to Proposition 3.6 for each :

 (8) JνGn(x)=λne−Snϕ(x).

Consider a hyperbolic cylinder of length and . By the Proposition 3.4:

 ∣∣ ∣∣log(JνGn(x)JνGn(y))∣∣ ∣∣=|Snϕ(x)−Snϕ(y)|≤K0.

To finish the proof, consider . ∎

Given , let be the collection of cylinders generated by words .

Let be the collection of cylinders generated by words of length belonging to .

###### Proposition 3.7.

The measure of the cylinders in , with respect to , decreases exponentially when goes to infinity. Therefore, almost every point belongs to elements of for infinitely many values of .

###### Proof.

Let and take a cylinder. Using (8), we have:

 ν(Gn(Rn)) = ∫RnJνGn dν = ∫Rnλne−Snϕ dν ≥ λne−nsupϕν(Rn).

Whereby it implies . If we take the sum over all cylinders generated by words and use the Proposition 3.2, we obtain:

 ∑I(γ,n)ν(Rn)≤λ−nesupϕ+β.

On the other hand, Lemma 3.1 gives us . Since , we have the result. ∎

We use the notation to refer to hyperbolic cylinders of length . In the Proposition 3.3 we obtained plenty of hyperbolic times associated to words which do not belong to the set .

Let be the closure of the set of points which belong to infinitely many hyperbolic cylinders . Next, we prove that is a full measure set with respect to .

###### Proposition 3.8.

Let be the first hyperbolic time of . We have that is integrable and .

###### Proof.

By the Proposition 3.3, for each we have . Therefore, if is such that we have . Then:

 ∫n1dν=∞∑n=0ν({x:n1(x)>n})≤1+∞∑n=1ν(Pn).

According to the Proposition 3.7, the measure of the sets in decreases exponentially with , which implies that the sum in the last inequality is finite. Then, the first hyperbolic time is integrable with respect to . Since consists of cylinders generated by words in the complement of , the Proposition 3.3 guarantees that every point belongs to at least hyperbolic cylinders, where . Besides that, by the Proposition 3.7, for large enough, almost every point belongs to a set in . Thus, every point in belongs to a infinitely many hyperbolic cylinders, then, . ∎

We consider for each , the -th hyperbolic time of . We also consider as in the Proposition 3.3. According to the proof of Proposition 3.8 we have the following corollary:

###### Corollary 3.1.

For every point and large enough the following holds:

 #{k≥1:nk(x)≤n}≥θn.

In the next proposition we estimate the measure of hyperbolic cylinders with respect to . This property will be useful later.

###### Proposition 3.9.

There exists a constant such that for every , a hyperbolic cylinder of length , and every , we have:

 K−12≤ν(Rn)exp(Snϕ(x)−Pn)≤K2,

where

###### Proof.

Let , and . By the Proposition 3.1 we have

 (L∗ϕ)nν=λnν∀n∈N.

And then, we obtain

 ePnν(Rn) = ePn∫χRndν = ePnλn∫Lnϕ(χRn)dν = ∫Lnϕ(χRn)dν,

where denotes the characteristic function on and . According to the transfer operator definition we can write:

 LnϕχRn(x)=∑y∈G−n(x)eSnϕ(y)χRn(y)=∑y∈G−n(x)∩RneSnϕ(y).

Therefore, given for each there exists a unique element . We define and then:

 ∣∣∣ν(Rn)exp(Snϕ(x)−Pn)∣∣∣=∣∣ ∣∣∫A(n)eSnϕdνeSnϕ(x)∣∣ ∣∣.

By the Proposition 3.4 there exists a constant satisfying:

 ∣Snϕ(x)−Snϕ(y)∣

for all x, y that belong to the same hyperbolic cylinder, which is enough to finish the proof. ∎

### 3.3. Transfer Operator Eigenfunction

In this section we will show the existence of an eigenfunction to the transfer operator of Ruelle-Perron-Fröbenius. We will, therefore, show the second part of the Theorem B and finish its proof. Besides that, we will show that any eigenfunction is positive and Hölder continuous.

###### Theorem 3.1.

There exists an eigenfunction associated to the operator . Moreover, it is positive and associated to the eigenvalue :

 Lϕh=λh.

We define, for each , the function by:

 Tn(x)=∑y∈G−n(x)∩RnRn
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