Equilibrium States for Random Nonuniformly Expanding Maps
Abstract.
We show that, for a robust (open) class of random nonuniformly expanding maps, there exists equilibrium states for a large class of potentials.In particular, these sytems have measures of maximal entropy. These results also give a partial answer to a question posed by LiuZhao. The proof of the main result uses an extension of techniques in recent works by AlvesAraújo, AlvesBonattiViana and Oliveira.
1. Introduction
Particles systems, as they appear in kinetic theory of gases, have been an important model motivating much development in the field of Dynamical Sytems and Ergodic Theory. While these are deterministic systems, ruled by Hamiltonian dynamics, the evolution law is too complicated, given the huge number of particles involved. Instead, one uses a stochastic approach to such systems.
More generally, ideais from statistical mechanics have been brought to the setting of dynamical systems, both discretetime and continuoustime, by Sinai, Ruelle, Bowen, leading to a beautiful and very complete theory of equilibrium states for uniformly hyperbolic diffeomorphisms and flows. In a few words, equilibrium states are invariant probabilities in the phase space which maximaze a certain variational principle (corresponding to the Gibbs free energy in the statistical mechanics context). The theory of SinaiRuelleBowen gives that for uniformly hyperbolic systems equilibrium states exist, and they are unique if the system is topologically transitive and the potential is Hölder continuous.
Several authors have worked on extending this theory beyond the uniformly hyperbolic case. See e.g. [5], [13], among other important authors. Our present work is more directly motivated by the results of Oliveira [12] where he constructed equilibrium states associated to potentials with nottoolarge variation, for a robust (open) class of nonuniformly expanding maps introduced by AlvesBonattiViana [2].
On the other hand, corresponding problems have been studied also in the context of the theory of random maps, which was much developed by Kifer [6] and Arnold [3], among other mathematicians. Indeed, Kifer [6] proved the existence of equilibrium states for random uniformly exapnding systems, and Liu [8] extended this to uniformly hyperbolic systems.
In the present work, we combine these two approaches to give a construction of equilibrium states for nonuniformly hyperbolic maps. In fact, some attempts to show the existence of equilibrium states beyond uniform hyperbolicity were made by KhaninKifer [7]. However, our point of view is quite different. Before stating the main result, we recall that a random map is a continuous map where is a compact manifold is a Polish space, and a measurably invertible continuous map with an invariant ergodic measure . The main result is the following :
“For a open set of nonuniformly expanding local diffeomorphisms, potentials with low variation and , there are equilibrium states for the random system associated to and . In particular, admits measures with maximal entropy.”
A potential has low variation if it is not far from being constant. See the precise definition in section . In particular, constant functions have low variation; their equilibrium states are measures of maximal entropy.
The proof, which we present in the next sections extends ideias from AlvesAraújo [1], AlvesBonattiViana [2] and Oliveira [12].
It is very natural to ask whether these equilibrium states we construct are unique and whether they are (weak) Gibbs states, Another very interesting question is whether existence (and uniqueness) of equilibrium states extends to (random r deterministic) nonuniformly hyperbolic maps with singularities, such as the Viana maps [1]. Although our present methods do not solve these questions, we believe the answers are affirmative.
2. Definitions
Random Transformations and Invariant Measures
Let be a compact dimensional Riemannian manifold and the space of local diffeomorphisms of . Let a measure preserving system, where is invariant ( is a Borel measure) and is a Polish space, i.e., is a complete separable metric space. By a random transformation we understand a continuous map . Then we define:
(1) 
We also define the skewproduct generating by :
We denote the space of probability measures on such that the marginal of on is . Let be the measures which are invariant.
Because is compact, invariant measures always exists and the property of be the marginal on of a invariant measures can be characterized by its disintegration:
An invariant measure is called ergodic if is ergodic, the set of all ergodic measures is denoted by . Furthermore, each invariant measure can be decomposed into its ergodic components by integration when the algebra on is countably generated and is ergodic.
In what follows, as usual, we always assume is a Lebesgue space, is ergodic and is measurably invertible and continuous. Observe that these assumptions are satified in the canonical case of leftshift operators , being or .
Entropy
We follow Liu [9] on the definition of the KolmogorovSinai entropy for random transformations:
Let an invariant measure like above. Let a finite Borel partition of . We set:
(2) 
where (and ), for a finite partition and a probability on (and are the sample measures of ).
Definition 2.1.
The entropy of is:
with the supremum taken over all finite Borel partitions of .
Definition 2.2.
The topological entropy of is
Theorem 2.3 (“Random” KolmogorovSinai theorem).
If is the Borel algebra of and is a generating partition of , i.e.,
then
Equilibrium States
Let the set of all families such that the map is a measurable map and .
For a , and , we define:
where .
Definition 2.4.
The map given by:
is called the pressure map.
It is well know that the variational principle occurs (see [9]):
Theorem 2.5.
If is a Lebesgue space, then for any we have:
(3) 
Remark 1.
If is ergodic then we can take the supremum over the set of ergodic measures.
Definition 2.6.
A measure is an equilibrium state for , if attains the supremum of (3).
Physical Measures
As in the deterministic case, we follow [1] on the definition of physical measure in the context of random transformations :
Definition 2.7.
A measure is a physical measure if for positive Lebesgue measure set of points (called the basin of ),
(4) 
for ae .
3. Statement of the results
Before starting abstract definitions, we comment that in next section, it is showed that there are examples of random transformations satisfying our hypothesis below.
We say that a local diffeomorphism of is in if is in and satisfies, for positive constants , , , , and , , the following properties :

There exists a covering of such that every is injective and

is uniformly expanding at every :

is never too contracting: for every .


is everywhere volumeexpanding: with .
Define

There exists a set containing such that
where and are the infimum and the supremum of on , respectively, and and are the infimum and the supremum of on , respectively.
This kind of transformations was considered by [2], [12], [1], where they construct open sets of such maps.
We will consider a subset such that :

There is a uniform constant s.t. for any and the constants are uniform on ;
From now on, our random transformations will be given by a map , and satisfy the following condition :

admits an ergodic absolutely continuous physical measure (see section 2).
Remark 2.
We will show in the appendix that implies the following property:

There exists some such that the random orbits of Lebesgue almost every point spends at most a fraction of time inside , depending only on , , . I.e., for a.e. and Lebesgue almost every
Then we analyse the existence of an equilibrium state for lowvariation potentials:
Definition 3.1.
A potential has low variation if
(5) 
Remark 3.
We call above a low variation potential because in the deterministic case (i.e., ), if then satisfies (5).
The main result is :
Theorem A. Assume hypotheses (H1), (H2), (H3) hold, with and sufficiently small and assume also conditions (C1), (C2). Then, there exists such that if is a continuous potential with low variation then has some equilibrium state. Moreover, these equilibrium states are hyperbolic measures, with all Lyapunov exponents bigger than some .
4. Examples
In this section we exhibit a open class of diffeomorphism which are contained in . To start the construction, we now follow [12] ipsisliteris and construct examples of ‘deterministic’ nonuniformly expanding maps. After this, we construct the desired random nonuniformly expanding maps in a neighborhood of a fixed diffeomorphism of .
We observe that the class contains an open set of nonuniformly exapanding which are not uniformly exapnding.
We start by considering any Riemannian manifold that supports an expanding map . For simplicity, choose the dimensional torus, and an endomorphism induced from a linear map with eigenvalues Denote by the eigenspace associated to the eigenvalue in .
Since is an expanding map, admits a transitive Markov partition with arbitrary small diameter. We may suppose that is injective for every . Replacing by a iterate if necessary, we may suppose that there exists a fixed point of and, renumbering if necessary, this point is contained in the interior of the rectangle of the Markov partition.
Considering a small neighborhood of we deform inside along the direction . This deformation consists essentially in rescaling the expansion along the invariant manifold associated to by a real function . Let us be more precise:
Considering small, we may identify with a neighborhood of in and with . Without loss of generality, suppose that , where is the ball or radius and center in . Consider a function such for every and for small constants :


for every ;

is close to : ,
Also, we consider a bump function such for every and for every . Suppose that for every . Considering coordinates such that , define by:
Observe that by the definition of and we can extend smoothly to as outside . Now, is not difficult to prove that satisfies the conditions (H1), (H2), (H3) above.
First, we have that Observe that:
Then, since for every , and we have that for every . Moreover, by condition 1, if we choose small in such way that then:
Notice also that This prove that:
Since coincides with outside , we have for every . Together with the above inequality, this proves condition (H1), with .
Choosing small and , for every , condition (H2) is immediate. Indeed, observe that the Jacobian of is given by the formula:
Then, if we choose :
Therefore, we may take
To verify property (H3) for , observe that if we denote by
with then Indeed, since is constant equal to outside we have that , for every . Given close to 0, we may choose close to 0 and satisfying the conditions above in such way that,
If and are the infimum and the supremum of on , respectively,
where . Then, we may take in (H3). If is the infimum of on , , since .
The arguments above show that the hypotheses are satisfied by . Moreover, if we one takes , then is fixed point for , which is not a repeller, since . Therefore, is not a uniformly expanding map.
It is not difficult to see that this construction may be carried out in such way that does not satisfy the expansiveness property: there is a fixed hyperbolic saddle point such that the stable manifold of is contained in the unstable manifold of two other fixed points.
Now, if denotes a small neighborhood of in , and is a continuous map, AlvesAraújo [1] shows that if is such that and is a sequence of measures, then for small there are physical measures for the RDS , . This concludes the construction of examples satisfying .
5. Proof of the theorem A
We now precise the conditions on and . We consider given in condition . By condition , there exists s.t. for any and a.e. holds :
where is such that for some , we have and (), if and are sufficiently small. Now, the constants fixed above allows us to prove good properties for the objects defined below, which are of fundamental interest in the proof of theorem A.
Expansive Measures and Hyperbolic Times
Definition 5.1.
We say that a measure is expanding with exponent if for almost every we have:
Definition 5.2.
We say that is a hyperbolic time for with exponent , if for every :
As in lemma 3.1 of [2], lemma 4.8 of [12] and lemma 2.2 of [1], we have infinity many hyperbolic times for expanding measures. For this we need a lemma due to Pliss (see [2]).
Lemma 5.3.
Let and . Given real numbers satisfying:
there are and such that:
Lemma 5.4.
For every invariant measure which is expanding, there exists a full measure set such that every has infinitely many hyperbolic times with exponent and, in fact, the density of hyperbolic times at infinity is larger than some :

for every

.
Proof.
Let with full measure. For any and large enough, we have:
Now, by we can apply lemma 5.3 with , , and and the statement follows. ∎
Lemma 5.5.
such that for a.e. , if is a hyperbolic time of and then , .
Proof.
By we know that there exists such that for any we have:
In fact, this hold in the orbit of ae. Indeed, let and , then has full measure and the estimate follows. Because , by the estimative above, we have that ae if we take the inverse branch of which sends to (restricted to )) and has derivative with norm less than , then we have . Using the estimate along the orbit (and induction), we have: for any
The statement follows. ∎
Now we define a set of measures where the “bad set” has small measure.
Definition 5.6.
We define the convex set by
Lemma 5.7.
is a compact set.
Proof.
Let . By compacity, we can assume that . Since is open then . This implies compacity. The physical measure given by condition (C2) (see equation (4)) is in , because a.e. random orbit stay at most inside (by ). By definition of physical measure (limit of average of Dirac measures supported on random orbits) and the absolute continuity with respect to the Lebesgue measure, for a.e. holds. In particular, . ∎
We recall that the ergodic decomposition theorem holds for RDS. With this in mind, we distinguish a set :
Definition 5.8.
( is the ergodic decomposition of ).
Lemma 5.9.
Every measure is expanding with exponent :
for a.e. .
Proof.
We assume first that is ergodic. By definition of , we have . But Birkhoff’s Ergodic Theorem applied to says that in the random orbit of a.e. we have:
Now, we use hypothesis (H1): for any and for any , obtaining:
a.e.
Entropy lemmas
Definition 5.10.
Given , we define :
Lemma 5.11.
Suppose that is ergodic and let given by lemma 5.5. Then, for almost every and any ,
Proof.
Let be a partition of in measurable sets with diameter less than . From the above lemma, we get :
Lemma 5.12.
is a generating partition for every .
Proof.
As usual we will write:
where is an element of the partition . By the previous lemma, we know that for a.e. , we have for a.e. Let a measurable set of and . Given and two compact sets such that and . Now if , the previous lemma says that if is big enough then for in a set of measure bigger than . The sets that intersects satisfy:
This end the proof.
∎
Corollary 5.13.
For every ,
We have that the map is upper semicontinuous at measure s.t. for a.e. , . In fact, we have :
But, if for any and a.e. , then the function given by is upper semicontinuous at . Indeed, since we are assuming that is continuous, the same argument in the proof of theorem of [11] shows this result. In particular, because the infimum of a sequence of upper semicontinuous functions is itself upper semicontinuous, this proves the claim.
Lemma 5.14.
All ergodic measures outside have small entropy : there exists such that
Proof.
By the random versions of Oseledet’s theorem and Ruelle’s inequality (see [9]), we have:
where and are the Lyapunov exponents of at and its multiplicity respectively (and are the positive Lyapunov exponents). Furthermore, by hypothesis the measure is ergodic, then these objects are constant a.e. then and . Since we have . By the definitions of , and the above estimates, we have by (C1):
Now the physical measure given by condition (C2) satisfy (by ). The Random Pesin’s formulae gives:
But then . Using that , and (C1) we have:
Then, we can choose such that