Pesin Entropy Formula for Diffeomorphisms with Dominated Splitting
For any diffeomorphism with dominated splitting we consider a nonempty set of invariant measures which describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating subbundle. As a consequence, if those exponents are non negative, and if the exponents on the dominated subbundle are non positive, those measures satisfy the Pesin Entropy Formula.
As pointed out by [P84, BCS13] and other authors, there is a gap between the and the Pesin Theory. To find new results that hold for maps relatively recent research started assuming some uniformly dominated conditions (see [ABC11, BCS13, ST10, ST12, T02]).
Let us consider , where is a compact and connected Riemannian manifold of finite dimension. We denote by the set of all Borel probability measures endowed with the weak topology, and by the set of -invariant probabilities. We denote by a normalized Lebesgue measure, i.e. . For any , the orbit of is regular for -a.e. (see for instance [BP07, Theorem 5.4.1]). We denote the Lyapunov exponents of the orbit of by
Theorem (Ruelle’s Inequality) [R78]
For all and for all
where denotes the metric theoretical entropy of .
Let and . We say that satisfies the Pesin Entropy Formula, and write , if
We denote by the Lebesgue measure along the unstable manifolds of the regular points for which positive Lyapunov exponents and unstable manifolds exist. We denote the (zero dimensional) unstable manifold of by , and in this case we have . For any invariant measure for which local unstable manifolds exist -a.e. we denote by the conditional measures of along the unstable manifolds, after applying the local Rohlin decomposition [R62].
The following are well known results of the Pesin Theory under the hypothesis :
Still in the -scenario, non uniformly partially hyperbolic diffeomorphisms possess invariant measures such that ; and hence (see for instance [BDV05, Theorem 11.16]).
The general purpose of this paper is to look for adequate reformulations of some of the above results which hold for all . That is, we would like to know when an invariant measure under satisfies Pesin Entropy Formula.
We first recall some definitions and previous results taken from [CE11].
Fix . The sequence of empirical probabilities of is , where
The -limit of is
We say that describes the asymptotic statistics of the orbit of .
(Basins of statistical attraction)
For any given the basin of (strong) statistical attraction of is
Consider a metric in that induces the weak topology and denote it by .
The basin of -weak statistical attraction of is
(SRB, physical and SRB-like measures)
An invariant probability measure is called SRB (and we denote SRB) if the local unstable manifolds exist - a.e. and .
The probability measure is called physical if
If then any hyperbolic ergodic SRB measure is physical. Nevertheless, if , the definition of SRB measure may not be meaningful since there may not exist local unstable manifolds ([P84, BCS13]). However, it still makes sense to define when a measure is physical.
In the -scenario, we call a probability measure SRB-like or pseudo-physical (and we denote SRB-like) if for all
It is standard to check that the set of SRB-like measures is independent of the metric chosen in and that it is contained in .
(Minimal description of the asymptotic statistics of the system)
Given , we say that a weak-compact set describes the asymptotic statistics of Lebesgue-almost all orbits of if for Lebesgue-almost all .
Theorems 1.3 and 1.5 of [CE11] prove that, for any continuous map the set of SRB-like measures is nonempty, it contains for Lebesgue-almost all , and it is the minimal weak-compact set such that for Lebesgue-almost all . Therefore, the set of SRB-like measures minimally describes the asymptotic statistics of Lebesgue-almost all orbits.
Our focus is to find relations, for diffeomorphisms, between:
Physical measures and, more generally, SRB-like measures.
Invariant measures such that .
Several interesting results were already obtained for . First, in [T02] Tahzibi proved the Pesin Entropy Formula for -generic area preserving diffeomorphisms on surfaces. More recently, Qiu [Q11] proved that if is a transitive Anosov, then -generically there exists a unique satisfying Pesin Entropy Formula. Moreover is physical and mutually singular with respect to Lebesgue (cf. [AB06]). Finally, we cite:
Sun-Tian Theorem [ST12]: If has an invariant measure , and if there exists a dominated splitting -a.e. such that , then .
To prove this theorem Sun and Tian use an approach introduced by Mañé [M81]. In that approach he gave a new proof of Pesin Entropy Formula for and hyperbolic . Mañé’s proof does not directly require the absolute continuity of the invariant foliations. So, it is reasonable to expect that it is adaptable to the -scenario.
We reformulate the technique of Mañé [M81] to obtain an exact lower bound of the entropy for non necessarily conservative , provided that there exists a dominated splitting.
Let be a diffeomorphism on a compact Riemannian manifold. Let be a continuous and -invariant splitting such that . We call a dominated splitting if there exist and such that
We will prove the following results:
Let with a dominated splitting . Let be an SRB-like measure for . Then:
Under the hypothesis of Theorem 1, if , then satisfies the Pesin Entropy Formula.
The proof of Corollary 2 is immediate: inequality (1) and Ruelle’s Inequality imply that satisfies Pesin Entropy Formula. Moreover, as said in Remark 1.5, the set of SRB-like measures is nonempty. So, under the hypothesis of Corollary 2, there are invariant measures that satisfy the Pesin Entropy Formula. Besides, they minimally describe the asymptotic statistics of Lebesgue-almost all orbits.
The same arguments of the proof of Theorem 1 also work under hypothesis that are more general than the global dominated splitting assumption. In fact, if is an invariant and compact topological attractor, and if is a compact neighborhood with dominated splitting , then the same statements and proofs of Theorem 1 and Corollary 2 hold for .
Now, let us pose an example for which Theorem 1 and Corollary 2 do not hold. Consider the simple eight-figure diffeomorphism in [BP07, Figure 10.1]. In this example, the Dirac-delta measure supported on a fixed hyperbolic point is physical. Thus is SRB-like. Besides, there exists a dominated splitting -a.e. because is hyperbolic. Nevertheless, inequality (1) does not hold because and the Lyapunov exponent along the unstable subspace of is strictly positive. So, the presence of a dominated splitting just -a.e. is not enough to obtain Theorem 1.
The following question arises from the statements of our results: Does the SRB-like property characterize all the measures that satisfy Pesin Entropy Formula? The answer is negative. In fact, the converse statement of Corollary 2 is false. As a counter-example consider a non transitive uniformly hyperbolic attractor, with a finite set () of distinct SRB ergodic measures (hence each is physical) such that statistically attracts Lebesgue-almost every orbit. Therefore, the set of all SRB-like measures coincides with (see Remark 1.5). So, is not an SRB-like measure. After Corollary 2, and satisfy Pesin Entropy Formula. It is well known that any convex combination of measures that satisfy Pesin Entropy Formula also satisfies it (see Theorem 5.3.1 and Lemma 5.2.2. of [K98]). We conclude that satisfies Pesin Formula but it is not SRB-like.
2 Reduction of the proof of Theorem 1
For the diffeomorphism with dominated splitting , we denote:
Consider a metric in the space of all Borel probability measures inducing its weak topology. For all , for all and for all , we denote:
Let with a dominated splitting . There exists a weak metric in , such that for any -invariant probability measure the following inequality holds:
where is the Lebesgue measure.
We note that the term is non negative due to Ruelle’s Inequality. Nevertheless, it is bounded from below by inequality (5), which relates it with the Lebesgue measure .
Let be -invariant. Assume that does not satisfy inequality (1). In other words,
From Proposition 2.1, for all small enough there exists such that
To prove Proposition 2.1, we take from [ST12] the idea of using Mañé’s approach [M81]. Nevertheless, we use this approach in a distinct context (i.e. we do not assume ) and apply different arguments. In [M81] Mañé considers and constructs a foliation , which is not necessarily invariant, but approximates the unstable invariant foliation. On the one hand, the given invariant measure has absolutely continuous conditional measures along the leaves of , because has. On the other hand, the hypothesis allows Mañé to use the Bounded Distortion Lemma. So, he obtained Pesin Entropy Formula after taking convergent to the unstable foliation.
In our case these arguments fail to work, except one. There still exists a (non invariant) foliation whose tangent sub-bundle approximates the dominating sub-bundle . Besides, since is , the conditional measures of (not of ) along the leaves of are absolutely continuous. But we have neither the hypothesis nor the regularity of . Also an invariant foliation to which would converge, may fail to exist. The role of the following Lemmas 2.2 and 2.3 is to overcome these problems. Before stating them, we adopt the following:
Notation. Let be the Borel -algebra on the manifold . We denote by a finite partition of , namely:
for all ,
We write .
For any pair of finite partitions and we denote
(Upper bound of the Lebesgue measure )
For all there exists such that for every finite partition with there exist a sequence of finite measures and a constant such that:
(i) for all , for all .
(ii) The following inequality holds for all and for all
Before stating the second lemma, recall equality (4).
(Lower bound of the metric entropy)
There exists a metric in with the following property:
For all and for all there exist a finite partition satisfying , and a real number such that:
For all , and for any sequence of finite measures such that there exists satisfying for all for all , the following inequality holds:
Apply again Lemma 2.2 to obtain the sequence of finite measures and the constant .
We now apply again Lemma 2.3 to deduce:
Besides, by Lemma 2.2:
Taking we obtain:
Since is arbitrary, we deduce inequality (5), as wanted.
3 Proof of Lemma 2.2
Notation: First take a fixed value of small enough such that is a diffeomorphism from onto its image in for all . Fix . Denote
Denote by (resp. ) the projection of on along (resp. on along ), and .
For any we denote
is a Hadamard graph (or simply “a graph”) if
for all and
is a -diffeomorphism onto its image. (See Figure 1.)
The foliation associated to the graph is the foliation whose leaves are parametrized on , with constant , by the diffeomorphism:
In Figure 1 we draw the foliation . To simplify the notation we omit the exponential map and denote . The leaf containing is denoted by .
The dispersion of the graph is
where and denotes the Fréchet derivative of
with a constant value of .
We denote by the Lebesgue measure along an embedded local submanifold .
For all there exists such that, if , then
For such a value of (depending on ), there exists such that, if , then and
Proof: The assertion follows from the properties that were established in the definition of Hadamard graphs and their associated foliations, and from the definition of dispersion. In particular (11) holds because of the continuous dependence of the splitting on the point .
Iterating the local foliation
Denote by the dynamical ball defined as
Take any graph in such that , and consider its associated local foliation . Construct the image in the dynamical ball , i.e.:
for all such that
(Reformulation of Lemma 4 of [M81])
There exists depending only on , such that for all there are such that for any point , if is the local foliation associated to a graph defined on with
then for all the iterated foliation is contained in the associated foliation of a graph defined on , and
Besides, for all the image is contained in a single leaf of the foliation associated to .
Step 1. Choose such that
For such a fixed value of , take so that for all , for all , and for any graph defined in with , there exists a graph defined on satisfying the following condition:
In Assertion 5.1 of the appendix we show that such exists. We note that the above assertion is true for any initial graph with dispersion smaller than and that Statement (14) a priori only holds if . The assertion that depends only on implies that the image of any point in the leaf associated to the graph , is contained in the leaf of associated to the graph .
Step 2. With fixed as above, there exists such that for any graph with and for all , if is the graph defined in satisfying (14), then:
We prove this statement in Assertion 5.2 of the appendix.
Moreover, if , then
Step 4. From the construction of in Step 1 and using that , we deduce that the graph exists for all . Moreover, for all . So, applying inequality (15) we obtain for all . Finally, applying inductively Assertions (13) and (14) we conclude that the graph exists for all and for all .
Once the constant of Lemma 3.5 is fixed, depending only on , one obtains the following property that allows to move the reference point (used to construct the graph on ), preserving the same associated local foliation and the uniformity of the upper bound of its dispersion:
For all there exists such that, for any and for any graph with defined in , the associated foliation in the neighborhood is also associated to a graph defined in for any . Besides .
Proof: The splitting depends continuously on . Then and also depend continuously on . Therefore, for all there exists such that
(For simplicity in the notation in the above inequalities we omit the derivative of which identifies with .)
We claim that if is small enough then, for any graph defined on , and for any point such that , there exists a graph defined on such that the local foliations associated to and coincide in an open set where both are defined. In fact, should satisfy the following equations:
Since by hypothesis is a graph, it is and
The above equations are solved by
The two equalities in (17) define a local diffeomorphism . In fact, on the one hand , where is a diffeomorphism (which is linear and uniformly near the identity map, independently of the graph ). On the other hand, for constant, the derivative with respect to of is , which is, independently of the graph , uniformly near . Thus, is a local diffeomorphism near the identity map provided that is chosen small enough (independently of the given graph ).
From the above construction we deduce that the composition of the mapping with the mapping defined by (18), is of class. Therefore is dependent on . Besides because . Due to Identity (16), the application defined by coincides with the application .
The diffeomorphism as constructed above, converges to the identity map in the topology, when , and uniformly for all graphs defined in . Thus, by Identity (16), converges uniformly to zero, independently of the given graph , when . This implies, in particular, that converges uniformly to when Thus, for any constant there exists , which is independent of the graph , such that . In other words, for all with , as wanted.
We are ready to prove the following Proposition, for all with a dominated splitting .
For all there are and a finite family of local foliations , each one defined in an open ball of a given finite covering of with -balls, such that:
(a) is - trivializable and its leaves are -dimensional,
(b) for all and for all ,
(c) the following assertion holds for all and for all such that :
(d) the following inequality holds for all and for all :
Proof: Consider the constant determined by Lemma 3.5. For each point construct a local foliation from a graph defined on , with dispersion smaller than a constant such that . The constant will be fixed later taking into account the given value of .
After Lemma 3.6, there exists such that, for all the graph defined on is redefined on , for any point , preserving the same associated foliation and having dispersion upper bounded by . Fix (depending on ) by Lemma 3.5 and such that . For any given finite covering of with balls , fix a finite family of local foliations so constructed, one in each ball of the covering.
By the definition of graph, each foliation of the finite family constructed above, is -trivializable and its leaves have the same dimension as the dominating subbundle . Thus Assertion (a) is proved.
From inequality (11), given (a fixed value of will be determined later), there exists such that, if then