A dichotomy for measures of maximal entropy near time-one maps of transitive Anosov flows
We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy, one with a positive central exponent and the other with a negative central exponent.
We establish this dichotomy for certain partially hyperbolic diffeomorphisms isotopic to the identity whenever both of their strong foliations are minimal. Our proof builds on the approach developed by Margulis for Anosov flows where he constructs suitable families of measures on the dynamical foliations.
2010 Mathematics Subject Classification:37C40, 37D30, 37A35, 37D35
In his pioneering work , Margulis studied measures of maximal entropy of geodesic flows in order to count closed geodesics for manifolds with variable negative curvature. More precisely, he constructed a family of measures such that for all the measure is carried by the unstable manifold at , and for all we have
He then built an invariant probability measure which was observed to be a measure of maximal entropy and is now called the Bowen-Margulis measure. It was then proved to be the unique measure of maximal entropy. We refer to Ledrappier  for an introduction.
In this paper, we will extend Margulis’ construction to a class of partially hyperbolic maps and obtain a striking dichotomy.
If is a transitive Anosov flow on a compact manifold , then there is an open set in which contains in its closure such that for any we have the following dichotomy:
either all the measures of maximal entropy have zero central Lyapunov exponents, or
there are exactly two ergodic measures of maximal entropy where one has a positive central exponent and the other has a negative central exponent, and both measures are Bernoulli.
These results are part of a larger program to understand properties of entropy beyond uniform hyperbolicity. In that classical setting, say for a transitive Anosov diffeomorphism, there is a unique measure of maximal entropy (MME). Even though there are a number of significant results beyond the hyperbolic setting [29, 9, 10] there are still many fundamental open questions beyond the uniformly hyperbolic setting. Partially hyperbolic diffeomorphisms with one-dimensional center have been studied as the “next nontrivial class”. A MME always exists in this setting by entropy expansivity (see [12, 15, 26]). Its uniqueness is a more delicate question.
Uniqueness of the MME has been shown for certain systems that are derived from Anosov, a subclass introduced by Mañé, first for specific constructions, then in greater and greater generality [7, 13, 38, 16, 8].
The partially hyperbolic diffeomorphisms with a center foliation into circles form another subclass with a more subtle behavior. Assuming accessibility,  has established the following dichotomy:
either the dynamics is isometric in the center direction and there exists a unique MME which is nonhyperbolic; or
there are multiple hyperbolic MMEs.
Strategy of proof
We introduce a new subclass of partially hyperbolic diffeomorphisms with one-dimensional center which we call flow type. They are isotopic to the identity and the fundamental examples are the perturbations of time-one maps of Anosov flows. Our main result is Theorem 3.9: the above dichotomy holds for partially hyperbolic flow type diffeomorphisms whose strong foliations are both minimal.
The uniqueness of the MME for a given sign of the central exponent (say nonpositive) follows from a variant of Margulis’ approach. Namely, we first build a family of measures on the center-unstable leaves. Then we construct measures on unstable leaves, which we call Margulis -conditionals. This is more difficult for maps than for flows.
We then use the entropy with respect to the unstable foliation as introduced by Ledrappier and Young  and an argument of Ledrappier  to show that, when its central exponent is nonpositive, a measure maximizes the entropy if and only if its disintegration along the unstable leaves is given by the Margulis conditionals.
A Hopf argument shows that if there is a MME with negative central exponent, then any MME with nonpositive central exponent must coincide with it. The symmetry between positive and negative central exponents in the hyperbolic case follows from the one-dimensionality of the central leaves: we associate to any measure with, say negative central exponent, an isomorphic one with nonnegative central exponent.
A hyperbolic ergodic MME is isomorphic to a Bernoulli shift times a circular permutation, according to a general result by Ben Ovadia . The triviality of the permutation follows by considering iterates. This concludes the proof of Theorem 3.9.
Finally, to prove Theorem 1.1 we establish Theorem 3.10, i.e., we find open sets of partially hyperbolic flow type diffeomorphisms with both strong foliations minimal near any time-one map of a transitive Anosov flow. We first show that such time-one maps are robustly flow type partially hyperbolic diffeomorphisms. Then Bonatti and Díaz  provide a perturbation ensuring robust transitivity. Lastly, by a further perturbation following Bonatti, Díaz, and Urès  we can ensure robust minimality of both strong foliations. Theorem 1.1 now follows from Theorem 3.9.
The use of Margulis conditionals
The construction of Margulis has given rise to a large body of work, mainly devoted to the estimation of the number of periodic orbits, sometimes beyond the uniformly hyperbolic setting . We refer to Sharp’s survey in , the long awaited publication of Margulis’ thesis. The works of Hamenstädt  and Hasselblatt  that give a geometric description of the Margulis conditionals are perhaps closer to our concerns.
While this work was being written, we learned that a different but related approach has been developed in . This approach can deal with equilibrium measures (i.e., generalizations of measures of maximal entropy taking into account a weight function) but requires non-expansion along the center. Separately, Jiagang Yang has told us that he also has results on the MMEs for the same type of diffeomorphisms as we consider.
Let us first note that part of our results could be obtained from symbolic dynamics, using generalizations of ideas going back to the classical works of Sinai, Ruelle, and Bowen (see, e.g., [36, 5, 34]). More precisely, existence of at most one MME with, say, positive central exponent can be deduced from [9, Section 1.6] since, in the terminology of this work, our minimality assumption implies that there is a unique homoclinic class of measures with a given sign of the central exponent. However, the dichotomy does not seem to follow from this approach which is blind to nonhyperbolic measures.
Second, one usually expects that results such as ours can be extended to smoothness, for any , and generalized to equilibrium measures with respect to Hölder-continuous potentials (although uniqueness holds for generic potentials ).
Our techniques demand a very strong form of irreducibility and the flow type property is somewhat technical. Hence we ask:
In Theorem 3.9, can one replace minimality of both strong foliations by minimality of just one or by robust transitivity? Can one replace flow type by isotopic to the identity?
In the volume-preserving setting there is a rigidity result . We think that some version of it may hold for MMEs in the dissipative setting.
In the setting of Theorem 3.9, is the hyperbolic case open and dense? When the MME is nonhyperbolic, does this imply that the diffeomorphism is the time one map of a flow? does it at least exclude the existence of hyperbolic periodic points?
Though we will identify the disintegrations of nonhyperbolic MMEs along both strong foliations, their analysis remains incomplete:
Consider a partially hyperbolic diffeomorphism with flow type and with minimality of both strong foliations. Can its disintegration along the center be atomic like in the hyperbolic case? Can there be more than one nonhyperbolic MME? Are nonhyperbolic MMEs Bernoulli?
We prove that the hyperbolic MMEs are Bernoulli, hence strongly mixing. One can try to establish some speed (see  for a related result).
If is a hyperbolic MME for a flow type diffeomorphism with minimality of both strong foliations, does it satisfy exponential decay of correlations for Hölder-continuous functions, i.e., for any Hölder-continuous functions , does there exist a number such that:
For Anosov flows, the topological entropy can obviously be changed by perturbations whereas it is locally constant for Anosov diffeomorphisms. What is the situation for the maps we consider?
Consider a flow type diffeomorphism whose strong foliations are robustly both minimal. Is it true that the volume growth of each strong leaf is equal to the topological entropy? Does the topological entropy have a homological interpretation? Can an arbitrarily small perturbation make the topological entropy locally constant as a function of the diffeomorphism?
In this section we review concepts of partial hyperbolicity, Lyapunov exponents, and disintegration of measures.
2.1. Partial hyperbolicity
For a diffeomorphism of a compact manifold to itself recall the norm and conorm with respect to a subspace of for some : and
A splitting is dominated111This is sometimes called pointwise domination, see . if it is nontrivial, invariant, and if there is some such that, for all :
A diffeomorphism is (strongly) partially hyperbolic if there is an invariant splitting of the tangent bundle: such that and are dominated, is uniformly contracted, and is uniformly expanded.
The stable and unstable bundles and of a partially hyperbolic diffeomorphism are always uniquely integrable into stable and unstable foliations, respectively, denoted by and . The bundles , , and fail to be integrable for some strongly partially hyperbolic diffeomorphisms.
A strongly partially hyperbolic diffeomorphism is dynamically coherent if there exists invariant foliations and that are tangent to the and bundles respectively. In this case there is a center foliation given by for .
We refer to  for various other definitions of dynamical coherence and their relationships.
For a dynamically coherent diffeomorphism each leaf of is subfoliated by the leaves of and the leaves of . A similar statement holds for the center-unstable foliation. Then for any points where there is a neighborhood of in the leaf and a homeomorphism such that
The map is the (local) stable holonomy map. We can similarly define the unstable holonomy map.
2.2. Center Lyapunov exponents
For a strongly partially hyperbolic diffeomorphism a real number is a center Lyapunov exponent at if there exists a nonzero vector such that
If , then the limit above only depends on and exists -almost everywhere for any -invariant Borel probability measure . For an ergodic -invariant Borel probability measure the limit takes on a single value for -almost every .
2.3. Disintegration of a measure
Let be a Polish space and be a finite Borel measure on . Let be a partition of into measurable sets. Let be the induced measure on the -algebra generated by . A system of conditional measures of with respect to is a family of probability measures on such that
for -almost every , and
given any continuous function , the function is integrable, and
We will consider partitions given by foliations of a manifold. If a foliation has a positive measure set of noncompact leaves, then the result of Rokhlin does not immediately apply. However, one can extend the result of Rokhlin by disintegrating into measures defined up to scaling (see Avila, Viana, and Wilkinson ).
Let be a manifold where and be a locally finite measure on . Let be a small foliation box. Then Rokhlin’s result implies there is a disintegration of the restriction of to the foliation box into conditional probability measures along the local leaves of the foliation, i.e., the connected components of for . From [2, Lemma 3.2] we know that if and are foliation boxes and -almost any , then the restriction of and coincide up to a constant factor.
We then know that for -almost every there is a projective measure (i.e., defined up to some scaling possibly depending on ) such that . Furthermore, the function is constant along the leaves of , and the conditional probabilities along the local leaves of any foliation box coincide almost everywhere with the normalized restriction of the to the local leaves of .
Finally, we note that if the foliation is fixed by some diffeomorphism (i.e., ) without fixed points, one can replace the projective measures by true measures using the global normalization: for all .
2.4. Continuous systems of measures
We will work with families of measures carried by the leaves of the dynamical foliations up to a union of exceptional leaves.
Given a foliation of some manifold and some -saturated subset a continuous system of measures on is a family such that:
for all , is a Radon measure on ;
for all , if ;
is covered by foliation charts such that: is continuous on for any .
The Radon property (i) means that each is a Borel measure and is finite on compact subsets of the leaf (here, and elsewhere, we consider the intrinsic topology on each leaf).
If is the disintegration of some probability measure along a foliation as defined in the previous definition and if is a continuous system of measures on , we will say that they coincide if and for -a.e. , and are proportional.
Assume that is a foliation which is invariant under some diffeomorphism , i.e., for all , . Let be -saturated. A continuous system of measures on is dilated if there is some number such that for all :
is called the dilation factor. We call the family a Margulis system on and the measures the Margulis -conditionals.
Our construction (following Margulis) relies on properties of the holonomy between foliations defined as follows:
Let be foliations which are invariant under some diffeomorphism . Let be an -saturated subset of . Assume that is a Margulis system of measures on and that is transverse to . The system is invariant, respectively quasi-invariant, along if, for all -holonomies with contained in -leaves included in :
The quasi-invariance in (3) can be characterized by the absolute continuity of the holonomies along with respect to a class of transversal measures defined by the Margulis system on .
Though an arbitrary continuous system of measures along the strong unstable foliation does not need to correspond to the disintegration of any invariant probability measure, those we construct in this paper will (see Proposition 5.1.)
3. Main Results
This section collects our main results. Our techniques deal with the following type of diffeomorphisms. For convenience, we fix some Riemannian structure on the compact manifold .
A diffeomorphism has flow type if:
partial hyperbolicity: is strongly partially hyperbolic with splitting and ;
Dynamical coherence: there are invariant foliations and tangent to and ;
Let be the foliation whose leaves are the connected components of the intersections , .
Center leaves: The center foliation is oriented and has at least one compact leaf.
Let be the continuous flow along with unit positive speed.
Flow like dynamics: there is a continuous such that, for all , and .
Following Margulis, we build special measures on most strong stable and strong unstable leaves. Let where is the union of the unstable leaves that intersect some compact center leaf. Define and likewise.
Let be a diffeomorphism with flow type and minimal stable and unstable foliations on a compact manifold . Then there is a continuous system of measures on such that:
each is atomless, locally finite, with full support;
is a Margulis system along with dilation factor ;
the system is -quasi-invariant;
is dense with full measure for any ergodic measure with positive entropy.
In the setting of the previous theorem, there is a unique system of measures satisfying the above items (1) and (2). Moreover its dilation factor is .
We call the system of measures the unstable Margulis system and the measures the unstable Margulis conditionals.
(1) The above theorem and addendum, applied to , defines a stable Margulis system with dilation factor .
(2) The smoothness assumption is only used by Theorem 4.3 to establish absolute continuity of the -holonomy but it is probably enough to assume smoothness. We do not know how to deal with smoothness.
Let be a diffeomorphism with flow type and minimal stable and unstable foliations on a compact manifold . Let be an ergodic MME.
If , then the disintegration of along is given by the unstable Margulis system from Theorem 3.2. In particular, the measure has have full support.
The above applied to shows that an ergodic MME with has disintegration along given by . In particular, any MME has full support.
The above theorem gives more information in the non-hyperbolic case. Indeed, if is an ergodic measure of maximal entropy with , then the disintegrations, along both strong foliations and , are given by the corresponding Margulis systems from Theorem 3.2.
The dichotomy will follow from two results about hyperbolic measures of maximal entropy. The first is a uniqueness result, based on the Hopf argument.
Let be a diffeomorphism with flow type and minimal stable and unstable strong foliations on a compact manifold . Let be some ergodic MME. If is hyperbolic, say , then there is no other ergodic MME with .
Let be a diffeomorphism of a compact manifold . Let be an orientable one-dimensional foliation (with continuously varying leaves). Assume that, for all , maps to itself in an orientation-preserving way. Let satisfy:
its Lyapunov exponent along is ;
for -a.e. , the following set is relatively compact in the intrinsic topology of :
and for -a.e. , the leaf is noncompact and contains no fixed point.
Then there is another invariant probability measure which is isomorphic to and with exponent .
Finally, we state the abstract version of our main result:
For any diffeomorphism with flow type and minimal stable and unstable foliations on a compact manifold , we have the following dichotomy:
either all the measures of maximal entropy have zero central Lyapunov exponents, or
there are exactly two ergodic measures of maximal entropy where one has a positive central exponent and one has a negative central exponent, and both are Bernoulli.
The next theorem shows that there is an abundance of diffeomorphisms satisfying the above assumptions. It follows from properties of perturbations of time-one maps of transitive Anosov flows established in  and , as discussed in Section 6.
If is a transitive Anosov flow on a compact manifold , then for all there exists a open set in such that belongs to the -closure of and every has flow type with both stable and unstable foliations minimal.
4. Building Margulis systems of measures
In this section, we consider flow type diffeomorphisms whose strong foliations are both minimal. To begin with, we follow Margulis’ construction of a system of measures on the -leaves that are invariant under stable holonomies. We then deduce from this a system of -conditionals that are quasi-invariant under center-stable holonomies. This proves Theorem 3.2, except for the uniqueness of the Margulis -system and the equality that will be deduced in Section 5 from the analysis of MMEs.
4.1. The -conditionals
We following Margulis’ construction.
Let on a compact manifold with a dominated splitting with uniformly contracted. Assume that:
there are foliations and which are tangent to, resp. and ;
Then there is a Margulis system on which is invariant under -holonomy and such that each is atomless, Radon, and fully supported on .
We introduce some convenient definitions. Let . We denote by the intrinsic Riemannian volume on each -leaf: for any subset of a -leaf, is its volume with respect to the Riemannian structure on the leaf. We denote by the distance defined on each leaf by the induced Riemannian structure and defining the intrinsic topology.
The -balls are . For a subset of such a leaf, we set . A -subset is a -bounded subset of a -leaf. A -test function is a nonnegative function such that is a -subset and the restriction of to
is continuous. We write if and has non-empty interior in the intrinsic topology. We denote by the collection of all -test functions.
Given a -holonomy its size is , and the two subsets are called equivalent along through provided . We say that they are -equivalent if the holonomy has size at most . Two functions are -equivalent along if their supports are equivalent through a -holonomy with size at most and satisfying . Two submanifolds are -transverse if they are transverse and if for every , the angle between any two nonzero vectors in and is at least .
Following Margulis, we consider functionals . Note that is one such functional. The map acts on them by:
A key class of such functionals are for any . That is, for any ,
To normalize, we fix some with . Considering the topology of pointwise convergence (i.e., working in with the product topology), let be the closure of the following set:
We will use the following covering numbers. For , a -subset, and , we denote by the smallest integer such that there are with .
We build the -system as a functional.
There exist and such that
and, for some positive numbers , for any :
if , then ; and
if is -equivalent to , then .
To prove this, we will show that is a convex and compact set and then apply the Schauder-Tychonoff fixed point Theorem to a normalized action of on .
We will relate the iterations of different -test functions by using the invariance under -holonomy and especially the following theorem (see, e.g., [1, Theorem C]).
For convenience let . Observe that for , and .
Let be a diffeomorphism on a compact Riemannian manifold . Assume that there is a dominated splitting with uniformly contracting. Fix and let be two submanifolds -transverse to and -equivalent through . Then is absolutely continuous.
More precisely, writing for the Riemannian volume on , the measures and are equivalent and there are constants and depending only on , , and such that, letting be the size of the holonomy :
The second term simply ensures that the above bound holds even for large .
We need two additional lemmas. The first one will give a uniform bound on the volume growth in the center unstable leaves.
For any open, non-empty -subset , there are constants and such that,
Fix so small that is not empty. The minimality implies that any is -equivalent to some point in . The continuity of the foliation and its transversality to yield and such that, for any , is -equivalent to a subset of . The compactness of yields , such that, for any point , is -equivalent to a subset of .
Since is contracted, there are numbers and such that the set is -equivalent to a subset of . Theorem 4.3 proves the claim. ∎
For any with , there are numbers and such that for any and any :
The left hand side is bounded by . Fix some . The previous lemma with yields and . Since is compact in its -leaf , there are with such that . Now
Summing over the cover of , the claim follows with and . ∎
The next lemma establishes approximate holonomy invariance.
There are numbers and with the following properties. Let be -equivalent for some .
First, if , then, for all :
Second, for any with , there are numbers and such that, for any we have
Since and are -equivalent, there is with size at most such that . Since is uniformly contracted, and are -equivalent through for some and . Theorem 4.3 yields:
Let so that . Using the identity , we get: Hence, if ,
proving eq. (6) for all by continuity.
Proof of Proposition 4.2.
We prove the first two claims (a) and (b) for arbitrary .
Step 1: Claim (a): .
Corollary 4.5 for yields numbers and such that for any : Therefore, for any :
This proves the claim since for . It extends to the closure , concluding Step 1.
Note that eq. (8) implies that is a subset of the compact set
Hence its closure is compact.
Step 2: Claim (b): if there is s.t. , .
We assume that and apply again Corollary 4.5, exchanging and . We get new numbers and defined by . Setting , we have . That is, so that, for any :
This again extends to , concluding Step 2 with the lower bound .
Step 3: Existence of with
We now build the functional as a fixed point of the map
We claim that is well-defined and continuous from to . Indeed, the map from to is well-defined since , is obviously continuous, and is positive by Step 2. Note that is well-defined and is continuous from to . The claim is proved.
Finally, it is obvious that , hence is a well-defined continuous map. Since is a convex, compact subset of the locally convex linear space , the Schauder-Tychonoff Theorem applies and yields with .
Therefore, with .
Step 4: Claim (c): -holonomy invariance of
Let . Writing and likewise for , we can assume . Assume that and are -equivalent. By compactness of their support, they are -equivalent for some and therefore and are -equivalent with . Using the dilation and the approximate holonomy invariance eq. (6) from Lemma 4.6, we get that, for any , for large enough :
As is arbitrarily small this implies , i.e., Claim (c). ∎
We deduce a Margulis system of -measures from the functional .
Proof of Proposition 4.1.
Proposition 4.2 yields a functional on , which contains for all . Note that is linear and positive (because this holds for all and extends by continuity to ). Hence, Riesz’s Representation Theorem gives a measure on , for each , by setting:
The local finiteness, full support, and -invariance of each follows from properties (a), (b), (d) of the functional from Proposition 4.2.
We deduce that each is atomless from the holonomy invariance. Assume by contradiction that there is with . Consider the stable leaf . By assumption it is dense in , hence by transversality in . By -invariance, must have a dense set of atoms , all of which have measure . This contradicts the finiteness of on compact sets.
We finally deduce the continuity from the holonomy invariance. As and are transverse, for any , there is a neighborhood of and a continuous map with (in particular, ). Let . By holonomy invariance, . Hence,
which converges to as goes to . This is the continuity property. ∎
The next lemma establishes that the constant is larger than .
Let have flow type with both strong foliations minimal. Let be a Margulis -system. The dilation of the Margulis system on satisfies: .
By assumption (III), there is a compact center leaf . It is contained in the -leaf . Since is a topological attractor for the restriction of to the invariant set , there is a relatively compact neighborhood of in such that has non empty interior. As has full support in , it follows that:
proving . ∎
4.2. Building the -conditionals
We start with the previously built Margulis -system and define the family of measures by extending subsets of -leaves to subsets of -leaves along the center foliation. For flows, Margulis used the formula