Weak and entropy approximation
of nonhyperbolic measures:
a geometrical approach
We study -robustly transitive and nonhyperbolic diffeomorphisms having a partially hyperbolic splitting with one-dimensional central bundle and a compact closed curve tangent to the central bundle. We prove that there is a -open and dense subset of such diffeomorphisms such that every nonhyperbolic ergodic measure (i.e. with zero central exponent) can be approximated in the weak topology and in entropy by measures supported in basic sets with positive (negative) central Lyapunov exponent. Our method also allows to show how entropy changes across measures with central Lyapunov exponent close to zero. We also prove that any nonhyperbolic ergodic measure is in the intersection of the convex hulls of the measures with positive central exponent and with negative central exponent.
Key words and phrases:blender-horseshoe, entropy, ergodic measures, Lyapunov exponents, minimal foliations, partial hyperbolicity, transitivity, weak topology
2000 Mathematics Subject Classification:37D25, 37D30, 28D20, 28D99
- 1 Introduction
- 2 Blender-horseshoes
- 3 Approximation of ergodic measures
- 4 Fake invariant foliations and distortion estimates
5 Construction of the hyperbolic sets in Theorem 1
- 5.1 Preliminaries
- 5.2 Skeletons
- 5.3 Postliminaries
- 5.4 Construction of the set
- 5.5 Hyperbolicity and entropy of the set
- 5.6 Controlling Birkhoff averages
- 6 Perturbation from negative to positive exponents: Proof of Theorem 4
- 7 Arcwise connectedness: Proof of Theorem 5
Consider a boundaryless Riemannian compact manifold and its space of -diffeomorphisms endowed with the uniform topology. A diffeomorphism is transitive if it has a dense orbit and it is -robustly transitive if it has a -neighborhood consisting of transitive diffeomorphisms. We consider the -open subset of robustly transitive diffeomorphisms and its subset formed by diffeomorphisms such that
has a -neighborhood consisting of nonhyperbolic diffeomorphisms,
there is a partially hyperbolic splitting with three non-trivial bundles such that is uniformly contracting, is one-dimensional, and is uniformly expanding, and
there is a closed curve tangent to .
Note that these conditions are open. Indeed, partially hyperbolic splittings have well defined continuations and any closed curve tangent to is normally hyperbolic and hence it has well defined continuations (see ).
There are two main types of examples of diffeomorphisms in : those having an invariant foliation tangent to consisting of circles (see [34, 19]) and those having simultaneously closed and non-closed leaves tangent to (the main example are some perturbations of the time-one map of a transitive Anosov flow, ). Note that there is a third important class of nonhyperbolic partially hyperbolic robustly transitive systems, called DA-diffeomorphisms , which do not fall into because they do not have closed curves tangent to the central bundle .
The diffeomorphisms in have invariant foliations and tangent to and , called the strong stable and strong unstable foliations, respectively (see ). A first crucial property here is that the existence of a closed central leaf implies that there is a -open and -dense subset of consisting of diffeomorphisms for which both foliations are minimal (every leaf of the foliation is dense in the whole space), see [10, 33].111This notation refers to minimality and existence of blender-horseshoes. In what follows, we will study the diffeomorphisms in . Let us observe that the existence of a pair of so-called un-/stable blender-horseshoes is a (implicit) key ingredient in the proof of the fact that is open and dense in in [10, 33] (even though the term “blender” was only coined later ). We will also explore the dynamics of these blenders, see Section 2, which will also be an important ingredient in our constructions. For details see Proposition 2.9 and Remark 2.10.
Nonhyperbolicity is closely related to the existence of zero Lyapunov exponents. Given , a point is Lyapunov regular if there are a positive integer , numbers , called the Lyapunov exponents of , and a -invariant splitting such that for all and , , we have
Note that in our partially hyperbolic setting there is some such that and we denote the corresponding Lyapunov exponent by .
We denote by the set of -invariant probability measures of and by the subset of ergodic measures. We equip the space with the weak topology. Given , Oseledets’ multiplicative ergodic theorem  claims that the set of Lyapunov regular points has full measure and and , , are constant -almost everywhere. The latter numbers are called the Lyapunov exponents of . If then is called nonhyperbolic. Note that in our setting the other exponents of are nonzero. We denote by the subset of of nonhyperbolic measures. Thus, the occurrence of a zero exponent is related to the central direction only and there is a natural decomposition
where and denote spaces of measures such that and , respectively.
The exploration of nonhyperbolic ergodic measures is a very active research field which started with the pioneering work in . Note that by  there is a -open and -dense subset of consisting of diffeomorphisms such that is nonempty and contains measures with positive entropy. The main focus of this paper is to study how nonhyperbolic ergodic measures insert in the space of ergodic measures. The main result is how nonhyperbolic measures are weak and in entropy are approached by hyperbolic ones which are supported on basic sets. It is a continuation of a line of arguments in [14, 16] where these questions were studied in a skew-product setting and where a general axiomatic framework to attack this problem was introduced, see the discussion after Corollary 2.
By a very classical result, mainly started by Katok [25, 26], every hyperbolic ergodic measure can be approximated by periodic ones. Here one can consider approximation in the weak topology. Moreover, one can approximate by means of ergodic measures supported on basic sets which converge weak and in entropy, that is, given hyperbolic ergodic, there is a sequence of basic sets such that in the weak topology and that . Katok’s result was first shown for surface diffeomorphisms [26, Supplement S.5], but extends also to higher-dimensional manifolds and - and dominated diffeomorphisms (see, for example, [13, 28, 18] and references therein and also ). Below we will present an analogous version for nonhyperbolic ergodic measures.
Given and a hyperbolic set of , denote by the subset of measures supported on . We define analogously . We say that a hyperbolic set is central contracting (central expanding) if on the bundle is stable ( is unstable). Recall that a set is basic if it is compact, -invariant, hyperbolic, locally maximal, and transitive.
Given a countable dense subset of continuous (nonzero) functions on , recall that
provides a metric which induces the weak topology on .
The following is a consequence of Theorem 5.1 which is stated under the minimal hypotheses which we require to construct central expanding (contracting) basic sets as stated.
Theorem 1 (Approximation in weak and entropy).
For every in the -open and -dense subset of every nonhyperbolic ergodic measure of has the following properties. For every and every there exist a pair of basic sets being central contracting and being central expanding such that the topological entropy of on satisfy
Moreover, every measure is -close to . In particular, there are hyperbolic measures satisfying
The program for proving the above result was laid out in [14, Section 8.3]. The result above is the corresponding version of [14, Theorem 1] (in a step skew-product setting with circle fiber maps) in the present setting. During the final preparation of this manuscript, we noticed that a preprint with a similar result was announced in .
We have the following straightforward consequence of the above.
Corollary 2 (Restricted variational principles).
Note that, in contrast to [15, Theorem 2] or , in general there are yet no general tools to establish the uniqueness of hyperbolic measures of maximal entropy. See also the results and discussion in .
Recall that an ergodic measure is periodic if it is supported on a periodic orbit. It is a classical result by Sigmund  that periodic measures are dense in for any basic set , and hence every hyperbolic ergodic measure is approximated by hyperbolic periodic ones. The above result then immediately implies that this is also true for nonhyperbolic ergodic measures.
Corollary 3 (Periodic approximation).
For every and every is approximated by hyperbolic periodic measures. Moreover, every is approximated by periodic measures in and in , respectively.
The following result shows how entropy “changes across measures with Lyapunov exponent close to zero”. As for Theorem 1, it will be an immediate consequence of a Theorem 6.1 correspondingly stated under the minimal hypotheses.
For every and every with , there is a positive constant such that for every , , and , there is a basic set being central expanding such that
its topological entropy satisfies
The same conclusion is true for and every , changing in the assertion to .
If then is a hyperbolic periodic orbit.
The result above corresponds to [14, Theorem 5].
Remark 1.2 (Continuations in the weak and in entropy of ergodic measures).
A consequence of Theorem 1 is that for every any ergodic measure of has a continuation in the following sense. Every diffeomorphism sufficiently -close to has an ergodic measure close to in the weak topology and with entropy close to the one of . If the measure is hyperbolic this is essentially a reformulation of Remark 1.1. In the nonhyperbolic case, just note that the measures supported on are close (in the weak and in entropy) to . Hence measures supported on the (well and uniquely defined) continuations of for diffeomorphisms nearby are close to . Note that these continuations of are hyperbolic. A much more interesting question, related to Theorem 4, is if for close to the diffeomorphism has a nonhyperbolic measure close to (in the weak and in entropy). This remains an open question. Note that by , -open and -densely, the diffeomorphisms close to have nonhyperbolic ergodic measures with positive entropy, but it is unclear and unknown if those can be chosen close to .
Finally, observe that our constructive method provides a way to obtain the hyperbolic sets (and hence their continuations) based on skeletons, see Section 5.2. Our notion of skeleton follows the one introduced in  and depends on a blender-horseshoe, two connection times to such a blender-horseshoe, and finitely many (long) finite segments of orbits (where the finite central exponent is close to zero). In our context, all these ingredients are persistent. Our concept of skeleton is different (although with somewhat similar flavor) from the one introduced in parallel in , that we call here DVY-skeleton. The latter is a finite collection of hyperbolic periodic points with no heteroclinic intersections such that the strong unstable leaf of any point in the manifold intersects transversally the stable manifold of the orbit of some point in the skeleton. Open and densely in , DVY-skeletons consist of just one point (this follows from the minimality of the strong foliations and by the fact that the manifold is a homoclinic class, see Section 7). Note that, in general, the DVY-skeletons may collapse by perturbations.
The space equipped with the weak topology is a Choquet simplex whose extreme points are the ergodic measures (see [38, Chapter 6.2]). In some cases the set of ergodic measures is dense in its closed convex hull in which case (assuming that is not just a singleton) one refers to it as the Poulsen simplex, see also . Although, in general, is very far from having such a property, it is a consequence of  that each of the subsets and is indeed a Poulsen simplex. We investigate further these simplices and study the remaining set of nonhyperbolic (ergodic) measures. Properties of this flavor were also studied in .
Theorem 5 (Arcwise connectedness).
There is an -open and -dense subset of consisting of diffeomorphisms for which the intersection of the closed convex hull of and the closed convex hull of is nonempty and contains . Each of the sets and is arcwise connected. Moreover, every measure in is arcwise connected with any measure in and , respectively.
Indeed, the open and dense subset in the above corollary is the subset of for which the entire manifold is simultaneously the homoclinic class of a saddle of index and of index , respectively. See the proof of Theorem 5.
The above theorem partially extends results in  to our partially hyperbolic setting. The results in  are stated for (i) measures supported on an isolated homoclinic class whose saddles of the same index are all homoclinically related and assuming that (ii) is . Concerning (ii), nowadays it is often used that the hypothesis can be replaced by plus domination. Concerning (i), we will see that these conditions are satisfied in our setting. Indeed, see Section 7, the set can be chosen such that these two hypotheses hold for every of its elements. Theorem 5 is proved in Section 7. See also [16, Section 3.1] for a proof of this type of results in a step skew product setting.
As we have mentioned, this paper is a continuation [14, 16], where the corresponding results where obtained for step skew-products with circle fiber maps. The axiomatic setting proposed in  considers three main hypotheses formulated for the underlying iterated function system (IFS) of the skew-product: transitivity, controlled expanding (contracting) forward covering relative to an interval (called blending interval), and forward (backward) accessibility relative to an interval. In [14, Section 8.3] it is explained how these conditions are in fact motivated by the setting of diffeomorphisms in : the controlled expanding (contracting) forward covering property mimics the existence of expanding (contracting) blenders, while the forward (backward) accessibility mimics the minimality of the strong unstable (stable) foliation. As discussed in , the axioms mentioned above capture the essential dynamical properties of diffeomorphisms in . In this paper, we complete the study initiated in [14, 16]. A key ingredient in the study of is the minimality of the strong invariant foliations. In  blender-horseshoes are used to prove this minimality, although at that time this concept was not yet introduced and the word blender does not appear in , and the authors refer to so-called complete sections (see Section 2.6 and Proposition 2.9). The next step, once these blender-horseshoes are obtained, is to study their dynamics and to state the precise correspondence of their expanding/contracting covering properties. This is done here in Section 2 and Proposition 2.3.
Idea of the proof
The proof is essentially based on the following ingredients. First we use blender-horseshoes with are just hyperbolic basic sets with an additional geometrical superposition property. The second ingredient are the minimal strong foliations. Our construction will use so-called skeletons. A skeleton consists of arbitrarily long orbit pieces that mimic the ergodic theoretical properties of the given nonhyperbolic measure . The cardinality of the skeletons is of order of , where is the length of each individual orbit segment in the skeleton. Using minimality, we see that these segments can be connected in uniformly bounded time to the “domain of the blender”. Technical difficulties are the control of distortion related to the central direction as well as the absence of a central foliation. This last difficulty is circumvented by the use of “fake local invariant foliations” introduced in .
The hyperbolic set in Theorem 1 is obtained as follows: using the segments of orbits provided by the skeleton property we construct pairwise disjoint full rectangles in the “domain of the blender” such that for a fixed iterate (which is of the order of ) the image of each rectangle intersects in a Markovian way each rectangle. This provides a hyperbolic basic set whose entropy is close to and its exponents are close to .
Organization of the paper
In Section 2, we review all ingredients to construct blender-horseshoes and state and prove Proposition 2.3 about the controlled expanding/contracting forward central covering property. In that section, we also prove their -open and -dense occurrence in . In Section 3, we state a general result on how to approximate the individual quantifiers of an ergodic measure by individual orbits. In Section 4, we recall fake invariant foliations to deal with the problem that in general there is no foliation tangent to the central bundle. Section 5 is dedicated to the proof of Theorem 1 and is the core of this paper. Theorem 4 is proven in Section 6, while Section 7 gives the proof of Theorem 5.
In this section, we review the construction of blender-horseshoes in  using the existing partially hyperbolic structure of the diffeomorphisms. Here, besides the topological properties of blender-horseshoes, we will also need an additional quantitative controlled expanding forward central covering, see Proposition 2.3. In Section 2.6, we state the open and dense occurrence of blender-horseshoes in our setting, see Proposition 2.9.
2.1. Definition of a blender-horseshoe
We will follow closely the presentation of blender-horseshoes in  based on ingredients such as hyperbolicity, cone fields, and Markov partitions, and sketch its main steps. We also provide some further information which is not explicitly stated in .
We say that a maximal invariant set of is an unstable blender-horseshoe if there exists a region diffeomorphic to such that
and is a hyperbolic set with -dimensional stable bundle and -dimensional unstable bundle which satisfies conditions (BH1)–(BH6) in [8, Section 3.2]. The set is the domain of the blender-horseshoe. A stable blender-horseshoe is an unstable blender-horseshoe for . Roughly speaking, it is a “horseshoe with two legs” having specific properties and being embedded in the ambient space in a especial way that it is has a “geometric superposition property”: stated in the simplest way, there is an interval such that for every any disk of the form intersects the local stable manifold of . A key feature is that this property also holds for perturbations of such disks.
To explain the simplest model, consider an affine horseshoe map such that in the central direction the map acts as a multiplication for some ; the maximal compact invariant set being contained in the rectangle . Refer to Figure 1 and the notation there. Note that this rectangle is not normally hyperbolic but is partially hyperbolic (consider the case of in Figure 1). We now perturb in such a way, keeping affinity, that “one of the legs is moved to the left” in the central direction changing the dynamics in the central direction in the rectangle to , small. This provides an example of an affine unstable blender-horseshoe where the domain is , small. A precise construction with all the details can be found in  (though the term blender is not used there). Indeed, this example corresponds to the prototypical blender-horseshoes in [8, Section 5.1]. Figure 1 shows a prototypical blender-horseshoe and illustrates at the same time all the elements in the (general) construction in this section.
The main result in this section is Proposition 2.3 which derives a controlled expanding forward central covering property, that is, the existence of some forward iteration along which any small enough unstable strip “crossing the domain of the blender-horseshoe” is uniformly expanded (in the central direction) and covers (in the central direction) the entire domain. This occurs with uniform control on iteration length and expansion strength which depend on the central size of only. This property has its correspondence to the Axiom CEC+ in  there stated for an IFS.
Recall that we assume that is a partially hyperbolic diffeomorphism with a globally defined splitting , where , , and . Here the hyperbolic structure of the blender-horseshoe fits nicely with the partially hyperbolic one of . In particular, and and the stable manifolds are the strong stable manifolds of .
Conditions (BH1) and (BH3) in  state the existence of a Markov partition and, in particular, imply that the set is conjugate to a full shift of two symbols, denoted by and . The Markov partition provides two disjoint “sub-rectangles” and of that codifies the dynamics, that is, and to each point the conjugation associates the sequence defined by . This implies that has a fixed point and a fixed point .
Condition (BH2) refers to the existence of strong stable , strong unstable , and unstable invariant cone fields (about , , and , respectively). More precisely, given we denote
We simply refer to if is not specified. Analogously for . Here we also consider a cone field contained in about the central bundle with the analogous definition. Note that is backward invariant, and are forward invariant, while is not invariant. In our case, due to the partial hyperbolicity, the (global) existence of these cone fields is automatic and the key point is the existence of (and some appropriate norm equivalent to the initial one, ) such that
This means that the, otherwise neutral, central direction is indeed expanding in and .
The explanation of the remaining conditions (BH4)–(BH6) demands some preliminary work. We consider the parts the boundary of the “rectangle” corresponding to and and call them strong stable and strong unstable boundaries, denoted by and , respectively.222Note that in , is called stable boundary and denoted by . As here simultaneously we have stable and strong stable bundles, we prefer this notation.
-complete and -complete disks. A -complete disk is a disk of dimension (that is a set diffeomorphic to ) contained in and tangent to the cone field whose boundary is contained in . Similarly, a -complete disk is a disk of dimension contained in and tangent to the cone field whose boundary is contained in . It turns out that - and -complete disks containing a point are not unique.
The local stable manifold of a point is the connected component of that contains .333Note that here . Similarly, for the local strong unstable manifold of . Note that is a -complete disk and is a -complete disk for every .
Condition (BH4) is a geometrical condition that claims that -complete disks cannot intersect simultaneously and .
-complete disk in-between. Condition (BH4) also implies there are two homotopy classes of -complete disks in disjoint from , called disks to the right and disks to the left of . Similarly for . A -complete disk that is to the right of and to the left of is called in-between and or shortly in-between. We denote these disks by . Choosing appropriately right and left, we have for every .
For each -complete disk in we consider the sets
Conditions (BH5)–(BH6) claims that for every then either or (and there are cases such that both sets are in-between). This concludes the sketch of the description of a blender-horseshoe.
Remark 2.1 (Orientation).
Recall again that there is a (global) partially hyperbolic splitting of the tangent bundle of the manifold . In the definition of a blender-horseshoe, we will also require that for restricted to , the tangent map preserves orientation in the bundle . Note that this is also implicitly assumed in .
2.2. -strips in-between and expanding central covering
Similarly as in [6, Section 1.a], we introduce the notion of a -strip. First, a curve in is called central if it is tangent to . A -strip is a closed disk of dimension tangent to the unstable cone field that is simultaneously foliated by -complete disks and by central curves (a central foliation of ). Given a -strip , a curve is called -complete if it is a curve (whole leaf444We define the strong unstable boundary of a strip in the same spirit of , a complete leaf joins the two components of that boundary.) of some central foliation of . To a -strip we associate its (inner) width defined by
We say that a -strip is in-between if it is foliated by -complete disks in-between. To each -strip in-between we associate the sets and . We say that a -strip is -complete if its intersects simultaneously and .
For our goals we need a more precise “quantitative” version of the “expanding” returns in the remark, that we call controlled expanding forward central covering stated below.
Proposition 2.3 (Controlled expanding forward central covering).
Let be as in (2.1). There is such that for every -strip in-between there is a positive integer ,
such that for every there is a subset such that
is contained in for all and
is a -complete -strip.
The proof of the above proposition will be completed in Section 2.5.
2.3. Further properties of blender-horseshoes
To get Proposition 2.3, we state additional properties (BH7), (BH8), and (BH9). Note that they are not additional hypotheses on the blender-horseshoe but rather straightforward consequences of (BH1)–(BH6) and the constructions in  and obtained taking a sufficiently thin strong unstable cone field.
The intersection consists of two connected components: and a second component , where is a homoclinic point555A point is a homoclinic point of if it belongs simultaneously to the stable and to the unstable manifold of . Note that, in our setting, a homoclinic point is automatically transverse, that is, those manifolds intersect transversally. of in . Similarly, consists of two connected components, and , where is a homoclinic point of in .
As above, we can speak of -complete disks to the left/right of and of . Similarly as in condition (BH4) the blender-horseshoe we have the following:
Every -complete disk which intersects is to the left of and every -complete disk intersecting is to the right of . In particular, any -complete disk intersecting and any -complete disk intersecting are disjoint. Moreover, there is so that every -strip intersecting and has minimal width bigger than .
The points and are auxiliary in order to quantify the size of the geometric superposition region (compare Figure 1). Note that, in order to prove Proposition 2.3 it is enough to show that given any -strip in-between there is a number of iterates (depending on only) so that we obtain a -strip which intersects the local stable manifold of and whose part to the right of has some least size (indeed, ). This in turn is guaranteed when intersects simultaneously and . This is a sketch of the content of Lemmas 2.6, 2.7, and 2.8.
Condition (BH8) implies that there are three possibilities for a -strip in-between:
it is to the left of ,
it is to the right of ,
it intersects simultaneously and , hence by (BH8) it has minimal width at least .
Consider a strip in-between. Then
If is to the left of then contains a -strip in-between with ,
If is to the right of then contains a -strip in-between with ,
There is a number with the following property:
Every -strip in-between to the right of has (inner) width less than .
Every -strip in-between to the left of has (inner) width less than .
In other words, any -strip in-between with (inner) width bigger than intersects simultaneously and . Compare Figure 1.
2.4. Iterations of -strips
The next step is the iteration of -strips to obtain covering properties. The key in this process is that here we have more accurate control of the image of the strips as in the (standard) blenders (compare with [6, Lemma 1.7]).
Lemma 2.6 (Simultaneous intersections).
Considering the strip in Lemma 2.6 and recalling that is a homoclinic point of and is a homoclinic point of , we have that intersects and that intersects .
Proof of Lemma 2.6.
The proof is by induction, using arguments as in [8, Lemma 4.5]. Let . If intersects simultaneously and we are done. Otherwise, by Remark 2.4, either is to the left of or is to the right of . If the first case consider and observe that by (BH9) we have that contains a -strip in-between with . In the second case, consider and observe that by (BH9) we have that contains a -strip in-between with . Note that .
We now proceed inductively, assume that we have defined -strips in-between that do not intersect simultaneously and , satisfy either or , according to the case, and . As in the first inductive step, we take if is to the left of or otherwise. In both cases, we have that
The choices of and imply that there is a first with such that intersects simultaneously and . Hence contains a -strip in-between that intersects simultaneously and . Note that by construction is contained in for all . This completes the proof of the lemma. ∎
In what follows, for convenience, we consider a -strip together with a family of -complete disks foliating (note that this foliation is not unique) and write . We say that a -strip is quasi to the right of if there is with and for every the disk is to the right of . Note that this means, in particular, that the intersection with occurs in the strong unstable boundary of the strip.
Let be a -strip in-between which intersects simultaneously and . Then contains a -strip quasi to the right of with minimal width , where was defined in (BH8).
Given a -strip whose interior intersects , we consider the -complete disk of intersecting . Note that, since the intersection of with is transverse, the disk is uniquely defined. Observe that has two connected components, a component consisting of -complete disks to the right of and a component consisting of -complete disks to the left of . We denote the closures of these components by and and observe that they intersect along the disk . Note that is quasi to the right of . We can argue similarly with strips which are quasi to the right of , in that case (thus ).
Finally, note that there is a number such that every -strip that is quasi to the right of with also intersects .
Consider a -strip with such that . Define as the first integer with , where is the expansion constant in (2.1). Then for every it holds that contains a -complete -strip such that for all .
The proof follows as in Lemma 2.6. Let and note that contains -strip that is quasi to the right of and satisfies . Now it is enough to argue inductively. ∎
2.5. Proof of Proposition 2.3
Consider a -strip in-between and let . By Lemma 2.6, there is a first
for some independent of , such that contains a -strip in-between that intersects simultaneously and . By Lemma 2.7, we have that contains a -strip quasi to the right of with . Note that . Take as in Lemma 2.8 and note that contains a -complete -strip . Note that, by the lemma,