A criterion for zero averages and support of ergodic measures

A criterion for zero averages and full support of ergodic measures

Christian Bonatti Institut de Mathématiques de Bourgogne bonatti@u-bourgogne.fr Lorenzo J. Díaz Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro lodiaz@mat.puc-rio.br  and  Jairo Bochi Facultad de Matemáticas, Pontificia Universidad Católica de Chile jairo.bochi@mat.puc.cl
Jan 26, 2012
Abstract.

Consider a homeomorphism defined on a compact metric space and a continuous map . We provide an abstract criterion, called control at any scale with a long sparse tail for a point and the map , that guarantees that any weak limit measure of the Birkhoff average of Dirac measures is such that -almost every point has a dense orbit in and the Birkhoff average of along the orbit of is zero.

As an illustration of the strength of this criterion, we prove that the diffeomorphisms with nonhyperbolic ergodic measures form a -open and dense subset of the set of robustly transitive partially hyperbolic diffeomorphisms with one dimensional nonhyperbolic central direction. We also obtain applications for nonhyperbolic homoclinic classes.

Key words and phrases:
Birkhoff average, ergodic measure, Lyapunov exponent, nonhyperbolic measure, partial hyperbolicity, transitivity
2000 Mathematics Subject Classification:
37D25, 37D35, 37D30, 28D99
This research has been supported [in part] by CAPES - Ciência sem fronteiras, CNE-Faperj, and CNPq-grants (Brazil), EU Marie-Curie IRSES “Brazilian-European partnership in Dynamical Systems” (FP7-PEOPLE-2012-IRSES 318999 BREUDS), and Fondecyt project 1140202 (Chile). The authors acknowledge the hospitality of PUC-Rio and IMB. LJD thanks the hospitality and support of ICERM - Brown University during the thematic semester “Fractal Geometry, Hyperbolic Dynamics, and Thermodynamical Formalism”.

1. Introduction

1.1. Motivation and general setting

This work is a part of a long-term project to attack the following general problem which rephrases the opening question in [GIKN] from a different perspective: To what extent does ergodic theory detect the nonhyperbolicity of a dynamical system?

More precisely, we say that a diffeomorphism is nonhyperbolic if its non-wandering set is nonhyperbolic. We aim to know if such possesses nonhyperbolic ergodic measures (i.e. with some zero Lyapunov exponent) and if some of them fully reflect the nonhyperbolic behaviour of . For instance, we would like to know

  • what is their support,

  • what is their entropy, and

  • how many Lyapunov exponents of the measures are zero.

In this generality, the answer to this question is negative. There are simple examples of (even analytic) nonhyperbolic dynamical systems whose invariant measures are all hyperbolic and even with Lyapunov exponents uniformly far from zero, see for instance the logistic map or the surgery examples in [BBS] where a saddle of a uniformly hyperbolic set is replaced by non-uniformly hyperbolic sets, among others (more examples of different nature can be found in [CLR, LOR]). Nevertheless, these examples are very specific and fragile. Thus, one hopes that the “great majority” of nonhyperbolic systems have nonhyperbolic ergodic measures which detect and truly reflect the nonhyperbolic behaviour of the dynamics.

Concerning this sort of questions, a first wave of results, initiated with [GIKN], continued in [DG, BDG], and culminated in [CCGWY], show that the existence of nonhyperbolic ergodic measures for nonhyperbolic dynamical systems is quite general in the -setting: for -generic diffeomorphisms, every nonhyperbolic homoclinic class supports a nonhyperbolic ergodic measure, furthermore under quite natural hypotheses the support of the measure is the whole homoclinic class111See [CCGWY, Main Theorem] and also [CCGWY, Theorem B and Proposition 1.1]. This last result states that the support of the nonhyperbolic measure in a nonhyperbolic homoclinic class of a saddle is the whole homoclinic class. This result requires neither that the stable/unstable splitting of the saddle extends to a dominated splitting on the class (compare with [BDG]) nor that the homoclinic class contains saddles of different type of hyperbolicity (compare with [DG])..

Given a periodic point of a diffeomorphism denote by the unique -invariant measure supported on the orbit of . We say that such a measure is periodic. The previous works follow the strategy of periodic approximations in [GIKN] for constructing a nonhyperbolic ergodic measure as weak limits of periodic measures supported on orbits of hyperbolic periodic points with decreasing “amount of hyperbolicity”. The main difficulty is to obtain the ergodicity of the limit measure. [GIKN] provides a criterion for ergodicity summarised in rough terms as follows. Each periodic orbit consists of two parts: a “shadowing part” where closely shadows the previous orbit and a “tail” where the orbit is far from the previous one. To get an ergodic limit measure one needs some balance between the “shadowing” and the “tail” parts of the orbits. The “tail part” is used to decrease the amount of hyperbolicity of a given Lyapunov exponent (see [GIKN]) and also to spread the support of the limit measure, (see [BDG]).

Nonhyperbolic measures seem very fragile as small perturbations may turn the zero Lyapunov exponent into a nonzero one. However, in [KN] there are obtained (using the method in [GIKN]) certain -open sets of diffeomorphisms having nonhyperbolic ergodic measures. Bearing this result in mind, it is natural to ask if the existence of nonhyperbolic measures is a -open and dense property in the space of nonhyperbolic diffeomorphisms. In this direction, [BBD2, Theorem 4] formulates an abstract criterion called control at any scale222This construction also involves the so-called flip-flop families, we will review these notions below as they play an important role in our constructions. that leads to the following result (see [BBD2, Theorems 1 and 3]): The -interior of the set of diffeomorphisms having a nonhyperbolic ergodic measure contains an open and dense subset of the set of -diffeomorphisms having a pair of hyperbolic periodic points of different indices robustly in the same chain recurrence class.

The method in [BBD2] provides a partially hyperbolic invariant set with positive topological entropy whose central Lyapunov exponent vanishes uniformly. This set only supports nonhyperbolic measures and the existence of a measure with positive entropy is a consequence of the variational principle for entropy [W]. A con of this method is that the “completely” nonhyperbolic nature of the (obtained) set where a Lyapunov exponent vanishes uniformly prevents the measures to have full support in nonhyperbolic chain recurrence classes. This shows that, in some sense, the criterion in [BBD2] may be “too demanding” and “rigid”.

The aim of this paper is to introduce a new criterion that relaxes the “control at any scale criterion” and allows to get nonhyperbolic measures with “full support” (in the appropriate ambient space: homoclinic class, chain recurrence class, the whole manifold, according to the case). To be a bit more precise, given a point and a diffeomorphism consider the empirical measures , , associated to defined as the averages of the Dirac measures in the orbit segment ,

(1.1)

The criterion in this paper, called control at any scale with a long sparse tail with respect to a continuous map of a point , allows to construct ergodic measures with full support (in the appropriate ambient space) and a prescribed average with respect to , see Theorem 1. This construction involves two main aspects of different nature: density of the orbits of -generic points and control of averages. The existence of ergodic measures satisfying both properties is a consequence of the construction.

A specially interesting case occurs when the map is the derivative of a diffeomorphism with respect to a continuous one-dimensional center direction (taking positive and negative values). In such a case we get that every measure that is a weak limit of a sequence of empirical measures of is such that -almost every point has a zero Lyapunov exponent and a dense orbit (in the corresponding ambient space), see Theorems 7 and 8.

To state more precisely the dynamical consequences of the criterion let us introduce some notation (the precise definitions can be found below). In what follows we consider a boundaryless Riemannian compact manifold and the following two -open subsets of diffeomorphisms:

  • The set of all robustly transitive diffeomorphisms333A diffeomorphism is called transitive if it has a dense orbit. The diffeomorphism is -robustly transitive if -nearby diffeomorphisms are also transitive. with a partially hyperbolic splitting with one-dimensional (nonhyperbolic) center,

  • The set defined as the -interior of the set of -diffeomorphisms having a nonhyperbolic ergodic measure with full support in .

As an application of our criterion we get that set is -open and -dense in , see Theorem 9. We also get semi-local versions of this result formulated in terms of nonhyperbolic homoclinic classes or/and chain recurrence classes, see Theorems 7 and 8. These results turn the -generic statements in [BDG] into -open and -dense ones. We observe that a similar result involving different methods was announced in [BZ]444The construction in [BZ] combines the criteria of periodic approximations in [GIKN] and of the control at any scale in [BBD2] and a shadowing lemma by Gan-Liao, [G].. Applications of the criterion in hyperbolic-like contexts, as for instance full shifts and horseshoes, are discussed in Section 1.4.

In this paper we restrict ourselves to the control of the support and the averages of the measures, omitting questions related to the entropy of these measures. Nevertheless it seems that our method is well suited to construct nonhyperbolic ergodic measures with positive entropy and full support. This is the next step of an ongoing project whose ingredients involve tools of a very different nature beyond the scope of this paper.

In the dynamical applications we focus on partially hyperbolic diffeomorphisms with a one-dimensional center bundle and therefore the measures may have at most one zero Lyapunov exponent. Here we do not consider the case of higher dimensional central bundles and the possible occurrence of multiple zero exponents. Up to now, there are quite few results on multiple zero Lyapunov exponents. The simultaneous control of several exponents is much more difficult, essentially due to the non-commutativity of for . We refer to [BBD] for examples of ( and ) robust existence of ergodic measures with multiple zero exponents in the context of iterated function systems. Recently, [WZ] announces the locally -generic vanishing of several Lyapunov exponents in homoclinic classes of diffeomorphisms.

We now describe our methods and results in a more detailed way.

1.2. A criterion for controlling averages of continuous maps

Consider a compact metric space , a homeomorphism defined on , and a continuous map . Given a point consider the set of empirical measures associated to defined as in (1.1). Consider the following notation for finite Birkhoff averages of ,

(1.2)

and limit averages of

(1.3)

if such a limit exists.

Consider a measure that is a weak limit of empirical measures of and a subsequence with in the weak topology. The convergence of the sequence of Birkhoff averages to some limit implies that . But since may be non-ergodic this does not provide any information about the Birkhoff averages of -generic points . We aim for a criterion guaranteeing that -generic points have the same limit average as . Naively, in [BBD2] the way to get this property is to require that “all large orbit intervals of the forward orbit of have average close to the limit average (say) ”. This was formalised in the criterion control of Birkhoff averages at any scale of a point with respect to a map in [BBD2]. This criterion implies that there are sequences of times and of “errors” such that every orbit interval with length of the forward orbit of has -Birkhoff average in . When -Birkhoff averages are controlled at any scale then the -Birkhoff averages of any -limit point of converge uniformly to (see [BBD2, Lemma 2.2]).

To get a limit measure whose support is the whole ambient space the requirement “all long orbit intervals satisfy the limit average property” is extremely restrictive. Roughly, in the criterion in this paper we only require that “most of large orbit intervals of the forward orbit of have average close to the limit average ”. Let us explain a little more precisely this rough idea.

If the limit measure has full support then the orbit of the point must necessarily visit “all regions” of the ambient space and these visits require an arbitrary large time. Moreover, to get limit measures whose generic points have dense orbits in the ambient space these “long visits” must occur with some frequency. During these long visits the control of the averages can be lost.

To control simultaneously Birkhoff averages and support of the limit measure, one needs some “balance” between the part of the orbit where there is a “good control of the averages” and the part of the orbit used for spreading the support of the measure to get its density (roughly, these parts play the roles of the “shadowing” and “tail parts” of the method in [GIKN]). The criterion in this paper formalizes an abstract notion for this balance that we call control at any scale with a long sparse tail with respect to and (see Definitions 2.10 and 2.11). Our main technical result is that this criterion provides ergodic measures having simultaneously a prescribed average and a prescribed support.

Theorem 1.

Let be a compact metric space, a homeomorphism, and a continuous map. Consider

  • a point that is controlled at any scale with a long sparse tail with respect and and

  • a measure that is a weak limit of the sequence of empirical measures of .

Then for -almost every point the following holds:

  1. the forward orbit of for is dense in and

  2. .

In particular, these two assertions hold for almost every ergodic measure of the ergodic decomposition of .

We now exhibit some dynamical configurations where the criterion holds. Indeed, we see that such configurations are quite “frequent”.

1.3. Flip-flop families with sojourns: control at any scale with a long sparse tail

To present a mechanism providing orbits controlled at any scale we borrow the following definition from [BBD2]:

Definition 1.1 (Flip-flop family).

Let be a compact metric space, a homeomorphism, and a continuous function.

A flip-flop family associated to and is a family of compact subsets of such that there are and a sequence of numbers , and as , such that:

  1. for every (resp. ) and every it holds (resp. );

  2. for every , there are sets and contained in ;

  3. for every and every family of sets , with it holds

    for every and every pair of points .

We call plaques555We pay special attention to the case when the sets of the flip-flop family are discs tangent to a strong unstable cone field. This justifies this name. the sets of the flip-flop family .

With the notation in Definition 1.1, [BBD2, Theorem 2.1] claims that for every number and every set there is a point whose orbit is controlled at any scale for the function . Hence the Birkhoff average of along the orbit of any point is . Furthermore, the -limit set of has positive topological entropy.

Since we aim to obtain measures with full support we need to relax the control of the averages. For that we introduce a “sojourn condition” for the returns of the sets of the flip-flop family (item (a) in the definition below). These“sojourns” will be used to get dense orbits and to spread the support of the measures and play a role similar to the “tails” in [GIKN].

Definition 1.2 (Flip-flop family with sojourns).

Let be a compact metric space, a compact subset of , a homeomorphism, and a continuous function.

Consider a flip-flop family associated to and . We say that the flip-flop family has sojourns along (or that sojourns along ) if for every there is an integer such that every plaque contains subsets such that:

  1. for every the orbit segment is -dense in (i.e., the -neighbourhood of the orbit segment contains );

  2. and ;

  3. for every and every pair of points or it holds

    where is a sequence as in Definition 1.1.

The conditions in Definition 1.2 are depicted in Figure 1.

\setkeys Ginscale=.60, \OVP@calc
Figure 1. A flip-flop family with sojourns: the plaques , , and (the plaque is omitted for visual simplicity).

Next theorem corresponds to [BBD2, Theorem 2.1] in our setting:

Theorem 2.

Let be a compact metric space, a compact subset of , a homeomorphism, and a continuous function. Consider a flip-flop family associated to and having sojourns along .

Then every plaque contains a point that is controlled at any scale with a long sparse tail with respect to and .

As a corollary of Theorems 1 and 2 we get (recall the notation for Birkhoff limits in (1.3)):

Corollary 3.

Under the hypotheses of Theorem 2 and with the same notation, any measure that is a weak limit of the empirical measures satisfies the following properties:

  • the orbit of -almost every point is dense in and

  • for -almost every point it holds .

As a consequence, almost every measure in the ergodic decomposition of has full support in and satisfies .

We now explore some consequences of the results above.

1.4. Birkhoff averages in homoclinic classes

An important property of our methods is that they can be used in nonhyperbolic and non-Markovian settings. We now present two applications of our criteria in the “hyperbolic” setting of a mixing sub-shift of finite type that are, as far as we are aware, unknown. The key point of Proposition 4 is that it only requires continuity of the potential . When the potential is Hölder continuous this sort of result is well-known666For instance, techniques from multifractal analysis provide the following: Given a Hölder continuous function , there is a parametrised family of Gibbs states , , where are as above, such that . Each has full support and positive entropy. The conclusion in this statement is stronger than the than the one in b) as it guarantees also positive entropy. For a survey of this topic see for instance [PW]..

Proposition 4.

Let be a mixing sub-shift of finite type and a continuous function. Let and be the infimum and maximum, respectively, of over the set of -invariant probability measures (or equivalently of the Birkhoff averages along periodic orbits). Then for every the following holds:

  1. (Application of the criterion in [BBD2]) There is a -invariant compact set with positive topological entropy such that the Birkhoff average of along the orbit of any point in is .

  2. (Application of the new criterion) There is an ergodic measure with full support in such that .

This proposition deals with systems satisfying specification properties. An important property of our two criteria is that they do not involve and do not depend on specification-like properties. Indeed, they are introduced to control averages of functions in partially hyperbolic settings where specification fails. We now present an application of our criterion in settings without specification properties.

In what follows let be a boundaryless compact Riemannian manifold and the space of -diffeomorphisms endowed with the standard uniform topology. The homoclinic class of a hyperbolic periodic point of a diffeomorphism , denoted by , is the closure of the set of transverse intersection points of the stable and unstable manifolds of the orbit of . Two hyperbolic periodic points and of are homoclinically related if the stable and unstable manifolds of their orbits intersect cyclically and transversely. The homoclinic class of can also be defined as the closure of the periodic points of that are homoclinically related to . A homoclinic class is a transitive set (existence of a dense orbit) whose periodic points form a dense subset of it. Homoclinic classes are in many cases the “elementary pieces of the dynamics” of a diffeomorphism and are used to structure its dynamics, playing a similar role of the basic sets of the hyperbolic theory (indeed each basic set is a homoclinic class), for a discussion see the survey in [B].

The -index of a hyperbolic periodic point is the dimension of its unstable bundle. We analogously define -index. Two saddles which are homoclinically related have necessarily the same - and -indices. However two saddles with different indices (it is not necessary to specify the index type) may be in the same homoclinic class. In such a case the class is necessarily nonhyperbolic. Indeed, the property of a homoclinic class containing saddles of different indices is a typical feature in the nonhyperbolic dynamics studied in this paper (see also [S, M1, BD1]).

The next result is a generalisation of the second part of Proposition 4 to a non-necessarily hyperbolic context, observe that we do not require hyperbolicity of the homoclinic class. Recall that if is a periodic point of we denote by the -invariant probability supported on the orbit of .

Theorem 5.

Let be a -diffeomorphism defined on a boundaryless compact manifold and a continuous function. Consider a pair of hyperbolic periodic points and of that are homoclinically related and satisfy

Then for every there is an ergodic measure whose support is the whole homoclinic class and satisfies

Note that the hypotheses in the theorem are -open. Observe that the difficulty in the theorem is to get simultaneously the three properties ergodicity, prescribed average, and full support. It is easier (and also known) to build measures satisfying simultaneously only two of these properties.

We also aim to apply the criterion in Theorem 1 to saddles and that have different indices and are in the same homoclinic class (or, more generally, chain recurrence class) and thus the saddles are not homoclinically related.

Before stating the next corollary let us recall the definition of a chain recurrence class. Given , a finite sequence of points is an -pseudo-orbit of a diffeomorphism if for every (here denotes the distance in ). A point is chain recurrent for if for every there is an -pseudo-orbit with . The chain recurrent set of , denoted by , is the union of the chain recurrent points of . The chain recurrence class of a point is the set of points such that for every there are -pseudo-orbits joining to and to . Two chain recurrence classes are either disjoint or equal. Thus the set is the union of pairwise disjoint chain recurrence classes. Let us observe that two points in the same homoclinic class are also in the same chain recurrence class (the converse is false in general, although -generically homoclinic classes and chain recurrence classes of periodic points coincide, see [BC]). Thus if is a hyperbolic periodic point then .

Corollary 6.

Let be a boundaryless compact manifold and be a -open set in such that every has a pair of hyperbolic periodic orbits and of different indices depending continuously on whose chain recurrence classes are equal. Let be a continuous function such that

Then there are two -open sets and whose union is -dense in such that every (resp. ) has an ergodic measure whose support is the homoclinic class (resp. ) and satisfies .

Note that the saddles in the corollary cannot be homoclinically related and hence Theorem  5 cannot be applied. We bypass this difficulty by transferring the desired averages to pairs of homoclinically related periodic points (then the proof follows from Theorem 5), see Section 5.3 for the proof of the corollary.

Remark 1.3.

By [BDPR, Theorem E], if in Corollary 6 we assume that the chain recurrence class is partially hyperbolic with one-dimensional center (see definition below) then there is a -open and dense subset of such that for all . Without this extra hypothesis the equality of the homoclinic classes is only guaranteed for a residual subset of , see [BC].

1.5. Nonhyperbolic ergodic measures with full support

In what follows we focus on partially hyperbolic diffeomorphisms with one-dimensional center. Our aim is to get results as above when is the “logarithm of the center derivative”. This will allow us to obtain nonhyperbolic ergodic measures with large support in quite general nonhyperbolic settings. Before going to the details we need some definitions.

Given a diffeomorphism we say that a compact -invariant set is partially hyperbolic with one-dimensional center if there is a -invariant dominated777A -invariant splitting is dominated if there are constants and such that for all and . In our case domination means that the bundles and are both dominated, where and . splitting with three non-trivial bundles

(1.4)

such that is uniformly expanding, has dimension , and is uniformly contracting. We say that and are the strong unstable and strong stable bundles, respectively, and that is the central bundle. We denote by and the dimensions of and , respectively.

Given an ergodic measure of a diffeomorphism the Oseledets’ Theorem gives numbers , the Lyapunov exponents, and a -invariant splitting , the Oseledets’ splitting, where , with the following property: for -almost every point

If the measure is supported on a partially hyperbolic set with one-dimensional center as above then

and . Let , we say that is the central exponent of . In this partially hyperbolic setting the logarithm of the center derivative map

(1.5)

is well defined and continuous, therefore the central Lyapunov exponent of the measure is given by the integral

This equality allows to use the methods in the previous sections to construct and control nonhyperbolic ergodic measures.

Let us explain some relevant points of our study. A (new) difficulty, compared with Theorem 5, is that the logarithm of the center derivative cannot take values with different signs at homoclinically related periodic points (by definition, such points have the same indices and thus the sign of is the same). To recover this signal property we consider chain recurrence classes containing saddles of different indices.

Theorem 7.

Let be a boundaryless compact manifold and a -open set of such that every has hyperbolic periodic orbits and such that:

  • they have different indices and depend continuously on ,

  • their chain recurrence classes and are equal and have a partially hyperbolic splitting with one-dimensional center.

Then there is a -open and dense subset such that every diffeomorphism has a nonhyperbolic ergodic measure whose support is the homoclinic class .

Let us first observe that Theorem 7 can be rephrased in terms of robust cycles instead of periodic points in the same chain recurrence class. For that we need to review the definition of a robust cycle. Recall that a hyperbolic set of has a well defined hyperbolic continuation for every close to . Two transitive hyperbolic basic sets and of a diffeomorphism have a -robust (heterodimensional) cycle if these sets have different indices and if there is a -neighbourhood of such that for every the invariant sets of and intersect cyclically. As discussed in [BBD2], the dynamical scenarios of “dynamics with -robust cycles” and “dynamics with chain recurrence classes containing -robustly saddles of different indices” are essentially equivalent (they coincide in a -open and dense subset of ).

We now describe explicitly the open and dense subset of in Theorem 7 using dynamical blenders and flip-flop configurations introduced in [BBD2], see Remark 1.4. Naively, a dynamical blender is a hyperbolic and partially hyperbolic set together with a strictly invariant family of discs (i.e., the image of any disc of the family contains another disc of the family) almost tangent to its strong unstable direction, see Definition 6.3. In very rough terms, a flip-flop configuration of a diffeomorphism and a continuous function is a -robust cycle associated to a hyperbolic periodic point and a dynamical blender such that is bigger than in the blender and smaller than on the orbit of . Important properties of flip-flop configurations are their -robustness, that they occur -open and densely in the set in Theorem 7, and that they yield flip-flop families. The latter allows to apply our criterion for zero averages. The set in Theorem 7 is described in the remark below.

Remark 1.4 (The set in Theorem 7).

The set is the subset of of diffeomorphisms with flip-flop configurations “containing” the saddle .

To state our next result recall that a filtrating region of a diffeomorphism is the intersection of an attracting region and a repelling region of . Let be a filtrating region of endowed with a strictly forward invariant unstable cone field of index and a strictly backward invariant cone field of index , see Section 6.1.2 for the precise definitions. Then the maximal -invariant set in has a partially hyperbolic splitting , with . As above this allows us to define the logarithm of the center derivative of . We have the following “variation” of Theorem 7.

Theorem 8.

Let be a boundaryless compact manifold. Consider with a a filtrating region endowed with a strictly -invariant unstable cone field of index and a strictly -invariant cone field of index .

Assume that has a flip-flop configuration associated to a dynamical blender and a hyperbolic periodic point both contained in .

Then there is a -neighbourhood of such that every has a nonhyperbolic ergodic measure whose support is the whole homoclinic class of the continuation of .

The hypothesis in this theorem imply that the blender and the saddle in the flip-flop configuration are in the same chain recurrence class. With the terminology of robust cycles, they have a -robust cycle.

Note that Theorem 8 is not a perturbation result: it holds for every diffeomorphism with such a flip-flop configuration. Moreover, and more important, the hypotheses in Theorem 8 are open (the set is also a filtrating set for every sufficiently close to , hence the homoclinic class is contained in and partially hyperbolic, and flip-flop configurations are robust). Thus Theorem 8 holds for the homoclinic class of the continuation of for diffeomorphisms close to .

Remark 1.5.

Theorem 8 does not require the continuous variation of the homoclinic class with respect to . Note also that, in general, homoclinic classes only depend lower semi-continuously on the diffeomorphism. As a consequence, the partial hyperbolicity of a homoclinic class is not (in general) a robust property. The relevant assumption is that the homoclinic classes are contained in a partially hyperbolic filtrating neighbourhood which guaranteed the robust partial hyperbolicity of the homoclinic class.

We can change the hypotheses in the theorem, omitting that is a filtrating neighbourhood and considering homoclinic classes depending continuously on the diffeomorphism (this occurs in a residual subset of diffeomorphisms). Then, by continuity, the class is robustly contained in the partially hyperbolic region and we can apply the previous arguments.

1.6. Applications to robustly nonhyperbolic transitive diffeomorphisms

A diffeomorphism is transitive if it has a dense orbit. The diffeomorphism is -robustly transitive if any diffeomorphism that is -close to is also transitive. In other words, a diffeomorphism is -robustly transitive if it belongs to the -interior of the set of transitive diffeomorphisms.

We denote by the (-open) subset of consisting of diffeomorphisms such that:

  • is robustly transitive,

  • has a pair of hyperbolic periodic points of different indices,

  • has a partially hyperbolic splitting , where is uniformly expanding, is uniformly contracting, and is one-dimensional.

Note that the last condition implies that the hyperbolic periodic points of have either -index or . Note also that our assumptions imply that (in lower dimensions , see [PS]).

In dimension and depending on the type of manifold , the set contains interesting examples. Chronologically, the first examples of such partially hyperbolic robustly transitive diffeomorphisms were obtained in [S] considering diffeomorphisms in obtained as skew products of Anosov diffeomorphisms on and derived from Anosov on ( stands for the -dimensional torus). Later, [M1] provides examples in considering derived from Anosov diffeomorphisms. Finally, [BD1] gives examples that include perturbations of time-one maps of transitive Anosov flows and perturbations of skew products of Anosov diffeomorphisms and isometries.

Theorem 9.

There is a -open and dense subset of such that every has an ergodic nonhyperbolic measure whose support is the whole manifold .

Let us mention some related results. First, by [BBD2], there is a -open and dense subset of formed by diffeomorphisms with an ergodic nonhyperbolic measure with positive entropy, but the support of these measures is not the whole ambient. By [BDG], there is a residual subset of of diffeomorphism with an ergodic nonhyperbolic measure with full support. Finally, a statement similar to our theorem is stated in [BZ], see Footnote 4.

Recall that given a periodic point of the measure is the unique -invariant measure supported on the orbit of .

Corollary 10.

Consider a continuous map . Suppose that has two hyperbolic periodic orbits and such that

Then there are a -neighbourhood of and a -open and dense subset of such that every has an ergodic measure with full support on such that

Remark 11.

By [C, Proposition 1.4], for diffeomorphisms in every hyperbolic ergodic measure is the weak limit of periodic measures supported on points whose orbits tend (in the Hausdorff topology) to the support of the measure . Thus, Corollary 10 holds after replacing the hypothesis by the existence of two hyperbolic ergodic measures and such that .

1.7. Organization of the paper

In Section 2 we introduce the concepts involved in the criterion of control at any scale with a long sparse tail and prove Theorem 1. In Section 3 we introduce the notion of a pattern and see how they are induced by long tails of scales. We study the concatenations of plaques of flip-flop families (associated to a map ) and the control of the averages of corresponding to these concatenations, see Theorem 3.9. In Section 4 we prove Theorem 2, Corollary 3, and Proposition 4. In Section 5 we prove Theorem 5 involving flip-flop families and homoclinic relations. In Section 6 we review some key ingredients as dynamical blenders and flip-flop configurations and prove Theorems 7 and 8. Finally, in Section 7 we apply our methods to construct nonhyperbolic ergodic measures with full support for some robustly transitive diffeomorphisms, proving Theorem 9 and Corollary 10.

2. A criterion for zero averages: control at any scale up to a long sparse tail

The construction that we present for controlling averages is probably too rigid but it is enough to achieve our goals and certain constraints perhaps could be relaxed. However, at this state of the art, we do not aim for full generality but prefer to present the ingredients of the construction in a simple as possible way. One may aim to extract a general conceptual principle behind the construction, but this is beyond the focus of this paper. In Sections 2.1 and 2.2 we introduce the concepts involved in the criterion for controlling averages and in Section 2.3 we prove Theorem 1.

2.1. Scales and long sparse tails

In what follows we introduce the definitions of scales and long sparse tails.

Definition 2.1 (Scale).

A sequence of strictly positive natural numbers is called a scale if there is a sequence (the sequence of factors of the scale) of natural numbers with for every such that

  • for every ;

  • .

We assume that the number , and hence every , is a multiple of .

We now introduce some notation. In what follows, given we let

Given a subset of a component of is an interval of integers such that and .

Definition 2.2 (Controling sequence).

Let be a sequence of positive numbers converging to . We say that is a controlling sequence if

Remark 2.3.

For a sequence of numbers with one has

Remark 2.4.

Let be a scale, , and . Then the sequence is a controlling one.

Definition 2.5 (Long sparse tail).

Consider a scale and a controlling sequence . A set is a -long -sparse tail if the following properties hold:

  1. Every component of is of the form , for some and (we say that such a component has size ).

Let be the union of the components of of size and let

the union of the components of of size larger than or equal to .

  1. , in particular

  2. Consider an interval of natural numbers of the form

    that is not contained in any component of then the following properties hold:

    • center position:

    • -sparseness:

The conditions in Definition 2.5 are depicted in Figure 2.

\setkeys Ginscale=.75, \OVP@calc-1
Figure 2. A long sparse tail
Definition 2.6 (Good and bad intervals).

With the notation of Definition 2.5, an interval of the form is called -bad if . The interval is called -good if .

Remark 2.7 (On the definition of a -long -sparse tail).

  1. It is assumed that . This implies that for every the initial interval is not a component of of size . Therefore is disjoint from and thus from for every . In other words, the interval is -good, that is,

  2. Let be an interval as in Item (c) of Definition 2.5. By Item (a) the interval is either contained in a component of whose size is larger than or equal to or is disjoint from . Thus, Item (c) considers the case where the interval is disjoint from .

    Now any component of of size less than is either disjoint from or contained in : just note that such a component has length , , and starts at a multiple of and is a multiple of .

    Item (c) describes the position and quantity of the components of size in the interval . For that, one splits the interval into tree parts of equal length . The following properties are required:

    • Every component of size contained in is contained in the middle third interval;

    • The middle third interval contains at least one component of size . Thus the intersection is not empty, but the the set has a small density in the interval that is upper bounded by .

  3. Item (c) does not consider the case . For and an interval of the form there are two possibilities: either is a component of (i.e., contained in ) or is disjoint from .

  4. Given any interval of the form there are two possibilities:

    • either and then and is -bad;

    • or and then is -good.

The definition of a long sparse tail involves many properties and conditions, thus its existence it is not obvious. We solve this difficulty in the next lemma.

Lemma 2.8 (Existence of long sparse tails).

Consider a scale and its sequence of factors . Write and let . Then there is a -long -sparse tail .

Proof.

First note that, by Remark 2.4, the sequence is a controlling one.

The construction of the set is done inductively. For each we define the intersection of the set with the intervals . We denote such an intersection by .

For , we let . Fix now and suppose that the sets has been constructed satisfying (in restriction to the interval ) the properties in Definition 2.5. We now proceed to define the set .

For any we denote by the union of the components of of length . We next define the family of subsets of by decreasing induction on as follows. We let

Let and assume that the sets are defined for every . The set is defined as follows:

  • if then

  • Otherwise we let

    (2.1)

Note that by construction,

Claim 2.9.

The set satisfies (in restriction to the interval ) the conditions of Definition 2.5.

Proof.

Property (a) in the definition follows from the construction: the components of have size and have no adjacent points with the components of .

For Property (b) one checks inductively that for every and .

Property (c) is a consequence of (2.1). If the set intersects a segment then it is contained in its middle third interval, implying the center position condition. For the sparseness note that by construction and the definition of , for each it holds

This completes the proof of the claim. ∎

Our construction also provides immediately the following properties: For every it holds:

  • if then ,

  • if the , and

  • .

The tail is now defined by

(2.2)

By construction, the set is an -sparse tail of . ∎

2.2. Control at any scale up a long sparse tail

In this section we give the definition of controlled points.

Definition 2.10.

Let be a compact set, a homeomorphism, and a continuous map. Consider

  • a scale , a controlling sequence , and a -long -sparse tail ;

  • decreasing sequences of positive numbers and , converging to .

The -orbit of a point is -dense along the tail if for every component of of length the segment of orbit is -dense in .

The Birkhoff averages of along the orbit of are -controlled for the scale with the tail if for every interval

such that (i.e., is either -good or is a component of ) it holds

Definition 2.11.

Let be a compact set, a homeomorphism, and a continuous map.

A point is controlled at any scale with a long sparse tail with respect to and if there are a scale , a controlling sequence , a -long -sparse tail , and sequences of positive numbers and converging to , such that

  • the -orbit of is -dense along the tail and

  • the Birkhoff averages of along the orbit of are -controlled for the scale with the tail .

In this definition we say that is the density forcing sequence, is the average forcing sequence, and the point is -controlled.

2.3. Proof of Theorem 1

In this section we prove Theorem 1, thus we use the assumptions and the notations in its statement. Consider a point that is controlled at any scale with a long sparse tail for and . Let

  • be the scale;

  • the -long -sparse tail; and

  • the density forcing sequence and the average forcing sequence.

Let be a measure that is a weak limit of the empirical measures . As remains fixed let us write . We need to prove that for -almost every point it holds:

  1. the forward orbit of is dense in

  2. the Birkhoff averages of satisfy