Nonuniform hyperbolicity for -generic diffeomorphisms
We study the ergodic theory of non-conservative -generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of -generic diffeomorphisms are nonuniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set of any -generic diffeomorphism exhibits many ergodic hyperbolic measures whose supports coincide with the whole set
In addition, confirming a claim made by R. Mañé in , we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin’s Stable Manifold Theorem, even if the diffeomorphism is only .
Keywords: dominated splitting, nonuniform hyperbolicity, generic dynamics, Pesin theory.
MSC 2000: 37C05, 37C20, 37C25, 37C29, 37D30.
In his address to the 1982 ICM, R. Mañé [M] speculated on the ergodic properties of -generic diffeomorphisms. He divided his discussion into two parts, the first dealing with non-conservative (i.e. “dissipative”) diffeomorphisms, the second with conservative diffeomorphisms.
In the first part, drawing inspiration from the work of K. Sigmund [Si] on generic measures supported on basic sets of Axiom A diffeomorphisms, Mañé first used his Ergodic Closing Lemma [M] to show that ergodic measures of generic diffeomorphisms are approached in the weak topology by measures associated to periodic orbits (this is item (i) of Theorem 3.8 of this paper; we include a detailed proof, since Mañé did not). He then went on to prove that the Oseledets splittings of generic ergodic measures222The space of ergodic measures of a diffeomorphism is a Baire space when endowed with the weak topology, so that its residual subsets are dense; see Subsection 5.1. of generic diffeomorphisms are in fact uniformly dominated, and to claim that such conditions – uniformly dominated Oseledets splittings – together with nonuniform hyperbolicity are sufficient to guarantee the existence of smooth local stable manifolds at -a.e. point, as in Pesin’s Stable Manifold Theorem [Pe].
In the second part, discussing the case of conservative diffeomorphisms, he stated a -generic dichotomy between (some form of) hyperbolicity and an abundance of orbits with zero Lyapunov exponents. In the two-dimensional setting this reduced to a dichotomy between Anosov diffeomorphisms and those having zero exponents at almost every orbit. Mañé never published a proof of this dichotomy.
For conservative diffeomorphisms much progress has been made. The generic dichotomy between hyperbolicity and zero Lyapunov exponents for surface diffeomorphisms, in particular, was proven by Bochi [Boc1] in 2000, later extended to higher dimensions by Bochi and Viana [BocV], and finally settled in the original (symplectic, in arbitrary dimension) statement of Mañé by Bochi [Boc2] in 2007. Many other important results have been obtained for -generic conservative diffeomorphisms, see for instance [ABC, DW, HT].
By contrast, there was for a long time after Mañé’s address little progress towards the development of the ergodic theory for -generic dissipative diffeomorphisms. This is in our view due to the two following obstacles:
Obstacle 1: The Absence of Natural Invariant Measures. Conservative diffeomorphisms are endowed with a natural invariant measure, namely the volume that is preserved. In the dissipative context, hyperbolic basic sets are endowed with some very interesting invariant measures, such as the measure of maximal entropy (see [Bow]), or, in the case of hyperbolic attractors, the Sinai-Ruelle-Bowen measure (see for instance [R]). In the case of -generic dissipative diffeomorphisms, however, it is difficult to guarantee the existence of measures describing most of the underlying dynamics. For instance, Avila-Bochi [AB] have recently shown that -generic maps do not admit absolutely continuous invariant measures.
Obstacle 2: The -Generic Lack of -Regularity. For much of differentiable ergodic theory the hypothesis of differentiability is insufficient; higher regularity, usually but at least +Hölder, is required. This is the case for instance of Pesin’s Stable Manifold Theorem [Pe] for nonuniformly hyperbolic dynamics 333Obstacle 2, unlike Obstacle 1, is of course also a problem in the conservative setting..
The aim of this paper is to realize some of Mañé’s vision of an ergodic theory for non-conservative -generic diffeomorphisms. Some of our results confirm claims made without proof by Mañé; others extend Sigmund’s work to the nonhyperbolic -generic setting; and still others go beyond the scope of both of these previous works. In any case, our results begin to tackle both of the aforementioned obstacles to a generic ergodic theory. We hope that this work will help the development of a rich ergodic theory for -generic dissipative diffeomorphisms.
Our starting point is the generic geometric theory for dissipative diffeomorphisms, that is, the study from the -generic viewpoint of non-statistical properties: transitivity, existence of dominated splittings, Newhouse phenomenon (coexistence of an infinite number of periodic sinks or sources)…There has been, especially since the mid-’s, an explosion of important generic geometric results, thanks largely to Hayashi’s Connecting Lemma [H]. It turns out, however, that many of these tools – especially from [ABCDW], [BDP], and [BDPR] – are also useful for the study of generic ergodic problems. Our results on generic ergodic theory follow largely from the combined use of these geometric tools with techniques by Sigmund and Mañé.
Some of our results hold for every diffeomorphism, some require a -generic assumption. We can group them into three types:
Approximations by Periodic Measures. A classical consequence of Mañé ergodic closing lemma [M] is that, for -generic diffeomorphisms, every invariant measure is the weak limit of a convex sum of dirac measures along periodic orbits. We propose some variation on this statement, for instance:
If is a -generic diffeomorphism then
The idea is to show that, analogously with what occurs from the “geometric” viewpoint with Pugh’s General Density Theorem [Pu], generically hyperbolic periodic measures are abundant (e.g., dense) among ergodic measures, and so provide a robust skeleton for studying the space of invariant measures.
Geometric Properties of Invariant Measures. Some of our results deal with the geometric and topological aspects of the invariant measures, such as the sizes of their supports, their Lyapunov exponents and corresponding Lyapunov spaces, and the structure of their stable and unstable sets. For instance:
Let be an isolated transitive set of a -generic diffeomorphism . Then every generic measure with support contained in is ergodic, has no zero Lyapunov exponents (i.e. is nonuniformly hyperbolic) and its support is equal to (Theorem 3.5 part ( b)).
Let be an ergodic measure without zero Lyapunov exponent, and whose support admits a dominated splitting corresponding to the stable/unstable spaces of . Then there exists stable and unstable manifolds a -almost every point (Theorem 3.11).
Ergodic Properties of Invariant Measures. Finally, many of our results deal with “statistical” properties such as ergodicity and entropy of the invariant measures. For instance:
Any homoclinic class coincides with the support of an ergodic measure with zero entropy (Theorem 3.1).
Some of our results may admit extensions to or analogues in the conservative setting, but we have not explored this direction.
2.1 General definitions
Given a compact boundaryless -dimensional manifold , denote by the space of diffeomorphisms of endowed with the usual topology.
Given a diffeomorphism , a point , and a constant , then the stable set of is
and the -local stable set of is
The unstable set and the -local unstable set are defined analogously.
Given , a compact -invariant set is isolated if there is some neighborhood of in such that
A compact -invariant set is transitive if there is some whose forward orbit is dense in . A transitive set is trivial if it consists of a periodic orbit.
We denote by the orbit of a periodic point and by its period.
For we denote by its minimal dilatation.
2.2 Homoclinic classes
The Spectral Decomposition Theorem splits the nonwandering set of any Axiom A diffeomorphism into basic sets which are pairwise disjoint isolated transitive sets. They are the homoclinic classes of periodic orbits. This notion of homoclinic class can be defined in a more general setting:
Let be a hyperbolic periodic orbit of . Then
the homoclinic class of is the set
given an open set containing , the homoclinic class of relative to is the set
Although the homoclinic class is associated to the periodic orbit of , we write sometimes instead of .
Relative homoclinic classes like full homoclinic classes are compact transitive sets with dense subsets of periodic orbits. There is another characterization of homoclinic classes:
Two hyperbolic periodic points and having the same stable dimension are homoclinically related if
If we define as the set of hyperbolic periodic points that are homocliically related to , then is -invariant and its closure coincides with .
In the relative case in an open set we denote by the set of hyperbolic periodic points whose orbit is contained in and which are homoclinically related with by orbits contained in . Once more is the closure of .
2.3 Invariant measures and nonuniform hyperbolicity
The statements of many of our results involve two different types of weak hyperbolicity: nonuniform hyperbolicity and dominated splittings. We now recall the first of these two notions.
– The support of a measure is denoted by . Given a compact -invariant set of some , set
endowed with the weak topology. Then, is a compact metric space hence a Baire space.
– We denote by the set of ergodic measures . This set is a subset of (see Proposition 5.1), and hence is a Baire space.
– Given a periodic orbit of , its associated periodic measure is defined by
Given a compact -invariant set of some , set
– Given any ergodic invariant probability of a diffeomorphism of a compact manifold of dimension the Lyapunov vector of denoted by is the -uple of the Lyapunov exponents of , with multiplicity, endowed with an increasing order.
An ergodic measure is nonuniformly hyperbolic if the Lyapunov exponents of -a.e. are all non-zero. The index of a nonuniformly hyperbolic measure is the sum of the dimensions of Lyapunov spaces corresponding to its negative exponents.
A measure is uniformly hyperbolic if is a hyperbolic set.
– Given a nonuniformly hyperbolic measure then its hyperbolic Oseledets splitting, defined at -a.e. , is the -invariant splitting given by
where is the Lyapunov space corresponding to the Lyapunov exponent at .
– A point is called irregular for positive iterations (or shortly irregular) if there is a continuous function such that the sequence is not convergent. A point is Lyapunov irregular if the Lyapunov exponents of are not well-defined for positive iteration. Irregular and Lyapunov irregular points are defined analogously, considering negative iterates instead.
A point is regular if it is regular and regular and if furthermore the positive and negative average of any given continuous function converge to the same limit.
2.4 Dominated splitting
We recall the definition and some properties of dominated splittings (see [BDV, Appendix B]).
A -invariant splitting of the tangent bundle over an -invariant set is dominated if there exists such that given any , any unitary vectors and , then
This will be denoted by .
More generally, a -invariant splitting of the tangent bundle is a dominated splitting if given any then the splitting
is dominated. A dominated splitting is non-trivial if contains at least two non-empty bundles.
If an invariant set admits a dominated splitting , then:
the splitting varies continuously with the point ;
the splitting extends to a dominated splitting (also denoted by ) over the closure of ;
there is a neighborhood of such that every -invariant subset of admits a dominated splitting with for each .
There always exists a (unique) finest dominated splitting over , characterized by the following property: given any dominated splitting over then there is some such that
That is, the finest dominated splitting is minimal in the sense that every dominated splitting over can be obtained by bunching together bundles of the finest dominated splitting. Equivalently, each of the bundles of the finest dominated splitting is indecomposable, in the sense that there exist no subbundles and such that and
is a dominated splitting. Roughly speaking: “there is no domination within each ”.
The finest dominated splitting “separates Lyapunov exponents”. That is, given an ergodic measure with Oseledets splitting and corresponding Lyapunov exponents defined at -a.e. , then there are numbers such that for each
at -a.e. , where the are the bundles of the finest dominated splitting. In other words, the bundles of the finest dominated splitting can be written as sums of the Lyapunov spaces of the increasing Lyapunov exponents of . So we speak of the Lyapunov spaces and of the Lyapunov exponents “inside” each bundle . We denote by the set of Lyapunov exponents of inside the bundle ; likewise, given a Lyapunov-regular point , we denote by the set of Lyapunov exponents of inside .
2.5 Semicontinuity and genericity
Given a compact metric space, we denote by the space of compact subsets of endowed with the Hausdorff distance: given two non-empty sets , set
where denotes the -ball centered on the set . (The distance from the empty set to any non-empty set is by convention equal to .)
Then the space is itself a compact (and hence a Baire) metric space.
Given a topological space and a compact metric space , a map is
lower-semicontinuous at if for any open with , there is a neighborhood of in such that for every ;
upper-semicontinuous at if for any open containing , there is a neighborhood of in such that contains for every ;
lower-semicontinuous (resp, upper-semicontinuous) if it is lower-semicontinuous (resp, upper-semicontinuous) at every .
Now, we can state a result from general topology (see for instance [K]) which is one of the keys to most of the genericity arguments in this paper:
Given a Baire space, a compact metric space, and a lower-semicontinuous (resp, upper-semicontinuous) map, then there is a residual subset of which consists of continuity points of .
In this paper is usually either (with the topology) or else (with the weak topology), while is usually or else .
In a Baire space, a set is residual if it contains a countable intersection of dense open sets. We establish a convention: the phrases “generic diffeomorphisms (resp., measures ) satisfy…” and “every generic diffeomorphism (resp., measure ) satisfies…” should be read as “there exists a residual subset of (resp., of ) such that every (resp., every ) satisfies…”
3 The Main Results
3.1 Homoclinic classes admit ergodic measures with full support
Let be a relative homoclinic class of a diffeomorphism . Then there is a measure which
has “full support”: ;
has zero entropy: .
One intriguing consequence of Theorem 3.1 is this: given a -generic conservative diffeomorphism, then admits at least one ergodic measure whose support coincides with all of . This follows from Theorem 3.1 and the fact that for -generic conservative diffeomorphisms the manifold is a homoclinic class (see [BC]).
We think furthermore that the (-invariant) volume is approached in the weak topology by ergodic measures with full support ; we have not checked this completely, the missing ingredient is a conservative version of the Transition Property Lemma in Subsection 4.2.
3.2 Generic ergodic measure of -generic diffeomorphisms
Methods similar to those used in the proof of Theorem 3.1’ yield an analogous result in the wider space of ergodic measures:
Given a -generic diffeomorphism then every generic measure in
has zero entropy: ;
is nonuniformly hyperbolic and its Oseledets splitting is dominated.
The splitting above is trivial when is supported on a periodic sink or source.
3.3 Isolated transitive sets of -generic diffeomorphisms
Isolated transitive sets are natural generalizations of hyperbolic basic sets. Bonatti-Diaz [BD] used Hayashi’s Connecting Lemma [H] to show that every isolated transitive set of a -generic diffeomorphisms is a relative homoclinic class (see also [Ab]). Though at this point it is not known whether every generic diffeomorphism exhibits some isolated transitive set, there are several examples of locally generic diffeomorphisms having some non-hyperbolic isolated transitive sets, for instance nonhyperbolic robustly transitive sets and diffeomorphisms.
Theorem 3.5 below presents a overview of -generic properties satisfied by measures contained in an isolated transitive set.
Let be an isolated non-trivial transitive set of a -generic diffeomorphism and let be the finest dominated splitting over . Then
The set of periodic measures supported in is a dense subset of the set of invariant measures supported in .
For every generic measure ,
has full support: ;
has zero entropy: ;
for -a.e. point the Oseledets splitting coincides with ;
is nonuniformly hyperbolic.
There exists a dense subset of such that every ,
has positive entropy: ;
is uniformly hyperbolic.
The conclusion of Theorem 3.5 does not apply to isolated transitive sets of arbitrary diffeomorphisms: consider for example a normally hyperbolic irrational rotation of the circle inside a two-dimensional manifold.
Recently Díaz and Gorodetski [DG] have shown that non-hyperbolic homoclinic classes of -generic diffeomorphisms always support at least one ergodic measure which is not nonuniformly hyperbolic.
Theorem 3.5 parts (a) and (b) is a nonhyperbolic, -generic version of the following theorem by Sigmund on hyperbolic basic sets:
Theorem (Sigmund, 1970).
Given a hyperbolic isolated transitive set of a diffeomorphism , then the set of periodic measures in is a dense subset of the set of invariant measures in . Moreover every generic measure is ergodic, , and .
Although this was not stated by Sigmund, the statement of Theorem 3.5 part (c) applies also to the space of measures over any non-trivial hyperbolic basic set.
3.4 Approximation by periodic measures
Given an ergodic measure of a -generic diffeomorphism , there is a sequence of periodic orbits such that
the measures converge to in the weak topology;
the periodic orbits converge to in the Hausdorff topology;
the Lyapunov vectors converge to the Lyapunov vector .
As already said, the main novelty here is that, at the same time, the Lyapunov exponents of the periodic measures converge to those of the measure . Theorem 3.8 is a generic consequence of the perturbative result Proposition 6.1 which refines Mañé’s Ergodic Closing Lemma.
Consider now the finest dominated splitting supported by the ergodic measure . Then [BGV] produces perturbations of the derivative of along the orbits of the periodic orbits which make all of the exponents inside a given subbundle coincide. One deduces:
Given an ergodic measure of a -generic diffeomorphism , let be the finest dominated splitting over . Then there is a sequence of periodic orbits which converges to in the weak topology, to in the Hausdorff topology, and such that for each the Lyapunov exponents of inside converge to the mean value of the Lyapunov exponents of inside the .
We state another result which allows to approximate measures by periodic measures contained in a homoclinic class.
For any -generic diffeomorphism , any open set and any relative homoclinic class of in , the closure (for the weak topology) of the set of periodic measures supported in is convex.
In other words, every convex sum of periodic measures in is the weak limit of periodic orbits in .
3.5 -Pesin theory for dominated splittings
Theorems 3.1, 3.3, and 3.5 constitute as a group an assault on Obstacle 1. Our next result deals with Obstacle 2. Pugh has built a -diffeomorphism which is a counter-example [Pu] to Pesin’s Stable Manifold Theorem. It turns out however that Pesin’s Stable Manifold Theorem does hold for maps which are only , as long as the +Hölder hypothesis is replaced by a uniform domination hypothesis on the measure’s Oseledets splitting. This has been already done by Pliss [Pl] in the case when all the exponents are strictly negative. The difficulty for applying Pliss argument when the measure has positive and negative exponents is that we have no control on the geometry of iterated disks tangent to the stable/unstable directions. The dominated splitting provides us this control solving this difficulty.
Theorem 3.11 below is a simpler statement of our complete result stated in Section 8, where we show that Pesin’s Stable Manifold Theorem applies to ergodic nonuniformly hyperbolic measures with dominated hyperbolic Oseledets splitting .
Let be an ergodic nonuniformly hyperbolic measure of a diffeomorphism . Assume that its hyperbolic Oseledets splitting is dominated.
Then, for -a.e. , there is such that the local stable set is an embedded disc, tangent at to and contained in the stable set . Furthermore, one can choose in such a way that is a measurable map and such that the family is a measurable family of discs.
In other words:
(Nonuniform hyperbolicity) + (Uniform domination) (Pesin Stable Manifold Theorem).
We note that its statement includes no genericity assumption on or on . It has already been used, in a -generic context, to obtain results on ergodic measures of diffeomorphisms far from homoclinic tangencies [Y].
Theorem 3.11 seems to be a folklore result. Indeed, R. Mañé [M] announced this result without proof444He did provide the following one-line proof: “This follows from the results of Hirsch, Pugh, and Shub.” Since the ingredients for the proof we provide in Section 8 are all classical and were available in , we believe that Mañé did indeed know how to prove it, but never wrote the proof (possibly because at the time there was little motivation for obtaining a Pesin theory for maps which are but not +Hölder). in his ICM address. Although no one seems to have written a full proof under our very general hypotheses, some authors have used the conclusion implicitly in their work. Gan [G], for instance, uses this kind of idea to extend Katok’s celebrated result on entropy and horseshoes of surface diffeomorphisms to a -diffeomorphisms.
Let be a -generic diffeomorphism. Then for any generic ergodic measure , -a.e. exhibits a stable local manifold tangent to at as in Theorem 3.11.
Let be an isolated transitive set of a -generic diffeomorphism . Then for any generic ergodic measure , -a.e. exhibits a stable local manifold tangent to at as in Theorem 3.11.
3.6 Genericity of irregular points
Our final two results make precise some informal statements of Mañé555“In general, regular points are few from the topological point of view – they form a set of first category”. [M, Page 264] regarding the irregularity of generic points of -generic diffeomorphisms.
Given any -generic diffeomorphism there is a residual subset such that every is irregular.
This result does not hold if we replace regular points by regular points: every point in the basin of a (periodic) sink is regular. We conjecture that if one excludes the basins of sinks, generic points of -generic diffeomorphisms are irregular. Our next result is that this conjecture is true in the setting of tame diffeomorphisms666Indeed a recent result by J. Yang [Y] allows us to extend Theorem 3.15 to -generic diffeomorphisms far from tangencies: Yang announced that, in this setting, generic points belongs to the stable set of homoclinic classes..
Recall that a diffeomorphism is called tame if all its chain recurrence classes are robustly isolated (see [BC]). The set of tame diffeomorphisms is a -open set which strictly contains the set of Axiom A+no cycle diffeomorphisms. The chain recurrent set of -generic tame diffeomorphisms consist of finitely many pairwise disjoint homoclinic classes. Our result is :
If is a -generic tame diffeomorphism then there is a residual subset such that if and is not a sink, then is both irregular and Lyapunov- irregular.
3.7 Layout of the Paper
The remainder of this paper is organized as follows:
In Section 4 we prove an ergodic analogue of Pugh’s General Density Theorem which we call Mañé’s Ergodic General Density Theorem. It implies items (i) and (ii) of Theorem 3.8. We also prove a “generalized specification property” satisfied by -generic diffeomorphisms inside homoclinic classes: this gives Theorem 3.10. One deduces from these results the parts (a) and (c) of Theorem 3.5.
The authors would like to thank the following people for useful suggestions and comments: A. Avila, A. Baraviera, J. Bochi, F. Béguin, L. J. Diaz, F. Le Roux, L.Wen, J. Yang.
This paper has been prepared during visits of the authors at the Université de Paris XIII, at the Université de Bourgogne and at IMPA, which were financed by the Brazil-France Cooperation Agreement on Mathematics. The text has been finished during the Workshop on Dynamical Systems at the ICTP in july 2008. We thank these institutions for their kind hospitality.
4 Approximation of invariant measures by periodic orbits
4.1 Mañé’s Ergodic General Density Theorem
In [M] Mañé states without proof the following fact (called Mañé’s Ergodic General Density Theorem):
For any -generic diffeomorphism , the convex hull of periodic measures is dense in .
More precisely, every measure is approached in the weak topology by a measure which is the convex sum of finitely many periodic measures and whose support is arbitrarily close to .
We now prove a more precise result which corresponds to items (i) and (ii) of Theorem 3.8: the ergodic measures are approached by periodic measures in the weak and Hausdorff senses. In Section 6 we shall modify the proof in order to include also the approximation of the mean Lyapunov exponents in each bundle of the finest dominated splitting (item (iii)).
Given an ergodic measure of a -generic diffeomorphism , then for every neighborhood of in and every neighborhood of in there is some periodic measure of such that and .
A (recurrent) point of is well-closable if given any and any neighborhood of in there is some such that and moreover
for all smaller than the period of by .
That is, a point is well-closable if its orbit can be closed via a small -perturbation in such a way that the resulting periodic point “shadows” the original orbit along the periodic point’s entire orbit. Mañé proved that almost every point of any invariant measure is well-closable:
Ergodic Closing Lemma.
Given and , -a.e. is well-closable.
Birkhoff’s ergodic theorem and Mañé’s ergodic closing lemma implies:
Given and an ergodic measure , then for any neighborhoods of in and of in and any there is having a periodic orbit such that and the Hausdorff distance between and is less than .
Proof of Theorem 4.2.
We consider endowed with the product metric. The space of compact subsets of is a compact metric space when endowed with the Hausdorff distance. Consider the map , which associates to each diffeomorphism the closure of the set of pairs where is a periodic orbit of .
Kupka-Smale Theorem asserts that there is a residual set of such that every periodic orbit of is hyperbolic. Then the robustness of hyperbolic periodic orbits implies that the map is lower-semicontinuous at . Applying the Semicontinuity Lemma (see Section 2.5) to , we obtain a residual subset of (and hence of ) such that every is a continuity point of . We shall now show that each such continuity point satisfies the conclusion of Lemma 4.2:
Consider and an ergodic measure of . Fix an open neighborhood of in . We need to prove that there exists a pair in , where is a periodic orbit of . Fix now a compact neighborhood of ; it is enough to prove that .
Applying the Corollary 4.4 to , we obtain an arbitrarily small -perturbation of having a periodic orbit such that simultaneously is weak-close to and is Hausdorff-close to . With another arbitrarily small -perturbation we make hyperbolic and hence robust, while keeping close to and close to . With yet another small -perturbation , using the robustness of , we guarantee that and .
By letting tends to , using the continuity of at and the compactnes of one gets that as announced. ∎
Ergodic Decomposition Theorem.
Given a homeomorphisms of a compact metric space and , then for any neighborhood of in there are a finite set of ergodic measures and positive numbers with such that
That is, any invariant measure may be approached by finite combinations of its ergodic components.
Proof of Theorem 4.1.
Let be an invariant measure of a -generic diffeomorphism as in the statement of Lemma 4.2. Fix a neighborhood of in and a number .
Let denote : it is a neighborhood of in . By the Ergodic Decomposition Theorem applied to and there is a convex combination
of ergodic measures which belongs to and supported in . Now, by Theorem 4.2, each ergodic component is weak-approached by periodic measures of whose support is contained in the -neighborhood of and hence of .
Now the convex sum is close to for the weak topology and its support is contained in the -neigborhood of . As the support of a measure varies lower-semicontinuously with the measure in the weak topology, we get that and are close in the Hausdorff distance.
4.2 Periodic measures in homoclinic classes of -generic diffeomorphisms
Through the use of Markov partitions, Bowen [Bow] showed that every hyperbolic basic set contains periodic orbits with an arbitrarily prescribed itinerary (this is known as the specification property). So the invariant probabilities supported in are approached in the weak topology by periodic orbits in . An intermediary step for this result consists in proving that every convex sum of periodic measures in is approached by periodic orbits in . One thus defines:
A set of periodic points has the barycenter property if, for any two points , any , and , there exists and pairwise disjoint sets such that
for every and for every .
The barycenter property implies that for any two periodic points in and there is some periodic point , of very high period, which spends a portion approximately equal to of its period shadowing the orbit of and a portion equal to shadowing the orbit of . As a consequence we get:
If a set has the barycenter property then the closure of is convex.
Consider now the set of periodic points homoclinically related to a hyperbolic periodic point of an arbitrary diffeomorphism . Then is contained in an increasing sequence of basic sets contained in the homoclinic class . For this reason, it remains true that every convex sum of periodic measure with is approached by a periodic orbit in the basic set. From the transition property in [BDP], we thus have:
For any open set and any hyperbolic periodic point whose orbit is contained in , the set of periodic points related to in satisfies the barycenter property.
Proposition 4.7 does not hold a priori for the set of periodic orbits in an homoclinic class in particular in the case where contains periodic points of different indices (which thus are not homoclinically related). However when two hyperbolic periodic orbits of different indices are related by a heterodimensional cycle, [ABCDW] shows that one can produce, by arbitrarily small -perturbations, periodic orbits which spend a prescribed proportion of time shadowing the orbit of , . Furthermore, if and are robustly in the same chain recurrence class, then the new orbits also belongs to the same class. This allows one to prove that the barycenter property holds generically:
Let be a -generic diffeomorphism and a hyperbolic periodic orbit and be an open set. Then the set of periodic orbits contained in satisfies the barycenter property.
We first give the proof for whole homoclinic classes (i.e. when ). According to [BD], for every -generic diffeomorphism and every periodic point of the homoclinic classes are either equal or disjoint; furthermore, if then there is an open neighborhood of such that for every generic the homoclinic classes of the continuations of and for are equal; moreover if and if and have the same index, then they are homoclinically related. Hence the barycenter property is satisfied for pairs of point of the same index in an homoclinic class.
Hence we now assume that contains periodic points and with different indices, and we fix some number . We want to prove the barycenter property for , and . Notice that the homoclinic classes of and are not trivial and from [BC], one may assume that they coincide with . The next lemma will allow us to assume that and have all their eigenvalues real, of different modulus, and of multiplicity equal to .
Let be a -generic diffeomorphism and be a periodic point of whose homoclinic class is non-trivial. Then for every and there is a periodic point homoclinically related with , and a segment with such that:
for every one has ,
the eigenvalues of are real; have different modulus, and multiplicity equal to ;
and have the same index of .
The proof consists in considering periodic orbits of very large period shadowing the orbit of an homoclinic intersection associated to . An arbitrarily small perturbation of the derivative of such orbits produces eigenvalues that are real, have different modulus and multiplicity . As this property is an open property, the genericity assumption implies that already exhibits the announced periodic orbits, without needing perturbations. ∎
Notice that if, for every , the barycenter property is satisfied for , and , then it also holds for , and . Hence we may assume that the points and have different indices and have all their eigenvalues real, of different modulus, and of multiplicity equal to . For fixing the idea one assume . Furthermore from [BD] and this property persists for any -generic diffeomorphism close to .
The end of the proof now follows from [ABCDW]; however there is no precise statement in this paper of the result we need. For this reason we recall here the steps of the proof. First by using Hayashi connecting lemma, one creates an heterodimensional cycle associated to the points and : one has and . Then [ABCDW, Lemma 3.4] linearizes the heterodimensional cycle producing an affine heterodimensional cycle. This heterodimensional cycle [ABCDW, Section 3.2] produces, for every large , a periodic point whose orbit spends exactly times shadowing the orbit of and times shadowing the orbit of and an bounded time outside a small neighborhood of these two orbits. So, we can choose and such that the orbit of spends a proportion of time close to the orbit of which is almost and a proportion of time close to the orbit of which is almost . Furthermore, one has and- . One deduces that, for any -generic diffeomorphism in an open set close to , . Since is generic, the class for already contained periodic orbits that satisfy the barycenter property.
In the proof for relative homoclinic classes, there are several new difficulties: the relative homoclinic class of in an open set is the closure of periodic orbits in related to by orbits in , but some periodic orbit may also be contained in the closure of . Furthermore the set of open sets is not countable: hence the set of relative homoclinic classes is not countable, leading to some difficulty for performing an argument of genericity. We solved these difficulties by considering the set of periodic orbits of which are contained in . Then, if is an increasing union of open subsets then
This argument allows us to deal with a countable family of open sets , . One now argues in a very similar way as before (just taking care that all the orbits we use are contained in the open set ). ∎
4.3 Approximation of measures in isolated transitive sets
One of the main remaining open question for -generic diffeomorphisms is
Given a -generic diffeomorphism and a homoclinic class of , is dense in ? In other words, is every measure supported on approached by periodic orbits inside the class?
The fact that we are not able to answer to this question is the main reason for which we will restrict the study to isolated transitive set classes, in this section.
An argument by Bonatti-Diaz [BD], based on Hayashi Connecting Lemma, shows that isolated transitive sets of -generic diffeomorphisms are relative homoclinic classes:
[BD] Given an isolated transitive set of a -generic diffeomorphism and let be an isolating open neighborhood of , then
for some periodic orbit of .
Proof of Theorem 3.5 part (a).
Let be an isolated transitive set of a -generic diffeomorphism and be an invariant measure supported on . According to Theorem 4.1, the measure is approached in the weak topology by a measure which is the convex sum of finitely many periodic measures and whose support is arbitrarily close to .
On the other hand is the relative homoclinic class of some periodic point in some isolating open neighborhood ; as the support of is close to the support of one gets that is contained in . As is an isolated neighborhood of the measure is in fact supported in : hence it is the convex sum of finitely many periodic measures in .
As is compact and contained in it does not contain periodic orbits on the boundary of . Hence Theorem 3.10 implies that the closure of the set is convex; this implies that belongs to the closure of , ending the proof. ∎
Proof of Theorem 3.5 part (c).
Let be a (non-trivial) isolated transitive set of a -generic diffeomorphism . By Theorem 4.10, every periodic point in has homoclinic class equal to , and hence exhibits some transverse homoclinic orbit. This implies that there are hyperbolic horseshoes arbitrarily close to this homoclinic orbit. The points in spend arbitrarily large fractions of their orbits shadowing the orbit of as closely as we want.
Every horseshoe supports ergodic measures which have positive entropy. Since each such is supported in a hyperbolic horseshoe, it follows that is also uniformly hyperbolic. Now, because the periodic horseshoe shadows along most of its orbit, it follows that is close in the weak topology to the periodic measure associated to the orbit of .
Since by the Theorem 3.5 part (a) the set of periodic measures is dense in , then it follows that the set of ergodic, positive-entropy, and uniformly hyperbolic measures as above is also dense in . ∎
5 Ergodicity, Support, Entropy
In this section we prove three “abstract” results on generic measures, dealing respectively with their ergodicity, support, and entropy. These results, together with the Theorem 3.5 part (a), respectively imply items (b.i), (b.ii), and (b.iii) of Theorem 3.5. We also use these general results to obtain Theorem 3.1 and item (i) of Theorem 3.3.
Let be an isolated transitive set of a -generic diffeomorphism . By Theorem 3.5 (a), ergodicity is a dense property in , since periodic measures are ergodic.
Since dense sets are residual, we need only prove that ergodicity is in the weak topology in order to conclude that ergodicity is generic in . And indeed we have the following general result (which implies in particular item (b.i) of Theorem 3.5):
Let be a compact metric space, be a continuous map, and denote the space of -invariant Borel probabilities on , endowed with the weak topology. Then ergodicity is a property in . In particular, if there exists a dense subset of which consists of ergodic measures, then every generic measure in is ergodic.
Let be a continuous real-valued function on . The set
of measures which are “ergodic with respect to ” is given by
In particular is a set: the integral in the right-hand side of the bracket varies continuously with the measure , and so the set defined within the brackets is open in ; this shows that is a countable intersection of open sets.
Now let be a countable dense subset of . By the argument above, for each there is some subset of consisting of measures which are ergodic with respect to . The measures which belong to the residual subset of obtained by intersecting the ’s are precisely the measures which are simultaneously ergodic with respect to every . Using standard approximation arguments one can show that such are ergodic with respect to any , and hence are ergodic. ∎
Proposition 5.1 implies in particular that the space of ergodic measures of a diffeomorphism is a Baire space when endowed with the weak topology. Indeed, any subset of a compact metric space is Baire, since is residual in .
5.2 Full Support
Given an isolated transitive set of a -generic diffeomorphism , then is a homoclinic class, and hence has a dense subset of periodic points. This last fact suffices to prove that generic measures on have full support (item (b.ii) of Theorem 3.5), as the following general result shows:
Let be a compact metric space, be a continuous map, and denote the space of -invariant Borel probabilities on , endowed with the weak topology. Then every generic measure in satisfies
In particular, if the set of periodic points of is dense in , then every generic satisfies .
Consider the map
It is easy to see that is lower-semicontinuous. By the Semicontinuity Lemma, there is a residual subset of which consists of continuity points of . The following claim then concludes the proof:
Given a continuity point of , then .
Let us now prove the claim. One considers any measure . The measures