Partially hyperbolic sets with a dynamically minimal lamination

We study partially hyperbolic sets of \begin{document}$C^1$\end{document} -diffeomorphisms. For these sets there are defined the strong stable and strong unstable laminations.A lamination is called dynamically minimal when the orbit of each leaf intersects the set densely. We prove that partially hyperbolic sets having a dynamically minimal lamination have empty interior. We also study the Lebesgue measure and the spectral decomposition of these sets. These results can be applied to \begin{document}$C^1$\end{document} -generic/robustly transitive attractors with one-dimensional center bundle.


1.
Introduction. Hyperbolicity of a proper set imposes quite specific properties of its "size" and "structure", especially when the dynamics on it is transitive. For instance, it is well known that transitive hyperbolic proper sets have empty interior. This is proved using the saturation principle in [12] 1 . Bowen proved in [11] that C 2 hyperbolic horseshoes have zero Lebesgue measure. The proof of this result involves bounded distortion arguments as well as the absolute continuity of the foliations, ingredients which are not available for maps with less regularity. Indeed, [10] provided an example of C 1 hyperbolic horseshoe with positive Lebesgue measure.
Similar results were obtained for non-hyperbolic dynamics assuming a weaker form of hyperbolicity known as partial hyperbolicity. A set Λ M is partially hyperbolic for a diffeomorphism f : M Ñ M if the tangent bundle T Λ M over the set Λ has a dominated splitting into three where E s and E u are uniformly expanded by Df and Df ¡1 , respectively. When E s , E c , and E u are all nontrivial, we speak of strongly partially hyperbolic sets.
The results in [2] study the case when the non-wandering set Ωpf q is partially hyperbolic and has non-empty interior. Recall that C 1 -generically 2 the set Ωpf q 2718 LUIZ FELIPE NOBILI FRANÇ A splits into pairwise disjoint homoclinic classes 3 which are its elementary pieces and form its spectral decompositiom, see [5] and Definition 4.1. It is proved that a strongly partially hyperbolic homoclinic class with non-empty interior is the whole manifold. Moreover, when the whole manifold is partially hyperbolic, this result holds C 1 -openly. Similar results were obtained in [18] assuming that the homoclinic class is bi-Lyapunov stable, which is a slightly more general condition than having non-empty interior.
Finally, considering again the Lebesgue measure of invariant transitive sets and in the same spirit of [11], the results in [4] extended Bowen's result to the partially hyperbolic setting by showing that sufficiently regular diffeomorphisms (of a class of differentiability bigger than one) have no "horseshoe-like" partially hyperbolic sets with positive Lebesgue measure.
In this work we deal with partially hyperbolic transitive sets Λ of C 1 -diffeomorphisms. We provide sufficient conditions guaranteing that these sets have empty interior or zero Lebesgue measure. A key feature in this setting is the existence of invariant dynamically defined laminations integrating the bundles E s and E u , that we denote by F s and F u , respectively. When for each leaf of the lamination its orbit has a dense intersection with Λ, we say that Λ is a weak s-minimal or weak u-minimal set, according to which lamination (F s or F u ) holds that property. A stronger condition is when this dense intersection is obtained with only a finite number of iterates on each leaf, situation in which we call the set s-minimal or u-minimal (see Definition 3.1). In [17] we prove that there is a wide class of systems verifying this property: robustly/generically transitive attractors with one-dimensional center bundle (see also [6,15] for previous results in this direction).
Our main result (Theorem A) claims that weak u-minimal and weak s-minimal proper sets have empty interior. Assuming that the central bundle is one-dimensional we prove that, C 1 -generically, s-minimal proper attractors have zero Lebesgue measure (see Theorem C).
Another motivation of this paper concerns the spectral decomposition results for sets containing the relevant part of the dynamics (limit, non-wandering, chainrecurrent sets, etc.). In the classical hyperbolic case, this decomposition consists of finitely many sets, called basic pieces, which each is a homoclinic class, see [19]. Specially important sets in this decomposition are the attractors and the repellers, which are persistent and robustly transitive and whose basins form an open and dense subset of the ambient space. There are some non-hyperbolic counterparts for this decomposition based on Conley's theory (see [5,13]). More recently, [3] states a C 1 -generic spectral decomposition theorem for chain-transitive locally maximal sets. Here we prove a spectral decomposition theorem for weak s-minimal and weak u-minimal homoclinic classes, see Theorems D.
1.1. Statement of the results. The precise definitions and notations involved in the results in this section can be found in Section 2.
Theorem A. Every weak s-minimal or weak u-minimal proper set has empty interior.
From Theorem B in [17] (see also item (2) of Proposition 2 in this paper), we get immediately the following corollary.
Corollary B. A C 1 -generic robustly transitive partially hyperbolic proper attractor with one-dimensional center bundle has robustly empty interior.
In the next statement, Λ f pUq denotes the maximal invariant set of f in the open set U .
Theorem C. For a generic f Diff 1 pMq, let Λ f pUq be a partially hyperbolic sminimal proper attractor with one-dimensional center bundle. Then there are a neighborhood U of f , an open and dense subset V U, and a residual subset W of U such that: 1. Λ g pUq has empty interior for all g V. 2. Λ g pUq has zero Lebesgue measure for all g W. Moreover, the set W contains every C 1 α diffeomorphism in V, for every α ¡ 0.
Observe that item (1) of Theorem C is stronger than Corollary B, as we get robustly empty interior even if the attractor is not robustly transitive. Unfortunately, this is only obtained for the s-minimal case.
Note that the results in [2] (see Corollary 1) also implies that, C 1 -generically, these attractors have empty interior 4 . Our results in Corollary B and (1) of Theorem C differ from theirs by the fact that, in our more restrictive setting, empty interior is verified in a robust way (that is, it holds C 1 -open and densely).
Finally, we state a spectral decomposition theorem for weak s-minimal and weak u-minimal homoclinic classes. Here we denote by Hpp, f q the homoclinic class of the hyperbolic periodic point p and by indexppq the dimension of the stable manifold of p.
Theorem D. Let Λ Hpp, f q be a weak s-minimal (resp. weak u-minimal) partially hyperbolic homoclinic class of a hyperbolic periodic point p with indexppq dimpE s q (resp. indexppq dimpE s q dimpE c q). Then Λ admits a unique spectral decomposition.
In this case, we see that weak s-minimality or weak u-minimality in fact implies s-or u-minimality.
Corollary E. Let Λ Hpp, f q be a weak s-minimal (resp. weak u-minimal) partially hyperbolic homoclinic class of a hyperbolic periodic point p with indexppq dimpE s q (resp. indexppq dimpE s q dimpE c q). Then Λ is an s-minimal set (resp. u-minimal set).
As an immediate consequence of Theorem B in [17], we get a robust spectral decomposition for robustly transitive attractors, meaning that every g in a small neighborhood of f has a spectral decomposition whose pieces are the continuations of the pieces in the spectral decomposition of Λ f . Nevertheless, this result is also a direct consequence of [3] Corollary F. There is a residual subset R of Diff 1 pMq satisfying the following. For every f R and U M , if Λ f pUq is a partially hyperbolic robustly transitive attractor with one-dimensional center bundle, then Λ f pUq has a robust spectral decomposition.
This paper is organized as follows. In Section 2 we give the basic definitions, terminology, and state some results we use along the paper. Theorem A is proved in subsection 3.1, Thorem C is proved in section 3.2, and Theorems D and Corollary E are proved in section 4.

2.
Preliminaries. Let M be a Riemannian compact manifold without boundary and, for r ¥ 1, let Diff r pMq be the space of C r diffeomorphisms from M to itself endowed with the C r -topology.
When a compact set Λ is the maximal f -invariant set of some open set U M , we say that Λ is an isolated set. Isolated sets vary upper semicontinuously. By an abuse of terminology, we call the set Λ g pUq the continuation of the set Λ f pUq when g varies in a small neighborhood of f .
A special kind of isolated set are attractors. We say that a set Λ is an attractor if there is an open set U M such that Λ nN f n pUq and f pUq U . Observe that M itself is an attractor (by taking U M ). The interesting case is when Λ $ M , when Λ is called a proper attractor.
In this work we study isolated sets with highly recurrent dynamics. We say that In our setting, this is equivalent to the following property: For any pair V 1 , V 2 of (relative) non-empty open sets of Λ, there is n Z such that f n pV 1 q V 2 r. A stronger recurrence property is the mixing property: For any pair V 1 , V 2 of (relative) open sets of Λ, there is n N such that f m pV 1 q V 2 r for all m ¥ n.
We speak of a robustly transitive set Λ Λ f pUq when Λ is transitive and the transitivity is also verified for the continuations Λ g pUq of every g in a small neighborhood U of f . If the transitivity is verified only in a residual subset of U, then we say that Λ is a generically transitive set.
In our context the isolated sets Λ are always assumed to be partially hyperbolic with E s E c E u denoting the partially hyperbolic splitting of T Λ M . The values of dimpE s q, dimpE c q and dimpE u q, are designated by d s , d c , and d u , respectively.
We also assume that Λ is robustly non-hyperbolic, meaning that E c does not have uniform contraction nor expansion in a robust way. We also require that none of the three bundles are trivial, in which case the set is strongly partially hyperbolic. See Appendix B of [7] for a list of elementary properties and a more complete view on this topic. Partial hyperbolicity leads to the existence of dynamically defined immersed submanifolds F s pxq and F u pxq, through each point x in the set, tangent to the stable and unstable subbundles, respectively. The set of such submanifolds are known as the stable and unstable lamination of the set and are denoted by F s and F u , respectively. We direct the reader to section 3 of [17], where the precise definition and main properties of these laminations are provided. When dealing with perturbations of a diffeomorphism, as in the case of the continuations of isolated sets, we need to specify in these notations which diffeomorphism we are referring to. So, let Λ f pUq be an isolated partially hyperbolic set and U be a neighborhood of f such that, for every g U, the set Λ g pUq is partially hyperbolic with the same bundles dimensions. We denote by F s pgq and

PARTIALLY HYPERBOLIC SETS WITH A DYNAMICALLY MINIMAL LAMINATION 2721
by F s px, gq, respectively, the strong stable lamination of Λ g pUq (with respect to the partial hyperbolicity of g) and the leaf of this foliation that contains x. Similarly, given a hyperbolic periodic point x Λ g pUq and ε ¡ 0, we denote by W s ε px, gq and W s px, gq the local stable manifold (of size ε) and the global stable manifolds of x, respectively. The union of all local or all global stable manifolds along the orbit of x is denoted by W s ε pO g pxq, gq and W s pO g pxq, gq, respectively. Similarly, fixed r ¡ 0, we denote by F s r pxq the open ball of radius r centered at x, relative to the induced distance on F s pxq. When there is no risk of misunderstanding, we simplify these notation by omitting the diffeomorphism, as F s pxq for F s px, f q, W s pxq for W s px, f q, and W s ε pO g pxqq for W s ε pO g pxq, gq.
Similar notations are considered for the unstable foliation and manifold.
In particular, when the central bundle is one-dimensional (d c 1), the index of a hyperbolic periodic point p is either d s or d s 1.
Following [17], given a diffeomorphism f and an isolated set Λ Λ f pUq, we define the concept of compatible neighbourhood of f , where the continuations of Λ f pUq share it main properties. Definition 2.2. Let Λ be an isolated set of a diffeomorphism f Diff 1 pMq and U M an isolated block of Λ. We call a neighborhood U of f a compatible neighborhood (with respect to U ) if U is sufficiently small so that, for all g U: the set Λ g pUq is isolated; if Λ f pUq is an attractor of f , then Λ g pUq is an attractor of g; if Λ f pUq is a partially hyperbolic set then Λ g pUq is a partially hyperbolic set of g, with the same bundles dimensions; if Λ f pUq is a generically (resp. robustly) transitive set of f , then Λ g pUq is a generically (resp. robustly) transitive set of g.

Generic isolated sets and attractors.
In this section we gather some useful results that we invoke along our proofs. They were stablished in [1,5,8,17,16]. For convenience, we restate them here in a compact form.

Proposition 1.
There is a residual subset R of Diff 1 pMq such that, for every f R and every isolated set Λ f pUq, it hold: 1. if Λ f pUq is a transitive attractor, then there is a neighborhood U of f such that, for every g R U, the set Λ g pUq is a transitive attractor. 2. if Λ f pUq is non-hyperbolic, then it contains a pair of (hyperbolic) saddles of different indices.
3. if Λ f pUq is a transitive isolated set of f that is partially hyperbolic with onedimensional center bundle, then for every pair of hyperbolic periodic points p, q Λ f pUq with indices d s and d s 1, respectively, there is an open set V p,q Diff 1 pMq, with f V p,q , satisfying: W s pO g pq g qq W s pO g pp g qq and W u pO g pp g qq W u pO g pq g qq for every g V p,q . Moreover, if Λ f pUq is robustly transitive, then Λ g pUq Hpp g , gq. 4. if Γ Hpp, f q is a partially hyperbolic homoclinic class, then there is an extension of the partially hyperbolic splitting on Γ to a continuous splitting on a compact neighborhood W of Γ such that it is invariant in the following sense: for every x W with f pxq W , we have that Df x pE i pxqq E i pfpxqq, for any i ts, c, uu. 5. if Λ f pUq is an s-minimal partially hyperbolic set with one-dimensional center bundle and U is a compatible neighborhood of f , then for every hyperbolic periodic point p Λ f pUq, there is an open set W p U, with f W p , such that Hpp g , gq O ¡ g pDq for every strong stable disk D centered at some point x Λ g pUq and every g W p . Moreover, if indexppq d s , then W p is a neighborhood of f .
In the rest of this paper, R always refers to the residual subset in Proposition 1.
Fixed an open set U M , denote by RTPHA 1 pUq (resp. GTPHA 1 pUq) the subset of Diff 1 pMq of diffeomorphisms f for which the maximal f -invariant subset Λ f pUq of U is a robustly (resp. generically) transitive attractor that is robustly non-hyperbolic and partially hyperbolic with one-dimensional center bundle. Observe that RTPHA 1 pUq is an open subset of Diff 1 pMq, and that GTPHA 1 pUq is locally residual in Diff 1 pMq.
Next proposition summarises Theorem A, Theorem B, and Corollary 4.9 in [17]. 1. for every g A, the set Λ g pUq is either generically s-minimal or generically u-minimal.
2. for every g B, the attractor Λ g pUq is either robustly s-minimal or robustly u-minimal. Moreover, Λ g pUq is a homoclinic class and depends continuously on g B.

Lebesgue measure and genericity.
In what follows we consider the manifold M endowed with a Lebesgue measure m. We see how Lebesgue measure behaves for the perturbations of an isolated set. Observe that every isolated set Λ f pUq is m-measurable, as it is a contable intersection of open sets.
3. Let f be a diffeomorphism in Diff 1 pMq, Λ f pUq be an isolated set, and U be a compatible neighborhood of f with respect to Λ f pUq. The map ϕ : U Ñ R defined by ϕpgq mpΛ g pUqq is upper semicontinuous. Consequently, the set of continuity points of the map ϕ is a residual subset of U.
Proof. Fix g U and consider the nested sequence of open sets given by Λpg, kq : k n¡k g n pUq. Clearly, Λpg, kq × Λ g pUq as k Ñ V. Since m is a regular measure, we obtain lim kÑV mpΛpg, kqq mpΛ g pUqq. Thus, fixed ε ¡ 0, there is N N pg, εq N such that mpΛpg, kqq mpΛ g pUqq ε ϕpgq ε, for all k ¥ N .

PARTIALLY HYPERBOLIC SETS WITH A DYNAMICALLY MINIMAL LAMINATION 2723
Note that there is N 0 N such that the closure of Λpg, N N 0 q is contained in the open set Λpg, N q. Then, for every h sufficiently close to g, it holds that Λph, N N 0 q Λpg, N q. Hence, mpΛ h pUqq ¤ mpΛph, N N 0 qq ¤ mpΛpg, N qq ¤ mpΛ g pUqq ε. This means that ϕphq ¤ ϕpgq ε, implying the lemma.
By an standard result of topology, we get the following consequence.

Corollary 1.
Under the hypotheses and with the notation of Lemma 2.3, if there is a dense subset W of U such that ϕpgq 0 for all g W, then there is a residual subset G of U such that ϕpgq 0 for all g G.
Remark 2. Lemma 2.3 and Corollary 1 hold for attractors, as any attractor is an isolated set.

Dynamically minimal laminations.
3.1. u-and s-minimal sets. For notational simplicity, given a strongly partially hyperbolic set Λ we adopt the following notation.
Definition 3.1 (dynamically minimal lamination). Let Λ be a partially hyperbolic set of a diffeomorphism f with nontrivial stable bundle E s . We say that the lami- When Λ Λ f pUq is an isolated set, Λ is a robustly s-minimal set if Λ g pUq is s-minimal for all g in a neighborhood U of f . If s-minimality is verified only in a residual subset of U, then Λ f pUq is called a generically s-minimal set.
Also, we say that a set Λ is weak s-minimal if it holds that nZ F s Λ pxq is dense in Λ for any x Λ. Clearly, any s-minimal set is weak s-minimal. In fact, these two notions are equivalent in a wide range of settings 6 .
The definition of u-minimality or weak u-minimality is analogous, considering the strong unstable lamination F u .
The main result in this section is the following equivalence of Theorem A.
Theorem 3.2. Any weak u-minimal or weak s-minimal set with non-empty interior is the whole manifold.
In the rest of this section, all the results are stated for weak s-minimal sets, though similar statements (with similar proofs) also hold in the weak u-minimal case.
We start with some auxiliary lemmas and the following Remark, that gives two well known properties of the strong stable.
Remark 3. For every r ¡ 0 sufficiently small, it hold: i) F s pxq nN f ¡n pF s r pf n pxqqq ii) There is N N such that A n pxq f ¡n.N pF s r pf n.N pxqqq yield a nested sequence (that is, A n pxq A n 1 pxq for every n N). Given a set K M , we denote by B ε pKq the ε-neighborhood of K relative to some fixed Riemannian metric on M . Lemma 3.3. Let Λ be a weak s-minimal set of a diffeomorphism f . Given any ε ¡ 0 and r ¡ 0 sufficiently small, there are constants N and k in N such that Proof. Fix ε ¡ 0 and r ¡ 0. From weak s-minimality and Remark 3, given any y Λ, there are natural numbers N y and k y such that By the continuity of the foliation F s , there is a neighborhood V pyq of y such that the previous inclusion holds for all z V pyq Λ, with N z N y and k z k y . Consider the covering tV pyqu yΛ of Λ. Since Λ is a compact set, we may extract a finite subcovering tV py i qu m i1 and constants N i and k i such that, if y Λ V py j q for some j t1, . . . , mu, then Let N LCMpN 1 , N 2 , ¤ ¤ ¤ , N m q be the lest common multiple of these numbers and k maxtk 1 , k 2 , ¤ ¤ ¤ , k m u. By item iiq of Remark 3, we can replace N j by N , with n N, so we have Given x Λ we set y f p¡N k 1q pxq in the above inclusion, so we obtain the lemma.
Lemma 3.4. Let Λ be a weak s-minimal set of a diffeomorphism f . If Λ contains some strong stable disk, then Λ contains the strong stable leaf of every point in Λ.
Proof. Let r ¡ 0 and x 0 Λ be such that the strong stable disk D F s r px 0 q is contained in Λ, and let y Λ be an accumulation point of the backward orbit of x 0 .
Fix δ ¡ 0 sufficiently small so that, by the partial hyperbolicity on Λ, there is m 0 N such that for every stable disk S of length δ and m ¥ m 0 , the image f m pSq is contained inside a stable disk of radius r. Hence, there is an increasing sequence tn i u nN N, with n i ¥ m 0 , such that lim iÑV f ¡ni px 0 q y and, for every i N, the disk f ¡ni pDq has inner radius bigger than δ. By the continuity of the lamination, we obtain that F s δ pyq Λ. For every m N, the point f ¡m pyq is also an accumulation point of the backward orbit of x 0 , so the same argument leads to F s δ pf m pyqq Λ. Then we conclude that f ¡m pF s δ pf m pyqqq Λ for every m N, which implies that Fpyq Λ (see Remark 3). Now weak s-minimality gives that iZ f i pF s pyqq is a dense subset of Λ. Since the strong stable lamination is continuous and Λ is closed, we get that F s pzq Λ for any z Λ, ending the proof of this Lemma.

PARTIALLY HYPERBOLIC SETS WITH A DYNAMICALLY MINIMAL LAMINATION 2725
We are now ready to prove Theorem 3.2 Proof of Theorem 3.2. Observe that the interior of Λ, denoted by intpΛq, is an invariant subset of Λ. Moreover, if Λ has non-empty interior, then it contains some strong stable disk. By Lemma 3.4, the set Λ contains the strong stable leaf of every point in Λ.
Suppose that the boundary fΛ of Λ is non-empty. Let z fΛ and consider the disk D F s r pzq Λ. By Lemma 3.3, there is N N such that f ¡N pDq intersects intpΛq. The f -invariance of intpΛq implies that D intpΛq r. Now, choose some point x in this intersection and an open neighborhood B of x with B intpΛq. For each point y B we consider its entire strong stable leave F s pyq, that is contained in Λ (recall Lemma 3.4). By the continuity of the strong stable foliation, the set V yB F s pyq Λ is a neighborhood of F s pxq F s pzq. Thus V is a neighborhood of z that is contained in Λ, contradicting the fact that z fΛ. Therefore fΛ r, and consequently Λ M .

s-minimal attractors.
In what follows we study s-minimal attractors apart, with no similar statements to the case of u-minimal attractors 7 .
The main result presented here is Theorem C. Before proving it, we need some intermediate results that also hold for d c ¥ 1.
In the next statements, the notation Per σ pf |Λ q stands for the set of hyperbolic periodic points in Λ of index σ.

Lemma 3.5.
Let Λ Λ f pUq be a partially hyperbolic attractor that is s-minimal, contains some strong stable disk, and has a point p Per d s pf |Λ q. Then Λ is the whole manifold.
Proof. By Theorem 3.2, it suffices to prove that Λ has non-empty interior. Consider the periodic point p Per d s pf |Λ q. Then, for a small ε ¡ 0, its local unstable manifold W u ε ppq is a pd u d c q-dimensional embedded manifold contained in the attractor.
By Lemma 3.4, the strong stable leaf of any point in Λ is contained in Λ. Thus the saturation of W u ε ppq by its strong stable leaves contains an open subset of Λ, so Λ has non-empty interior.
The following proposition is a simplified version of Corollary B in [4] for the case of partially hyperbolic attractors.

Proposition 3 ([4]
). Fix α ¡ 0 and f Diff 1 α pMq. If Λ is a partially hyperbolic set of f with mpΛq ¡ 0, then Λ contain some strong stable disk and some strong unstable disk. Lemma 3.6. Let f Diff 1 α pMq and Λ Λ f pUq be partially hyperbolic attractor that is s-minimal. If Per ds pf |Λ q $ r and mpΛq ¡ 0, then Λ is the whole manifold.
Proof. By Proposition 3 there is a strong stable disk D contained in Λ. Now Lemma 3.5 implies the statement.
We are now ready to prove Theorem C.
Proof of theorem C. Since f is C 1 -generic and Λ f pUq is s-minimal, we can assume that Λ f pUq is generically s-minimal (see Proposition 2). Let U be a compatible neighbourhood of f and J 0 be the residual subset of U of diffeomorphisms g such that Λ g pUq is s-minimal. 7 Recall that by taking f ¡1 , the attractor becomes a repellor.

LUIZ FELIPE NOBILI FRANÇ A
Claim 3.7. For every g J 0 , ε ¡ 0, and every hyperbolic periodic point a Λ g pUq Per d s 1 pgq it holds that intpW s ε paq Λ g pUqq r.
Here the interior refers to the topology of W s ε paq. Proof of the claim. The proof is by contradiction. Assume that there are ε ¡ 0 and a Λ g pUq Per d s 1 pgq such that intpW s ε pa, gq Λ g pUqq contains an open ball B of W s ε pa, gq. By saturating B with strong unstable leaves (which are subsets of the attractor Λ g pUq) we get an open set (relative to the ambient manifold M ) contained in Λ g pUq. Thus Λ g pUq has non-empty interior and, by Theorem 3.2 it is the whole manifold, contradicting the fact that Λ g pUq is a proper attractor.
Consider a diffeomorphism f as in the statement of Theorem C and a pair of hyperbolic periodic points p, q Λ f pUq with indices d s and d s 1, respectively (these points exist by item (2) of Proposition 1 and Remark 1). Let W p and V p,q be the open sets given by items (3) and (5) of Proposition 1, respectively. By shrinking W p if necessary, we can assume that W p V p,q , so the continuation q g of q is well defined for every g W p . Claim 3.8. The map φ given by g Þ Ñ W s ε pq g , gq Λ g pUq, defined on W p , is upper semicontinuous.
Proof. Observe that, for every g W p , the set tF u ε pxq | x W s ε pq g , gqΛ g pUqu is an open subset of Λ g pUq. Since W s ε pp g , gq varies continuously, this observation shows that an upper discontinuity of φ would imply an upper discontinuity of Λ g pUq. However, such a discontinuity for Λ g pUq is not possible as attractors vary upper semicontinuously.
As a consequence of this claim, there is a residual subset J 1 W p consisting of continuity points of the map φ.
By Claim 3.7 and the definition of J 1 we conclude that, for every h J 0 J 1 (that is a subset of W p ), there is a neighborhood U h of h such that W s ε pq g , gq Λ g pUq for all g U h . (1) The set V p hJ0J1 U h is an open and dense subset of W p . Claim 3.9. For every g V p the attractor Λ g pUq does not contain any strong stable disk, and consequently it has empty interior.
Proof. Suppose that there is g V p for which Λ g pUq has a strong stable disk D Λ g pUq. By the invariance and closeness of Λ g pUq, any accumulation point of the backward orbit of D belongs to Λ g pUq. By item (4) of Proposition 1, the closure of the negative orbit of D contains Hpp g , gq, so we conclude that F s pp g , gq Λ g pUq. Now, item (3) of Proposition 1 implies that W s pq g , gq Λ g pUq, contradicting Equation (1).
Recall that V p depends on the choice of f Diff 1 pMq and, since f W p , we also have f V p . Hence, to obtain item (1) of Theorem C, we apply Claim 3.9 with respect to every diffeomorphism in R U. The union of all open sets obtained in this way is the announced open and dense subset V of U.
Fix α ¡ 0. To prove the second part of the theorem, observe that, if g V Diff 1 α pMq is such that mpΛ g pUqq ¡ 0, then it contains a strong stable disk (see Proposition 3). This contradicts Claim 3.9, since we have taken g V. This and intersects transversely W u ppq as well. For the same reason, F s pf pj nq ppqq intersects transversely W u ppq. This is enough to get that p is homoclinic related to f pi nq ppq and f pj nq ppq (see Lemma 2.2 in [3]). Consequently, f pi nq ppq and f pj nq ppq are homoclinic related and, by invariance, Γ i Γ j . It follows that i j.
Before proving Corollary E, let us state an auxiliary lemma.
Lemma 4.2. Let Γ be a partially hyperbollic pointwise homoclinic class of a periodic point p with indexppq d s (the dimension of E s ). If x Γ is such that F s pxq intersect transversely W u ppq at a point w, then w Γ. Proof. Since x Γ, there is a sequence of homoclinic points z n F s ppq Γ accumulating at x. Let w be a transverse intersection of F s pxq and W u ppq. Then F s pz n q intersects W u ppq in a sequence of points w n that accumulates at w. As F s pz n q F s ppq, the points w n are transverse homoclinic points of p, so w n Γ for any n N, which implies that w Γ. Proof of Corollary E. Now we show that Λ Hpp, f q is indeed an s-minimal set. Take Γ j in the spectral decomposition of Λ and x Γ j . Set g f πppq , where πppq is the period of p. We want to prove that F s Λ pxq is dense in Γ j . Let U be an open set with U Γ j $ r. Let r be a transverse homoclinic point of f j ppq in U . By the inclination lemma, there is an open neighborhood V of f j ppq and n 0 N such that if y V Λ, then F s pg m pyqq intersects transversely W u pf j ppqq in a small neighborhood of r for all n ¥ n 0 . By Lemma 4.2, all these intersections are in Λ. Now let z Λ be a point in the w-limit of x (with respect to g). Since the orbit of the stable leaves of z intersects Λ densely, we have that F s pzq intersects transversely W u pf j ppqq, say at the point w. By the inclination lemma, there is a neighborhood W of w and n 0 N such that for any point y W Λ, the stable manifold F s pg ¡n pyqq intersects transversely W u pf j ppqq inside U for any n ¥ n 0 . Now take n ¥ n 0 so that g n pxq is so close to z so that F s pg n pxqq contains a point in W Λ. Therefore, F s pxq intersects transversely W u pf j ppqq inside U , and by Lemma 4.2 this point is in Λ. Hence, F s Λ pxq U $ r.