Generic robustness of spectral decompositions

Abstract

We prove that given a compact n-dimensional boundaryless manifold M , n 2, there exists a residual subset R of Diff 1 (M) such that if f ∈ R admits a spectral decomposition (i.e., the nonwandering set Ω(f) admits a partition into a finite number of transitive compact sets), then this spectral decomposition is robust in a generic sense (tame behavior). This implies a C 1-generic trichotomy that generalizes some aspects of a two-dimensional theorem of Mañé [Topology 17 (1978) 386-396]. Lastly, Palis [Astérisque 261 (2000) 335-347] has conjectured that densely in Diff k (M) diffeomorphisms either are hyperbolic or exhibit homoclinic bifurcations. We use the aforementioned results to prove this conjecture in a large open region of Diff 1 (M).  2003 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Nous montrons qu'étant donnée une variété compacte M de dimension n, n 2, il existe un sous-ensemble résiduel R de Diff 1 (M) tel que si f ∈ R admet une décomposition spectrale (c'est-à-dire, Ω(f) admet une partition en un nombre fini d'ensembles compacts transitifs), alors cette décomposition spectrale est robuste dans un sens générique. Cela implique une trichotomie générique qui généralise certains aspects d'un théorème bi-dimensionnel de Mañé. Enfin, Palis a conjecturé que dans un sous-ensemble dense de Diff 1 (M), les difféomorphismes ou bien sont hyperboliques, ou bien admettent des bifurcations homoclines. Nous utilisons les résultats précédents pour prouver cette conjecture dans une grande région ouverte de Diff 1 (M).

SHADOWABLE CHAIN TRANSITIVE SETS OF C1-GENERIC DIFFEOMORPHISMS

“…It is explained in [1] that every C 1 -generic diffeomorphism comes in one of two types: tame diffeomorphisms, which have a finite number of homoclinic classes and whose nonwandering sets admit partitions into a finite number of disjoint transitive sets; and wild diffeomorphisms, which have an infinite number of (disjoint and different) homoclinic classes and whose nonwandering sets admit no such partitions. It is easy to show that if a diffeomorphism has a finite number of chain components, then every chain component is locally maximal, and so every chain component of a tame diffeomorphism is locally maximal.…”

Diffeomorphisms with thes-limit shadowing property

“…Thus, together with Lemma 1, it is not hard to show that C 1 -generically if f jR( f ) has the s-limit shadowing property, then the number of homoclinic classes is finite. It is proved by [11] that C 1 -generically the map from Diff(M) to N [ {1} assigning the number of homoclinic classes is well defined and locally constant. Thus, C 1 -generically if f jR( f ) has the s-limit shadowing property, then f is tame so that our theorem follows from [3, Theorem 5].…”

C1-stably shadowable chain components

“…In [1], two C 1 -open generic sets were introduced: tame diffeomorphisms, which have a finite number of homoclinic classes and whose non-wandering sets admit partitions into a finite number of disjoint transitive sets; and wild diffeomorphisms, which have an infinite number of (disjoint different) homoclinic classes (for a survey of this subject, see [9,Chapter 10]). It was also proved in [3] that there is a residual set R ⊂ Diff(M) such that if f ∈ R is tame, then the following two conditions are equivalent: (a) f satisfies both Axiom A and the no-cycle condition; (b) f has the shadowing property.…”