Partial hyperbolicity far from homoclinic bifurcations

Abstract

We prove that any diffeomorphism of a compact manifold can be C 1 -approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in [C 2 ]. RésuméHyperbolicité partielle loin des bifurcations homoclines. Nous montrons que tout difféomorphisme d'une variété compacte peutêtre approché en topologie C 1 par un difféomorphisme qui présente une bifurcation homocline (une tangence homocline ou un cycle hétérodimensionnel) ou bien par un difféomorphisme partiellement hyperbolique (son ensemble récurrent par chaînes se décompose en pièces partiellement hyperboliques dont les fibrés centraux sont de dimensions au pluségalesà 2). Nousétudionségalement d'un point de vue plus systématique les modèles de dynamiques centrales introduits en [C 2 ].

Codimension one structurally stable chain classes

“…(More detailed discussion on the orientations of the central models of Crovisier [Cro2] ensures that J can be chosen so that J and I are on the same side of z so that J ∩ I is not a single point z.) Then J still goes into K in the sense of chains.…”

“…To prove Theorem 4.1 we use the δ-interval argument taken from , combined with ideas from the more recent central model theory of Crovisier [Cro2].…”

SURVEY Towards a global view of dynamical systems, for the C1-topology

“…For instance, it was proved in [Ma, DPU, BDP] that every robustly transitive diffeomorphism admits a dominated splitting, and Crovisier [Cr2] proved the existence of a partially hyperbolic structure, with at most two one-dimensional central bundles for dynamical systems far from homoclinic bifurcations. For instance, it was proved in [Ma, DPU, BDP] that every robustly transitive diffeomorphism admits a dominated splitting, and Crovisier [Cr2] proved the existence of a partially hyperbolic structure, with at most two one-dimensional central bundles for dynamical systems far from homoclinic bifurcations.…”

“…Stated in any C r -topologies, some of these conjectures have been (at least partially) successful in the C 1 -topology: in this C 1 -setting, the weak-density conjecture has been proved in [PS,BGW,Cr3] and Crovisier [Cr2,CP] proved an essential part of Conjecture 3. Stated in any C r -topologies, some of these conjectures have been (at least partially) successful in the C 1 -topology: in this C 1 -setting, the weak-density conjecture has been proved in [PS,BGW,Cr3] and Crovisier [Cr2,CP] proved an essential part of Conjecture 3.…”

Tangências homoclínicas, entropia e medidas de SRB

“…Na última referência, Pujals exibe um conjunto atrator topologicamente hiperbólico de um difeomorfismo Kupka-Smale numa variedade tridimensional suave e prova que, sob algumas hipóteses de dissipatividade, ou o conjunto é hiperbólico ou o difeomorfismo é C 1 -aproximado por outro que exibe ou um ciclo heterodimensional ou uma tangência homoclínica. Outro resultado recente é um preprint de Sylvain Crovisier [Cro08], onde o autor prova que qualquer difeomorfismo de uma variedade compacta pode ser C 1 -aproximado por um difeomorfismo que exibe uma bifurcação homoclínica ou por um difeomorfismo que é parcialmente hiperbólico.…”