R�currence et g�n�ricit�

R-robustly measure expansive homoclinic classes are hyperbolic

“…Note that the chain component C f (p) of f is a equivalent class, it is a closed set and f-invariant set. The following was proved by Bonatti and Crovisier [2].…”

“…Let H f (p) be the homoclinic class of f associated to a hyperbolic periodic point p, and suppose that H f (p) is R-robustly measure expansive. Then for C, λ as in Theorem 3.1 and δ > 0 satisfying λ = λ(1 + δ) < 1 and q ∼ p, there exists 0 < 1 < such that if for all 0 n π(q) it holds that for some 2 …”

“…In the Lemma 3.3, we consider q ∈ homo p . Then we can extend x ∈ H f (p), that is, for any x ∈ H f (p) and 1 > 0 there exists 2 By Proposition 3.1 (3), there is δ > 0 such that for any q, r ∈ homo p ,…”

Generic Hamiltonian Dynamical Systems: An Overview

“…The problem of knowing if a given dynamical system exhibits only one "piece" or, in other words, if there is any dense orbit, is a central problem in the modern theory of dynamical systems. A partial answer to this problem was given by Bonatti and Crovisier in [19] for the volume-preserving discrete-time case and by the same authors and Arnaud in the symplectomorphism framework [5]. They proved that for some C 1 -residual subset any map has a dense orbit.…”

Entropy-Expansiveness and Domination for Surface Diffeomorphisms

“…In [2] it is proved that for generic C 1 diffeomorphisms, the elementary pieces are the chain recurrent classes, and when one of these sets contains a periodic point p it coincides with the homoclinic class of p. Moreover, [9] together with the Closing Lemma of [32], give that the homoclinic classes constitute a partition of a dense part of the limit set of generic diffeomorphisms, and [10] establishes that chain recurrent sets of generic diffeomorphisms are Hausdorff limits of homoclinic classes. All these results evidence the importance of understanding the dynamics restricted to homoclinic classes.…”