On C 1 -Persistently Expansive Homoclinic Classes Mart´ ın Sambarino, Jos´ e Vieitez ∗ December 20, 2004 Abstract Let f : M → M be a diffeomorphism defined in a d-dimensional com- pact boundary-less manifold M . We prove that C 1 -persistently expansive homoclinic classes H(p), p an f -hyperbolic periodic point, have a dominated splitting E ⊕ F , dim(E) = index(p). Moreover, we prove that if the H(p)-germ of f is expansive (in particular if H(p) is an attractor, repeller or maximal invariant) then it is hyperbolic. 1 Introduction It has been a problem in differentiable dynamical systems during the last decades to understand the influence of a robust dynamic property (i.e. a property that holds for a system and all nearby ones) on the behavior of the tangent map of the system. For instance, to begin with the context of this paper, Ma˜ n´ e ([Ma1]) proved that if M is a compact manifold and a diffeomorphism f : M → M and all C 1 nearby ones are expansive then f is a quasi-Anosov system (any non vanishing vector growth exponentially in norm by forward or backward iterations of the tangent map), and in particular is a hyperbolic system (Axiom A). In this paper we are concerned when some natural invariant subset is robustly expansive. A first generalization would be to assume that the non-wandering set is robustly expansive. However, this immediately yields that f is in the C 1 interior of those systems having all the periodic points hyperbolic, and it follows by ([Ao],[Ha]) that the system is Axiom A. In this paper we study the case when a homoclinic class is robustly or persistently expansive (see definitions below) generalizing previous results in [PPV]. The difference of this case with that of [Ma1] is that no regards is done on ∗ Partially supported by PEDECIBA and CSIC-Uy 2000 Mathematics Subject Classification: 37D30 1