Entropy-Expansiveness and Domination for Surface Diffeomorphisms

Abstract

Let f : M → M be a Cr-diffeomorphism, r ≥ 1, deffined on a closed manifold M. We prove that if M is a surface and K ⊂ M is a compact invariant set such that TKM = E ⊕ F is a dominated splitting then f/K is entropy expansive. Moreover C¹ generically in any dimension, isolated homoclinic classes H(p), p hyperbolic, are entropy expansive. Conversely, if there exists a C1 neighborhood U of a surface diffeomorphism f and a homoclinic class H(p), p hyperbolic, such that for every g ∈ U the continuation H(pg) of H(p) is entropy-expansive then there is a dominated splitting for f/H(p).