Robust Entropy Expansiveness Implies Generic Domination

Let f : M → M be a C r-diffeomorphism, r ≥ 1, defined on a compact boundaryless d-dimensional manifold M , d ≥ 2, and let H(p) be the homoclinic class associated to the hyperbolic periodic point p. We prove that if there exists a C 1 neighborhood U of f such that for every g ∈ U the continuation H(p g) of H(p) is entropy-expansive then there is a Df-invariant dominated splitting for H(p) of the form E ⊕ F 1 ⊕ · · · ⊕ F c ⊕ G where E is contracting, G is expanding and all F j are one dimensional and not hyperbolic.