Entropy-expansiveness and domination

Maria Jose Pacifico | Jose Vieitez

**Keywords:**Entropy-expansiveness | h-expansiveness | dominated splitting

Let $f: M \to M$ be a $C^r$-diffeomorphism, $r\geq 1$, defined in a compact boundary-less surface $M$. We prove that if $K$ is a compact $f$-invariant subset of $M$ with a dominated splitting then $f/K$ is $h$-expansive. Reciprocally, if there exists a $C^r$ neighborhood of $f$, ${\cal U}$, such that for $g\in {\cal U}$ there exists $K_g$ compact invariant such that $g/K_g$ is $h$-expansive then there is a dominated splitting for $K_g$.