Preprint D029/2007
Entropy-expansiveness and domination
J.L. Vieitez
Keywords: Entropy-expansiveness | Dominated splitting | homoclinic class
Let $f: M \to M$ be a $C^r$-diffeomorphism, $r\geq 1$, defined on a compact boundary-less manifold $M$. We prove that $C^1$-generically if $H(p)$, the $f$-homoclinic class of a hyperbolic periodic point $p$, has a dominated splitting then $f/H(p)$ is entropy-expansive. Conversely, if there exists a $C^1$ neighborhood $\mathcal{U}$ of a diffeomorphism $f$ defined on a compact surface and a homoclinic class $H(p)$ of an $f$- hyperbolic periodic point $p$, such that for every $g\in {\cal U}$ the continuation $H(p_g)$ of $H(p)$ is entropy-expansive then there is a dominated splitting for $H(p)$.