Preprint D59/2009
Robust entropy expansiveness implies generic domination.
José L. Vieitez | Pacifico, Maria Jose
Keywords: entropy expansiveness | hyperbolicity | heterodimensional cycles | Dominated splitting | generic dominated splitting
\begin{abstract} Let $f: M \to M$ be a $C^r$-diffeomorphism, $r\geq 1$, defined on a compact boundaryless $d$-dimensional manifold $M$, $d\geq 2$, 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 $\mathcal{U}$ of $f$ such that for every $g\in {\cal U}$ 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 $E\oplus F_1\oplus\cdots \oplus F_c\oplus G$. If $H(p)$ is isolated then $E$ is contracting, $G$ is expanding and all $F_j$ are one dimensional and not hyperbolic. Moreover, in that case for an open dense subset of $\mathcal{U}$ either the class is hyperbolic or has robust heterodimensional cycles. \end{abstract}