Robust entropy expansiveness implies generic domination

TL;DR

This paper shows that robust entropy-expansiveness in homoclinic classes of diffeomorphisms implies the existence of a specific type of dominated splitting with certain hyperbolic and non-hyperbolic components.

Contribution

It establishes a link between robust entropy-expansiveness and the structure of dominated splittings in homoclinic classes of $C^r$-diffeomorphisms.

Findings

Robust entropy-expansiveness implies a specific dominated splitting.

The splitting includes contracting, expanding, and one-dimensional non-hyperbolic subbundles.

This structure is stable under small $C^1$ perturbations.

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 $g\in {\mathcal U}$ 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 $E\oplus F_1\oplus... \oplus F_c\oplus G$ where $E$ is contracting, $G$ is expanding and all $F_j$ are one dimensional and not hyperbolic.