Robust entropy expansiveness implies generic domination

Let $f:M\to M$ be a $C^r$-diffeomorphism, $r\ge 1$, defined on a compact boundaryless $d$-dimensional manifold $M$, $d\ge 2$, 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 $E\oplus F_1\oplus \mathrm{.}\mathrm{.}\mathrm{.}\oplus F_c\oplus G$ where $E$ is contracting, $G$ is expanding and all $F_j$ are one dimensional and not hyperbolic.