On the dynamics of dominated splitting

Abstract

Let $f:M\to M$ be a $C^2$ diffeomorphism of a compact surface. We give a complete description of the dynamics of any compact invariant set having dominated splitting. In particular, we prove a Spectral Decomposition Theorem for the limit set $L(f)$ under the assumption of dominated splitting. Moreover, we describe all the bifurcations that these systems can exhibit and the different types of dynamics that could follow for small $C^r-$perturbations.