On Critical Point for Two Dimensional Holomorphics Systems

Abstract

Let $f:M\rightarrow M$ be a biholomorphisms on two--dimensional a complex manifold, and let $X\subseteq M$ be a compact $f$--invariant set such that $f|X$ is asymptotically dissipative and without sinks periodic points. We introduce a solely dynamical obstruction to dominated splitting, namely critical point. Critical point is a dynamical object and capture many of the dynamical properties of their one--dimensional counterpart.