On critical point for two-dimensional holomorphic systems

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