On critical point for two-dimensional holomorphic systems

Let f: M -> M be a biholomorphism on a two-dimensional complex manifold, and let X subset of M be a compact f -invariant set such that f vertical bar(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.