On Critical Point for Two Dimensional Holomorphics Systems

TL;DR

This paper introduces the concept of critical points as a dynamical obstruction to dominated splitting in two-dimensional holomorphic systems, linking complex dynamics with real dynamical properties.

Contribution

It defines and explores the notion of critical points in two-dimensional holomorphic systems, providing a new tool to understand their dynamical behavior.

Findings

Critical points serve as a dynamical obstruction to dominated splitting.

The concept captures many properties of one-dimensional dynamical systems.

Provides a new perspective on complex holomorphic dynamics.

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.