Dynamics of generic endomorphisms of Oka-Stein manifolds

Abstract

We study the dynamics of a generic endomorphism $f$ of an Oka-Stein manifold $X$. Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. We give several descriptions of the Fatou set and the Julia set of $f$. In particular, we show that the Julia set is the derived set of the set of attracting periodic points of $f$ and that it is also the closure of the set of repelling periodic points of $f$. Among other results, we prove that $f$ is chaotic on the Julia set and that every periodic point of $f$ is hyperbolic. We also give an explicit description of the "Conley decomposition" of $X$ induced by $f$ into chain-recurrence classes and basins of attractors. For $X=\mathbb{C}$, we prove that every Fatou component is a disc and that every point in the Fatou set is attracted to an attracting cycle or lies in a dynamically bounded wandering domain (whether such domains exist is an open question).