Generic aspects of holomorphic dynamics on highly flexible complex manifolds

TL;DR

This paper establishes new closing lemmas for automorphisms and endomorphisms on Stein and Oka-Stein manifolds, demonstrating density of hyperbolic periodic points and extending results on chaotic automorphisms in holomorphic dynamics.

Contribution

It introduces a new tameness condition for automorphisms and proves the density of hyperbolic periodic points on a broad class of complex manifolds, extending prior work on holomorphic dynamics.

Findings

Hyperbolic periodic points are dense in the tame non-wandering set.

First results on holomorphic dynamics on Oka manifolds.

Generalization of previous theorems on chaotic volume-preserving automorphisms.

Abstract

We prove closing lemmas for automorphisms of a Stein manifold with the density property and for endomorphisms of an Oka-Stein manifold. In the former case we need to impose a new tameness condition. It follows that hyperbolic periodic points are dense in the tame non-wandering set of a generic automorphism of a Stein manifold with the density property and in the non-wandering set of a generic endomorphism of an Oka-Stein manifold. These are the first results about holomorphic dynamics on Oka manifolds. We strengthen previous results of ours on the existence and genericity of chaotic volume-preserving automorphisms of Stein manifolds with the volume density property. We build on work of Fornaess and Sibony: our main results generalise theorems of theirs and we use their methods of proof.