Generic aspects of holomorphic dynamics on highly flexible complex manifolds

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.