Dynamics of generic automorphisms of Stein manifolds with the density property

Abstract

We study the dynamics of a generic automorphism $f$ of a Stein manifold with the density property. Such manifolds include all linear algebraic groups. Even in the special case of $\mathbb C^n$, $n\geq 2$, most of our results are new. We study the Julia set, non-wandering set, and chain-recurrent set of $f$. We show that the closure of the set of saddle periodic points of $f$ is the largest forward invariant set on which $f$ is chaotic. This subset of the Julia set of $f$ is also characterised as the closure of the set of transverse homoclinic points of $f$, and equals the Julia set if and only if a certain closing lemma holds. Among the other results in the paper is a generalisation of Buzzard's holomorphic Kupka-Smale theorem to our setting.