Dynamics of generic automorphisms of Stein manifolds with the density property
Leandro Arosio, Finnur Larusson
TL;DR
This work studies the dynamics of generic automorphisms of Stein manifolds with the density property, combining Andersén–Lempert perturbations with holomorphic dynamics to obtain a comprehensive dynamical picture. It proves that all periodic points are hyperbolic and that homoclinic/heteroclinic intersections are transverse, establishing a robust Kupka–Smale-type framework and a detailed description of Julia, Fatou, non-wandering, and chain-recurrent sets. A central contribution is the introduction of the chaotic Julia set $J_f^* = \overline{{\operatorname{sad}}(f)}$, shown to be not compact and to coincide with the closure of transverse homoclinic points, thereby isolating the maximal chaotic core in $J_f^*$. The paper also generalizes Buzzard’s holomorphic Kupka–Smale theorem to this setting and formulates open problems on closing lemmas and the equality $J_f=J_f^*$, linking chaotic dynamics to Conley-type structures. Collectively, these results unify attracted and recurrent dynamical views with holomorphic chaos for a broad class of Stein manifolds, providing a rigorous framework for hyperbolicity, transversality, and the chaotic core $J_f^*$.
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.