Generic conservative dynamics on Stein manifolds with the volume density property

TL;DR

This paper investigates the dynamics of generic volume-preserving automorphisms on Stein manifolds with the volume density property, revealing chaotic behavior, dense homoclinic points, and infinite entropy, with results extending to non-conservative cases.

Contribution

It establishes that generic volume-preserving automorphisms on Stein manifolds are chaotic with dense homoclinic points and infinite entropy, and proves the Kupka-Smale theorem in this setting.

Findings

Generic automorphisms are chaotic with infinite topological entropy.

Transverse homoclinic points are dense for saddle periodic points.

Results extend to non-conservative automorphisms.

Abstract

We study the dynamics of generic volume-preserving automorphisms $f$ of a Stein manifold $X$ of dimension at least 2 with the volume density property. Among such $X$ are all connected linear algebraic groups (except $\mathbb{C}$ and $\mathbb{C}^*$) with a left- or right-invariant Haar form. We show that a generic $f$ is chaotic and of infinite topological entropy, and that the transverse homoclinic points of each of its saddle periodic points are dense in $X$. We present analogous results with similar proofs in the non-conservative case. We also prove the Kupka-Smale theorem in the conservative setting.