Chaotic holomorphic automorphisms of Stein manifolds with the volume density property

TL;DR

This paper proves that chaotic automorphisms are common among volume-preserving holomorphic automorphisms of certain Stein manifolds with the volume density property, extending known results from complex Euclidean spaces to more general manifolds.

Contribution

It generalizes the generic chaos results for volume-preserving automorphisms from C2^n to a broad class of Stein manifolds with the volume density property.

Findings

Chaotic automorphisms are generic among volume-preserving holomorphic automorphisms.

Existence of chaotic automorphisms on these Stein manifolds.

Extension of chaos results from C2^n to general Stein manifolds with the volume density property.

Abstract

Let $X$ be a Stein manifold of dimension $n\geq 2$ satisfying the volume density property with respect to an exact holomorphic volume form. For example, $X$ could be $\mathbb{C}^n$, any connected linear algebraic group that is not reductive, the Koras-Russell cubic, or a product $Y\times\mathbb{C}$, where $Y$ is any Stein manifold with the volume density property. We prove that chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of $X$. In particular, $X$ has a chaotic holomorphic automorphism. A proof for $X=\mathbb{C}^n$ may be found in work of Forn\ae ss and Sibony. We follow their approach closely. Peters, Vivas, and Wold showed that a generic volume-preserving automorphism of $\mathbb{C}^n$, $n\geq 2$, has a hyperbolic fixed point whose stable manifold is dense in $\mathbb{C}^n$. This property can be interpreted as a kind of chaos. We generalise their theorem to a Stein manifold as above.