Chaotic holomorphic automorphisms of 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.