Chaotic Holomorphic Automorphisms of Stein Manifolds with the Volume Density Property

Let X be a Stein manifold of dimension n >= 2 satisfying the volume density property with respect to an exact holomorphic volume form. For example, X could be C-n, any connected linear algebraic group that is not reductive, the Koras-Russell cubic, or a product Y x 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 = C-n may be found in work of Fornaess and Sibony. We follow their approach closely. Peters, Vivas, and Wold showed that a generic volume-preserving automorphism of C-n, n >= 2, has a hyperbolic fixed point whose stable manifold is dense in C-n. This property can be interpreted as a kind of chaos. We generalise their theorem to a Stein manifold as above.