Chaotic holomorphic automorphisms of Stein manifolds with the volume density property

Abstract

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 Cn, any connected linear algebraic group that is not reductive, the Koras–Russell cubic, or a product Y×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=Cn may be found in work of Fornæss and Sibony. We follow their approach closely. Peters, Vivas, and Wold showed that a generic volume-preserving automorphism of Cn, n≥2, has a hyperbolic fixed point whose stable manifold is dense in Cn. This property can be interpreted as a kind of chaos. We generalise their theorem to a Stein manifold as above.