ATTRACTING BASINS OF VOLUME PRESERVING AUTOMORPHISMS OF ℂ^k

Abstract

We study topological properties of attracting sets for automorphisms of C k . Our main result is that a generic volume preserving automorphism has a hyperbolic fixed point with a dense stable manifold. On the other hand, we show that an attracting set can only contain a neighborhood of the fixed point if it is an attracting fixed point. We will see that the latter does not hold in the non-autonomous setting.

Chaotic Holomorphic Automorphisms of Stein Manifolds with the Volume Density Property

“…Peters, Vivas, and Wold point out that polynomial automorphisms with an attracting fixed point at infinity are dense in Aut ω 0 (C n ) [17,Remark 3.1]. It follows that the residual subset of chaotic automorphisms has empty interior in Aut ω 0 (C n ).…”

Dynamics of transcendental Hénon maps-II

“…The analogous statement was proved for volume preserving holomorphic automorphisms of C n in [11], and generalized to Stein manifolds with the volume density property in [2].…”

The first thirty years of Andersen-Lempert theory

“…For X = C n , n ≥ 2, with the standard volume form this was proved by Fornaess and Sibony [69], and the authors follow their approach. They also showed that a generic volume preserving automorphism has a hyperbolic fixed point whose stable manifold is dense in X, generalizing a result of Peters, Vivas, and Wold on C n [144]. In their second paper [25], they proved closing lemmas for automorphisms of a Stein manifold with the density property and for endomorphisms of an Oka Stein manifold.…”