The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1744–1762 | Cite as

Chaotic Holomorphic Automorphisms of Stein Manifolds with the Volume Density Property

  • Leandro Arosio
  • Finnur LárussonEmail author


Let X be a Stein manifold of dimension \(n\ge 2\) 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 Fornæss and Sibony. We follow their approach closely. Peters, Vivas, and Wold showed that a generic volume-preserving automorphism of \(\mathbb {C}^n\), \(n\ge 2\), 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.


Stein manifold Linear algebraic group Homogeneous space Holomorphic automorphism Volume-preserving automorphism Chaotic automorphism Andersén–Lempert theory Volume density property Algebraic volume density property Stable manifold 

Mathematics Subject Classification

Primary 32M17 Secondary 14L17 14R10 14R20 32H50 32M05 37F99 


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Authors and Affiliations

  1. 1.Dipartimento Di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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