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Dynamics of generic automorphisms of Stein manifolds with the density property

Part of: Holomorphic mappings and correspondences Complex spaces with a group of automorphisms Affine geometry Automorphic functions Complex manifolds

Published online by Cambridge University Press:  09 June 2025

LEANDRO AROSIO Open the ORCID record for LEANDRO AROSIO [Opens in a new window]  and
FINNUR LÁRUSSON Open the ORCID record for FINNUR LÁRUSSON [Opens in a new window]
LEANDRO AROSIO*
Affiliation:
Dipartimento Di Matematica, https://ror.org/02p77k626Università di Roma ‘Tor Vergata’, Via Della Ricerca Scientifica 1, 00133 Roma, Italy
FINNUR LÁRUSSON
Affiliation:
Discipline of Mathematical Sciences, https://ror.org/00892tw58University of Adelaide, South Australia 5005, Australia (e-mail: finnur.larusson@adelaide.edu.au)
*
e-mail: arosio@mat.uniroma2.it
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Abstract

We study the dynamics of a generic automorphism f of a Stein manifold with the density property. Such manifolds include almost all linear algebraic groups. Even in the special case of ${\mathbb {C}}^n$, $n\geq 2$, most of our results are new. We study the Julia set, non-wandering set and chain-recurrent set of f. We show that the closure of the set of saddle periodic points of f is the largest forward invariant set on which f is chaotic. This subset of the Julia set of f is also characterized as the closure of the set of transverse homoclinic points of f, and equals the Julia set if and only if a certain closing lemma holds. Among the other results in the paper is a generalization of Buzzard’s holomorphic Kupka–Smale theorem to our setting.

Keywords

dynamicsStein manifolddensity propertyJulia sethomoclinic point

MSC classification

Primary: 32H50: Iteration problems
Secondary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) 32M17: Automorphism groups of $^n$ and affine manifolds 32Q28: Stein manifolds
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 20
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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