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Topological and geometric hyperbolicity criteria for polynomial automorphisms of ${\mathbb {C}^2}$

Part of: Complex dynamical systems

Published online by Cambridge University Press:  04 May 2021

ERIC BEDFORD  and
ROMAIN DUJARDIN
ERIC BEDFORD
Affiliation:
Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY 11794, USA (e-mail: ebedford@math.stonybrook.edu)
ROMAIN DUJARDIN*
Affiliation:
Sorbonne Université, CNRS, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM), F-75005 Paris, France
*
e-mail: romain.dujardin@sorbonne-universite.fr
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Abstract

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We prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of $\mathbb {C}^2$. Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.

Keywords

polynomial automorphismscomplex Hénon maphyperbolicitynon-uniform hyperbolicity

MSC classification

Primary: 37F15: Expanding maps; hyperbolicity; structural stability 37F10: Polynomials; rational maps; entire and meromorphic functions

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 7 , July 2022 , pp. 2151 - 2171
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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