Polynomial diffeomorphisms of \({\mathbb C}^2\): Currents, equilibrium measure and hyperbolicity
type: Item
instance of: scholarly article
title: Polynomial diffeomorphisms of \({\mathbb C}^2\): Currents, equilibrium measure and hyperbolicity
zbMATH Open document ID: 0721.58037
DOI: 10.1007/BF01239509
author: Eric Bedford
author: John Smillie
published in: Inventiones Mathematicae
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143853
review text: \textit{Sh. Friedland} and \textit{J. Milnor} [Ergodic Theory Dyn. Syst. 9, No.1, 67-99 (1989; Zbl 0651.58027)] have classified polynomial diffeomorphisms of \({\mathbb{C}}^ 2\) up to conjugation. One of these classes, noted by \({\mathcal G}\), consists of the finite compositions of generalized Hénon mappings, which have the form \(g(x,y)=(y,p(y)-ax)\) for a monic polynomial p of degree at least 2 and \(a\neq 0.\) The authors study the dynamics of maps \(g\in {\mathcal G}\), using the methods of plurisubharmonic Green functions and positive currents. An analogue of Brolin's theorem is proved. The notion of hyperbolicity for diffeomorphisms in \({\mathcal G}\) is introduced. If f is a hyperbolic diffeomorphism, then its periodic points are dense in the nonwandering set. It is shown that the set of hyperbolic polynomial diffeomorphisms is open in \({\mathcal G}\).
reviewed by: Ion Mihai
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 37D25
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 32C30
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32H50
zbMATH DE Number: 4188825
zbMATH Keywords: polynomial diffeomorphisms
zbMATH Keywords: generalized Hénon mappings
zbMATH Keywords: plurisubharmonic Green functions
zbMATH Keywords: currents
zbMATH Keywords: hyperbolic diffeomorphism
zbMATH Keywords: periodic points
MaRDI profile type: Publication
cites work: On the automorphism group of a Stein manifold
cites work: Fine topology, Šilov boundary, and \((dd^ c)^ n\)
cites work: Markov Partitions for Axiom A Diffeomorphisms
cites work: Unique ergodicity for horocycle foliations
cites work: Invariant sets under iteration of rational functions
cites work: Q5565773
cites work: Dynamical properties of plane polynomial automorphisms
cites work: A two-dimensional mapping with a strange attractor
cites work: Hénon mappings in the complex domain. I: The global topology of dynamical space
cites work: Q5590990
cites work: Q3888200
cites work: Currents, flows and diffeomorphisms
cites work: Q3744918
cites work: Cycles for the dynamical study of foliated manifolds and complex manifolds
description: scientific article