Holomorphic motions of hyperbolic sets
type: Item
instance of: scholarly article
title: Holomorphic motions of hyperbolic sets
zbMATH Open document ID: 0974.37019
author: Mattias Jonsson
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: A holomorphic family \(\{f_a\}_{a\in D}\) of diffeomorphisms of a Hermitian manifold \(M\), parametrised by the unit disk \(D\) is considered. It is shown that if \(f_0\) has a hyperbolic invariant set \(K\), then there exists a small disk \(D_r\) and a holomorphic map \(h:D_r\times M\to M\) such that \(K_a:= h(a,k)\) is a hyperbolic invariant set of \(f_a\), for all \(a\in D_r\), and \(h_a:K \to K_a\) is a homeomorphism verifying \(f_a\circ h_a=h_a\circ f_a\). The map \(h\) is called a holomorphic motion. Sometimes a hyperbolic set \(K\) moves holomorphically with the parameter if \(f_a\) are only endomorphisms (this is the case when \(K\) is a repellor). Most often this is not so and examples are given. The proof of the main theorem above is a kind of complexification of the real case treated in the book of \textit{D. Ruelle} [Elements of differentiable dynamics and bifurcation theory, Boston, Academic Press (1989; Zbl 0684.58001)]. An idea of another proof is given, applying Sullivan's telescope construction. The general setting is very well done and the paper is exceptionally clear.
reviewed by: Zofia Denkowska
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 32H50
zbMATH DE Number: 1548273
zbMATH Keywords: hyperbolic sets
zbMATH Keywords: holomorphic multifunctions
zbMATH Keywords: holomorphic families of endomorphisms
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132192
OpenAlex ID: W2044071273
DOI: 10.1307/MMJ/1030132192
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description: scientific article; zbMATH DE number 1548273