Rigidity of holomorphic Collet-Eckmann repellers
type: Item
instance of: scholarly article
title: Rigidity of holomorphic Collet-Eckmann repellers
zbMATH Open document ID: 1034.37026
DOI: 10.1007/BF02412220
author: Feliks Przytycki
author: Steffen Rohde
published in: Arkiv för Matematik
publication date: 2004-03-15T00:00:00Z
review text: Let \(X\) be a compact subset of the Riemann sphere \(P_1( {\mathbb C})\), let \(U\) be a neighborhood of \(X\) and consider a holomorphic map \(f: U \to P_1({\mathbb C})\) with \(f(X) =X\). The pair \((X,f)\) is called a holomorphic repeller if the forward orbits of only the points of \(X\) remain in a neighborhood of \(X\). Following a definition introduced by P. Collet and J.-P. Eckmann about nonuniformly hyperbolic one-dimensional maps, the authors call a holomorphic repeller \((X,f)\) Coller-Eckmann if there are constants \(C>0\) and \(\lambda > 1\) such that for every critical point \(c\in X\) of \(f\) whose forward orbit does not meet other critical points of \(f\), one has the estimate \(| (f^n)' (f(c))| \geq C \lambda^n\) for all integers \(n\geq 0\). Also, the authors call a holomorphic repeller \((X,f)\) topological Collet-Eckmann if there exists a constant \(d\geq 1\) and for each \(x\in X\) a set \(G(x)\) of positive integers of lower density \(\geq { 1\over 2}\) such that for every \(n\in G(x)\), there exists a connected neighborhood of \(x\) which is mapped properly by \(f^n\) to a large disc, with topological degree at most \(d\). Their first theorem extends a result due to \textit{C. T. McMullen} and \textit{D. Sullivan} for rational maps [Adv. Math. 135, 351--395 (1998; Zbl 0926.30028)]: If \((X,f)\) and \((Y,g)\) are holomorphic repellers which are conjugate by an orientation-preserving homeomorphism \(h_0\) and if \((X,f)\) is topological Collet-Eckmann, then there exists a semi-local quasiconformal conjugacy \(h\) between \(f\) and \(g\) extending the restriction \(h_0|_X\) to a neighborhood of \(X\). In the proof, they apply the classical characterization of quasiconformal mappings due to \textit{J. Heinonen} and \textit{P. Koskela } [Invent. Math. 120, 61--79 (1995; Zbl 0832.30013)]. As one of the corollaries (also obtained independently by \textit{P. W. Jones} and \textit{S. Smirnov} [Ark. Mat. 38, 263--279 (2000)]) they deduce that if \(f\) …
reviewed by: Joël Merker
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37D25
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 37C15
zbMATH DE Number: 2057371
zbMATH Keywords: holomorphic map
zbMATH Keywords: holomorphic repeller
zbMATH Keywords: nonuniformly hyperbolic maps
zbMATH Keywords: conjugacy
zbMATH Keywords: quasiconformal mappings
zbMATH Keywords: holomorphic polynomials
zbMATH Keywords: shrinking of neighborhoods
zbMATH Keywords: distortion
zbMATH Keywords: reversed telescopic construction
MaRDI profile type: Publication
cites work: Julia and John
cites work: Positive Liapunov exponents and absolute continuity for maps of the interval
cites work: Collet, Eckmann and Hölder
cites work: Definitions of quasiconformality
cites work: Q4847893
cites work: Q4222592
cites work: Local connectivity of the Julia set of real polynomials
cites work: The Fibonacci Unimodal Map
cites work: On a theorem of Fatou
cites work: Quasiconformal homeomorphisms and dynamics. III: The Teichmüller space of a holomorphic dynamical system
cites work: Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials
cites work: Q4332755
cites work: Q4384023
Recommended article: Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps
Recommended article: Unbounded components in parameter space of rational maps
Recommended article: Q4332755
Recommended article: Quasiconformal homeomorphisms and dynamics. III: The Teichmüller space of a holomorphic dynamical system
Recommended article: Statistical properties of topological Collet–Eckmann maps
Recommended article: Equivalence of Collet–Eckmann conditions for slowly recurrent rational maps
Recommended article: Q4941622
Recommended article: A connecting lemma for rational maps satisfying a no-growth condition
Recommended article: Collet–Eckmann condition for recurrent critical orbits implies uniform hyperbolicity on periodic orbits
Recommended article: Extending rational maps
description: scientific article; zbMATH DE number 2057371