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Coarse Expanding Conformal Dynamics
Peter Haïssinsky, Université de Provence, Marseille, France, and Kevin M. Pilgrim, Indiana University, Bloomington, IN
A publication of the Société Mathématique de France.
 Astérisque 2009; 139 pp; softcover Number: 325 ISBN-10: 2-85629-266-6 ISBN-13: 978-2-85629-266-2 List Price: US$57 Member Price: US$45.60 Order Code: AST/325 Motivated by the dynamics of rational maps, the authors introduce a class of topological dynamical systems satisfying certain topological regularity, expansion, irreducibility, and finiteness conditions. The authors call such maps "topologically coarse expanding conformal" (top. CXC) dynamical systems. Given such a system $$f: X \to X$$ and a finite cover of $$X$$ by connected open sets, the authors construct a negatively curved infinite graph on which $$f$$ acts naturally by local isometries. The induced topological dynamical system on the boundary at infinity is naturally conjugate to the dynamics of $$f$$. This implies that $$X$$ inherits metrics in which the dynamics of $$f$$ satisfies the Principle of the Conformal Elevator: arbitrarily small balls may be blown up with bounded distortion to nearly round sets of definite size. This property is preserved under conjugation by a quasisymmetric map, and (top. CXC) dynamical systems on a metric space satisfying this property the authors call "metrically CXC". The ensuing results deepen the analogy between rational maps and Kleinian groups by extending it to analogies between metric CXC systems and hyperbolic groups. The authors give many examples and several applications. In particular, they provide a new interpretation of the characterization of rational functions among topological maps and of generalized Lattès examples among uniformly quasiregular maps. Via techniques in the spirit of those used to construct quasiconformal measures for hyperbolic groups, the authors also establish existence, uniqueness, naturality, and metric regularity properties for the measure of maximal entropy of such systems. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in geometry and topology. Table of Contents Introduction Coarse expanding conformal dynamics Geometrization Expanding non-invertible dynamics A. Quasiconformal analysis B. Hyperbolic groups in a nutshell Bibliography Index Table of symbols
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