Serie : Astérisque. Vol 325
Paru le 30/04/2010 | Broché 138 pages
Professionnels
Motivated by the dynamics of rational maps, we introduce a class of topological dynamical systems satisfying certain topological regularity, expansion, irreducibility, and finiteness conditions. We call such maps "topologically coarse expanding conformal" (top. CXC) dynamical systems. Given such a system f : X Vecteur X and a finite cover of X by connected open sets, we 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 we 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. We give many examples and several applications. In particular, we 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, we also establish existence, uniqueness, naturality, and metric regularity properties for the measure of maximal entropy of such systems.
