# Coarse Expanding Conformal Dynamics

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*Peter Haïssinsky; Kevin M. Pilgrim*

A publication of the Société Mathématique de France

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.