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.