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Conformal Fractals: Ergodic Theory Methods
Introduction Basic examples and definitions 1. Measure preserving endomorphisms 2. Compact metric spaces 3. Distance expanding maps 4. Thermodynamical formalism 5. Expanding repellers in manifoldsExpand
Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps
Abstract.We show equivalence of several standard conditions for non-uniform hyperbolicity of complex rational functions, including the Topological Collet-Eckmann condition (TCE), UniformExpand
Lyapunov characteristic exponents are nonnegative
We prove that, for an arbitrary rational map f on the Riemann sphere and an arbitrary probability invariant measure on the Julia set, Lyapunov characteristic exponents are nonnegative a.e. InExpand
On the Perron-Frobenius-ruelle operator for rational maps on the riemann sphere and for Hölder continuous functions
AbstractLet be a rational function on the Riemann sphere, φ be a Hölder continuous function on the Julia set denote the Perron-Frobenius-Ruelle operator on the space of continuous functions:Expand
Conical limit set and Poincaré exponent for iterations of rational functions
We contribute to the dictionary between action of Kleinian groups and iteration of rational functions on the Riemann sphere. We define the Poincaré exponent δ(f, z) = inf{α ≥ 0 : P(z, α) ≤ 0}, whereExpand
External rays to periodic points
We prove that for every polynomial-like holomorphic mapP, ifaεK (filled-in Julia set) and the componentKaofK containinga is either a point ora is accessible along a continuous curve from theExpand
PREIMAGE ENTROPY FOR MAPPINGS
Several entropy-like invariants have been defined for noninvertible maps, based on various ways of measuring the dispersion of preimages and preimage sets in the past. We investigate basic propertiesExpand
Ergodicity of toral linked twist mappings
© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1983, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www.Expand
When do two rational functions have the same Julia set
It is proved that non-exceptional rational functions f and g on the Riemann sphere have the same measure of maximal entropy iff there exist iterates F of f and G of g and natural numbers M, N suchExpand
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