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Fine inducing and equilibrium measures for rational functions of the Riemann sphere

Artykuł
Czasopismo : ISRAEL JOURNAL OF MATHEMATICS   Tom: 210, Zeszyt: 1, Strony: 399–465
Michał Marcin Szostakiewicz [1] , [2] , Mariusz Urbański [3] , Anna Zdunik [4]
2015-09-01 angielski
Cechy publikacji
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  • Oryginalny artykuł naukowy
  • Zrecenzowana naukowo
Abstrakty ( angielski )
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Let f: ℂ̂ →ℂ̂ be an arbitrary rational map of degree larger than 1. Denote by J(f) its Julia set. Let φ: J(f) → ℝ be a Hölder continuous function such that P(φ) > sup(φ). It is known that there exists a unique equilibrium measure μφ for this potential. We introduce a special inducing scheme with fine recurrence properties. This construction allows us to prove four main results. Firstly, dimension rigidity, i.e., we characterize all maps and potentials for which HD(μφ)=HD(J(f)). As its consequence we obtain that HD(μφ)=2 if and only if both the function φ: J(f) → ℝ is cohomologous to a constant in the class of continuous functions on J(f), and the rational function f: ℂ̂ →ℂ̂ is a critically finite rational map with a parabolic orbifold. Secondly, real analyticity of topological pressure P(tφ) as a function of t. Third, some bold stochastic laws, namely, exponential decay of correlations, and, as its consequence, the Central Limit Theorem and the Law of Iterated Logarithm for Hölder continuous observables. Also, the Law of Iterated Logarithm for all linear combinations of Hölder continuous observables and the function log |f′|. Finally, its geometric consequences that allow us to compare equilibrium states with the appropriate generalized Hausdorff measures in the spirit of [PUZ].
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