Let f: be an arbitrary rational map of degree larger than 1. Denote by J(f) its Julia set. Let phi: J(f) -> a"e be a Holder continuous function such that P(phi) > sup(phi). It is known that there exists a unique equilibrium measure 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 . As its consequence we obtain that if and only if both the function phi: J(f) -> a"e is cohomologous to a constant in the class of continuous functions on J(f), and the rational function f: is a critically finite rational map with a parabolic orbifold. Secondly, real analyticity of topological pressure P(t phi) 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 Holder continuous observables. Also, the Law of Iterated Logarithm for all linear combinations of Holder 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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