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Article Dans Une Revue Stochastics and Dynamics Année : 2012

CENTRAL LIMIT THEOREM FOR DIMENSION OF GIBBS MEASURES IN HYPERBOLIC DYNAMICS

Résumé

For an equilibrium measure of a Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. A noticeable consequence is that when this measure is not absolutely continuous, the probability that a ball of radius ε chosen at random have a measure smaller (or larger) than ε δ is asymptotically equal to 1/2, where δ is the Hausdorff dimension of the measure. Our method applies to a class of non-conformal expanding maps on the d-dimensional torus. It also applies to conformal repellers and Axiom A surface diffeomorphisms and possibly to a class of one-dimensional non-uniformly expanding maps. These generalizations are presented at the end of the paper.
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Dates et versions

hal-03996436 , version 1 (19-02-2023)

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Renaud Leplaideur, Benoît Saussol. CENTRAL LIMIT THEOREM FOR DIMENSION OF GIBBS MEASURES IN HYPERBOLIC DYNAMICS. Stochastics and Dynamics, 2012, 12 (02), pp.1150019. ⟨10.1142/S0219493712003675⟩. ⟨hal-03996436⟩
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