seminar: Central limit theorem for dimension of Gibbs measures in hyperbolic dynamics
speaker: Benoît Saussol (Université de Bretagne Occidentale)abstract: For an equilibrium measure of an Hoelder 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.
An unexpected consequence is that when the measure is not absolutely continuous, then half of the balls of radius $eps$ have a measure smaller than $epsdelta$ and half of them have a measure larger than $epsdelta$, where $delta$ is the Hausdorff dimension of the measure.
We first show that the problem is equivalent to the study of the fluctuations of some Birkhoff sums. Then we use general results from probability theory as the weak invariance principle and random change of time to get our main theorem.
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.
(joint work with Renaud Leplaideur)
timetable:
Wed 12 May, 17:00 - 18:00, Sala Riunioni del Centro De Giorgi
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