Central limit theorem for dimension of Gibbs measures for skew expanding maps

TL;DR

This paper establishes a Central Limit Theorem for the measure of small balls under Gibbs measures for certain expanding maps, revealing statistical properties of measure fluctuations and their relation to Hausdorff dimension.

Contribution

It proves a CLT for measure fluctuations in non-conformal expanding maps and extends the approach to conformal repellers and Axiom A systems.

Findings

Half of the small balls have measure less than ε^δ when the measure is not absolutely continuous.

The fluctuations of measure are governed by a CLT derived from Birkhoff sums.

Method applies to various classes of expanding and hyperbolic systems.

Abstract

We consider a class of non-conformal expanding maps on the $d$-dimensional torus. For an equilibrium measure of an H\"older 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 $\eps^\delta$ and half of them have a measure larger than $\eps^\delta$, 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 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.