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We consider a class of non-conformal expanding maps on the $d$-dimensionaltorus. For an equilibrium measure of an H\"older potential, we prove ananalogue of the Central Limit Theorem for the fluctuations of the logarithm ofthe measure of balls as the radius goes to zero. An unexpected consequence isthat when the measure is not absolutely continuous, then half of the balls ofradius $\eps$ have a measure smaller than $\eps^\delta$ and half of them have ameasure larger than $\eps^\delta$, where $\delta$ is the Hausdorff dimension ofthe measure. We first show that the problem is equivalent to the study of thefluctuations of some Birkhoff sums. Then we use general results fromprobability theory as the weak invariance principle and random change of timeto get our main theorem. Our method also applies to conformal repellers andAxiom A surface diffeomorphisms and possibly to a class of one-dimensional nonuniformly expanding maps. These generalizations are presented at the end of thepaper.

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