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.