ABSTRACT:

We will study the limit theorems satisfied by the Birkhoff sums of the observables x->x^{-a} (a>0) under the iteration of the doubling map T(x)=2x mod 1 on the interval [0,1]. Depending on the value of a, such Birkhoff sums satisfy the classical central limit theorem, or converge to so-called stable laws. The focus of the talk will not be on the result itself, but rather on the techniques we will use: they were introduced to study intermittent maps, but it will be the occasion to show that they can easily be adapted to deal with different settings.