Phase transitions for the multifractal analysis of self-similar measures

Abstract

We are interested in the multifractal analysis of a class of self-similar measures with overlaps. This class, for which we obtain explicit formulae for the Lq-spectrum, τ(q), as well as the singularity spectrum f(α), is sufficiently large to point out new phenomena in the multifractal structure of self-similar measures. We show that, unlike in the classical quasi-Bernoulli case, the Lq-spectrum, τ(q), of the measures studied can have an arbitrarily large number of non-differentiability points (phase transitions). These singularities occur only for the negative values of q and yield to measures that do not satisfy the usual multifractal formalism. The weak quasi-Bernoulli property is the key point of most of the arguments.