Phase transitions for the multifractal analysis of self-similar measures

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We study the multifractal analysis of a class of self-similar measures with overlaps. This class, for which we obtain explicit formulae for the Lq spectrum tau(q) as well as the singularity spectrum f(alpha), is sufficiently large to point out new phenomena concerning the multifractal structure of self-similar measures. We show, that unlike the classical quasi-Bernoulli case, the Lq spectrum can have an arbitrarely 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 multifractal formalism. The weak quasi-Bernoulli property is the key point of most of the arguments.