Phase transitions for the multifractal analysis of self-similar measures

Details Phase transitions for the multifractal analysis of self-similar measures Phase transitions for the multifractal analysis of self-similar measures

2006-04-03

arxiv.org/abs/math/0604035v1

Mathematics

Classical Analysis and ODEs

Article accept\'{e} pour publication dans Nonlinearity. Article en ligne \`{a} http://www.iop.org/EJ/journal/Non

Scientific paper

10.1088/0951-7715/19/5/009

We study the multifractal analysis of a class of self-similar measures with overlaps. This class, for which we obtain explicit formulae for the L^q 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 L^q 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.

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