Eric Olivier

Multifractal analysis of weak Gibbs measures and phase transition-application to some Bernoulli convolutions

For a given expanding d-fold covering transformation of the one-dimensional torus, the notion of weak Gibbs measure is defined by a natural generalization of the classical Gibbs property. For these measures, we prove that the singularity spectrum and the L q -spectrum form a Legendre transform pair. The main difficulty comes from the possible existence of first-order phase transition points, that is, points where the L q -spectrum is not differentiable. We give examples of weak Gibbs measure with phase

Measures of Full Dimension on Self-Affine Graphs

For a compact subset of the 2-torus that is left invariant by an expanding diagonal endomorphism, the Hausdorff and the Minkowski dimensions may not coincide: this dimensional hiatus is possible whenever the x-axis and y-axis expansion rates differ. The variational principle for dimension ensures that the Hausdorff dimension of the invariant compact set is obtained as the Hausdorff dimension of invariant probability measures called the measures of full dimension: we shall investigate such measures on examples related to classical self-affine graphs.