How to prove that some Bernoulli convolution has the weak Gibbs property

Éric Olivier (1) , Alain Thomas (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

In this paper we give one example of uniform convergence of the sequence of column vectors $\displaystyle{A_1\dots A_nV\over\left\Vert A_1\dots A_nV\right\Vert}$, $A_i\in\{A,B,C\}$, $A,B,C$ being some $(0,1)$-matrices of order $7$ with much null entries, and $V$ a fixed positive column vector. This example comes from the study of a continuous singular measure defined by infinite convolution; in the last section we deduce from the uniform convergence result that this measure has the weak Gibbs property.

Mots clés: Infinite products of nonnegative matrices; Gibbs properties; multifractal analysis of mesures; Bernoulli convolutions

Domaines: Mathématiques générales [math.GM]

Éric Olivier, Alain Thomas. How to prove that some Bernoulli convolution has the weak Gibbs property. 2010. ⟨hal-00493068v1⟩