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 an 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. These matrices come from the study of the Bernoulli convolution in the base $\beta>1$ such that $\beta^3=2\beta^2-\beta+1$, that is, the (continuous singular) probability distribution of the random variable $\displaystyle(\beta-1)\sum_{n=1}^\infty{\omega_n\over\beta^n}$ when the independent random variables $\omega_n$ take the values $0$ and $1$ with probability $\displaystyle{1\over2}$. In the last section we deduce, from the uniform convergence of $\displaystyle{A_1\dots A_nV\over\left\Vert A_1\dots A_nV\right\Vert}$, the Gibbs and the multifractal properties of this measure.

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-00493068v2⟩