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

Abstract

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.