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

In this paper we give an example of uniform convergence of the sequence of column vectors $\displaystyle{A_1\dots A_nV\over\left\Vert A_1\dots A_nV\right\Vert}$,$A_i\in{A,B,C}$,$A,B,C$beingsome$(0,1)$-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 ${\displaystyle (\beta -1)\sum _{n=1}^{\mathrm{\infty }}\frac{{\omega }_n}{{\beta }^n}}$ when the independent random variables ${\omega }_n$ take the values $0$ and $1$ with probability ${\displaystyle \frac{1}{2}}$. 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.