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 $\frac{A_1\dots A_{n}V}{\Vert A_1\dots A_{n}V\Vert }$, $A_i\in \{A,B,C\}$, $A,B,C$ 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 $(\beta -1)\sum _{n=1}^{\infty }\frac{{\omega }_n}{{\beta }^n}$ when the independent random variables ${\omega }_n$ take the values $0$ and $1$ with probability $\frac{1}{2}$. In the last section we deduce, from the uniform convergence of $\frac{A_1\dots A_{n}V}{\Vert A_1\dots A_{n}V\Vert }$, the Gibbs and the multifractal properties of this measure.

We have included the content of this paper in arXiv:0908.4171

References in corpus (1)

• Can an infinite left-product of nonnegative matrices be expressed in terms of infinite left-products of stochastic ones?