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

TL;DR

This paper demonstrates the uniform convergence of certain matrix products related to Bernoulli convolutions with a specific algebraic base, leading to proofs of their Gibbs and multifractal properties.

Contribution

It provides a novel example linking matrix product convergence to the Gibbs property in Bernoulli convolutions with a specific algebraic base.

Findings

Established uniform convergence of matrix sequences

Proved the measure has the weak Gibbs property

Derived multifractal characteristics of the measure

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.