Normalized image of a vector by an infinite product of nonnegative matrices

TL;DR

This paper investigates the convergence of normalized products of nonnegative matrices associated with sofic measures, providing a sufficient condition that supports multifractal analysis and applications like Bernoulli convolutions.

Contribution

It introduces a new sufficient condition for the convergence of normalized matrix products, advancing the understanding of multifractal formalism for sofic measures.

Findings

Established a convergence criterion for matrix product sequences

Applied the criterion to analyze Bernoulli convolutions

Enhanced the theoretical framework for multifractal analysis

Abstract

A sofic measure is the image of a Markov probability measure by a continuous morphism, and can be represented by means of products of matrices $A_n$ that belong to a finite set of nonnegative matrices. To prove that the multifractal formalism holds for such a measure, it is necessary to know whenever the sequence $n\mapsto\frac{A_1\cdots A_nv}{\Vert A_1\cdots A_nv\Vert}$ converges when $v$ is a positive vector. We give a sufficient condition for this convergence, that we use for the study of one Bernoulli convolution.