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

Alain Thomas

Published 2018-08-22Version 1

Let $P_n=A_1\cdots A_n$~, where $\mathcal A=(A_n)_{n\in\mathbb N}$ is a sequence of $d\times d$ matrices, and let $V$ be a $d$-dimensional column-vector. We call "normalized image of~$V$ by the infinite product of the matrices $A_n$", the vector $\lim_{n\to\infty}P_nV/{\Vert P_nV\Vert}$ if exists. In general the sequence of matrices $n\mapsto P_n/{\Vert P_n\Vert}$ does not converge but, under some sufficient conditions specified in Theorem~A the sequence of vectors $n\mapsto P_nV/{\Vert P_nV\Vert}$ converges. We use Theorem~A to prove that certain sofic (i.e. linearly representable) measures satisfy the multifractal formalism.