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

Alain Thomas

Published 2022-05-17Version 1

Let $\nu$ be a measure on the set $\{0,1,\dots,b-1\}^\mathbb N$, linearly representable by means of a finite set $\mathcal M$ of square nonnegative matrices. To prove that $\nu$ has the weak Gibbs property, one generally use the uniform convergence on $\mathcal M^\mathbb N$ of the sequence of vectors $n\mapsto\frac{A_1\cdots A_nc}{\Vert A_1\cdots A_nc\Vert}$, where $c$ is a positive column-vector. We give a sufficient condition for this sequence to converge, that we apply to some measures defined by Bernoulli convolution. Secondarily we prove that the sequence of matrices $n\mapsto\frac{A_1\cdots A_n}{\Vert A_1\cdots A_n\Vert}$ in general diverges.