Infinite products of nonnegative $2\times 2$ matrices by nonnegative vectors

Abstract

Given a finite set $\{M_0,\dots ,M_{d-1}\}$ of nonnegative $2\times 2$ matrices and a nonnegative column-vector $V$, we associate to each $({\omega }_n)\in \{0,\dots ,d-1{\}}^{\mathbb{N}}$ the sequence of the column-vectors $\frac{M_{{\omega }_1}\dots M_{{\omega }_n}V}{\Vert M_{{\omega }_1}\dots M_{{\omega }_n}V\Vert }$. We give the necessary and sufficient condition on the matrices $M_k$ and the vector $V$ for this sequence to converge for all \hbox{$({\omega }_n)\in \{0,\dots ,d-1{\}}^{\mathbb{N}}$} such that $\forall n,M_{{\omega }_1}\dots M_{{\omega }_n}V\ne \left(\begin{array}{c}0\\ 0\end{array}\right)$.

8 pages