Infinite products of nonnegative $2\times2$ matrices by nonnegative vectors

Alain Thomas (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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 $\displaystyle{M_{\omega_1}\dots M_{\omega_n}V\over\Vert M_{\omega_1}\dots M_{\omega_n}V\Vert}$. 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 $\forall n,\ M_{\omega_1}\dots M_{\omega_n}V\ne\begin{pmatrix}0\\0\end{pmatrix}$.

Mots clés: Infinite products of nonnegative matrices

Domaines: Mathématiques générales [math.GM]

Alain Thomas. Infinite products of nonnegative $2\times2$ matrices by nonnegative vectors. 2010. ⟨hal-00493739⟩