Renewal theorems, products of random matrices, and toral endomorphisms

Abstract

We consider a subsemigroup $T$ of the linear group $G$ of the $d$-dimensional Euclidean space $V$, which is "sufficiently large". We study the orbit closures of $T$ in $V$ and we apply the results to semigroups of endomorphisms of the $d$-dimensional torus. The method uses the knowledge of the potential kernel of the Markov chain on $V$ defined by a probability measure supported on $T$. The condition of being "large" is satisfied for example by a subsemigroup of $SL(V)$, Zariski-dense in $SL(V)$.