Renewal theorems, products of random matrices, and toral endomorphisms.(English)Zbl 1119.37008

Aikawa, Hiroaki (ed.) et al., Potential theory in Matsue. Selected papers of the international workshop on potential theory, Matsue, Japan, August 23–28, 2004. Tokyo: Mathematical Society of Japan (ISBN 4-931469-33-7/hbk). Advanced Studies in Pure Mathematics 44, 53-66 (2006).
Let $$G$$ be the linear group of the Euclidean space $$V={\mathbb R}^d$$ and $${\mathbb T}^d$$ be the $$d$$-dimensional torus $${\mathbb T}^d={\mathbb R}^d/{\mathbb Z}^d,$$ where $${\mathbb Z}^d$$ is the lattice of integer points in $${\mathbb R}^d$$. In the paper a subsemigroup $$T$$ of $$G$$ is considered. The orbit closures of $$T$$ in $$V$$ are studied. The results are applied to semigroups of endomorphisms of the $$d$$-dimensional torus.
For the entire collection see [Zbl 1102.31001].

MSC:

 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 47D07 Markov semigroups and applications to diffusion processes 60K05 Renewal theory

Keywords:

random matrix; semigroup; endomorphisms