SEMIGROUPS CONTAINING PROXIMAL LINEAR-MAPS

A linear automorphism of a finite dimensional red vector space V is ca lled proximal if it has a unique eigenvalue-counting multiplicities-of maximal modulus. Goldsheid and Margulis have shown that if a subgroup G of GL(V) contains a proximal element then so does every Zariski den se subsemigroup H of G, provided V considered as a G-module is strongl y irreducible. We here show that H contains a finite subset M such tha t for every g is an element of GL(V) at least one of the elements gamm a g, gamma is an element of M, is proximal. We also give extensions an d refinements of this result in the following directions: a quantitati ve version of proximality, reducible representations, several eigenval ues of maximal modulus.