Semigroups containing proximal linear maps

A linear automorphism of a finite dimensional real vector space V is called 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 dense subsemigroup H of G, provided V considered as a G-module is strongly irreducible. We here show that H contains a finite subset M such that for every g∈GL(V) at least one of the elements γg, γ∈M, is proximal. We also give extensions and refinements of this result in the following directions: a quantitative version of proximality, reducible representations, several eigenvalues of maximal modulus.