On asymptotic properties of matrix semigroups with an invariant cone

Recently, several research efforts showed that the analysis of joint spectral characteristics of sets of matrices is greatly eased when these matrices share an invariant cone. In this short note we prove two new results in this direction. We prove that the joint spectral subradius is continuous in the neighborhood of sets of matrices that leave an embedded pair of cones invariant. We show that the (averaged) maximal spectral radius, as well as the maximal trace, of products of length t, converge towards the joint spectral radius when the matrices share an invariant cone, and addi- tionally one of them is primitive.