Matrix products with constraints on the sliding block relative frequencies of different factors

Abstract

One of fundamental results of the theory of joint/generalized spectral radius, the Berger-Wang theorem, establishes equality between the joint and generalized spectral radii of a set of matrices. Generalization of this theorem on products of matrices whose factors are applied not arbitrarily but are subjected to some constraints is connected with essential difficulties since known proofs of the Berger-Wang theorem rely on the arbitrariness of appearance of different matrices in the related matrix products. Recently, X. Dai proved an analog of the Berger-Wang theorem for the case when factors in matrix products are formed by some Markov law. We introduce the concepts of the joint and generalized spectral radii for products of matrices subjected to constraints on the sliding block relative frequencies of occurrences of different matrices, and prove an analog of the Berger-Wang theorem for this case.