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One of fundamental results of the theory of joint/generalized spectralradius, the Berger-Wang theorem, establishes equality between the joint andgeneralized spectral radii of a set of matrices. Generalization of this theoremon products of matrices whose factors are applied not arbitrarily but aresubjected to some constraints is connected with essential difficulties sinceknown proofs of the Berger-Wang theorem rely on the arbitrariness of appearanceof different matrices in the related matrix products. Recently, X. Dai provedan analog of the Berger-Wang theorem for the case when factors in matrixproducts are formed by some Markov law.  We introduce the concepts of the joint and generalized spectral radii forproducts of matrices subjected to constraints on the sliding block relativefrequencies of occurrences of different matrices, and prove an analog of theBerger-Wang theorem for this case.

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