Non-Sturmian sequences of matrices providing the maximum growth rate of matrix products

12/01/2021

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by   Victor Kozyakin, et al.

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One of the most pressing problems in modern analysis is the study of the growth rate of the norms of all possible matrix products A_i_n⋯ A_i_0 with factors from a set of matrices 𝒜. So far, only for a relatively small number of classes of matrices 𝒜 has it been possible to rigorously describe the sequences of matrices {A_i_n} that guarantee the maximal growth rate of the corresponding norms. Moreover, in almost all theoretically studied cases, the index sequences {i_n} of matrices maximizing the norms of the corresponding matrix products turned out to be periodic or so-called Sturmian sequences, which entails a whole set of "good" properties of the sequences {A_i_n}, in particular the existence of a limiting frequency of occurrence of each matrix factor A_i∈𝒜 in them. The paper determines a class of 2× 2 matrices consisting of two matrices similar to rotations of the plane in which the sequence {A_i_n} maximizing the growth rate of the norms A_i_n⋯ A_i_0 is not Sturmian. All considerations are based on numerical modeling and cannot be considered mathematically rigorous in this part. Rather, they should be interpreted as a set of questions for further comprehensive theoretical analysis.