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

TL;DR

This paper demonstrates that for certain matrix classes, the sequences maximizing growth rates are not Sturmian, challenging previous assumptions and opening new questions for theoretical analysis in matrix product growth dynamics.

Contribution

It introduces a class of matrices where the optimal growth sequences are non-Sturmian, contrary to prior findings that mostly involved Sturmian or periodic sequences.

Findings

Maximizing sequences can be non-Sturmian for specific matrix classes

Numerical modeling suggests non-Sturmian growth-maximizing sequences exist

Challenges existing beliefs about the nature of optimal matrix product sequences

Abstract

In the theory of linear switching systems with discrete time, as in other areas of mathematics, the problem of studying the growth rate of the norms of all possible matrix products $A_{\sigma_{n}}\cdots A_{\sigma_{0}}$ with factors from a set of matrices $\mathscr{A}$ arises. So far, only for a relatively small number of classes of matrices $\mathscr{A}$ has it been possible to accurately describe the sequences of matrices that guarantee the maximum rate of increase of the corresponding norms. Moreover, in almost all cases studied theoretically, the index sequences $\{\sigma_{n}\}$ of matrices maximizing the norms of the corresponding matrix products have been shown to be periodic or so-called Sturmian, which entails a whole set of "good" properties of the sequences $\{A_{\sigma_{n}}\}$, in particular the existence of a limiting frequency of occurrence of each matrix factor $A_{i}\in\mathscr{A}$ in them. In the paper it is shown that this is not always the case: a class of matrices is defined consisting of two $2\times 2$ matrices, similar to rotations in the plane, in which the sequence $\{A_{\sigma_{n}}\}$ maximizing the growth rate of the norms $\|A_{\sigma_{n}}\cdots A_{\sigma_{0}}\|$ 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.