Partial classification of spectrum maximizing products for pairs of $2\times2$ matrices

TL;DR

This paper investigates the existence and structure of spectrum maximizing products for pairs of 2x2 matrices, identifying regions with guaranteed simple SMPs and regions where SMPs may not exist but Sturmian measures do.

Contribution

It characterizes regions in the space of 2x2 matrix pairs where SMPs are guaranteed to exist and be simple, and identifies areas with potential non-existence of SMPs.

Findings

Regions with guaranteed SMP existence and simplicity

Existence of Sturmian maximizing measures in some regions

Identification of 'wild' regions with complex behavior

Abstract

Experiments suggest that typical finite sets of square matrices admit spectrum maximizing products (SMPs): that is, products that attain the joint spectral radius (JSR). Furthermore, those SMPs are often combinatorially "simple." In this paper, we consider pairs of real $2 \times 2$ matrices. We identify regions in the space of such pairs where SMPs are guaranteed to exist and to have a simple structure. We also identify another region where SMPs may fail to exist (in fact, this region includes all known counterexamples to the finiteness conjecture), but nevertheless a Sturmian maximizing measure exists. Though our results apply to a large chunk of the space of pairs of $2 \times 2$ matrices, including for instance all pairs of non-negative matrices, they leave out certain "wild" regions where more complicated behavior is possible.