Lyapunov maximizing measures for balanced pairs of matrices

TL;DR

This paper proves that for every balanced pair of 2x2 real matrices, there exists a unique Lyapunov maximizing measure, which is always Sturmian, revealing a specific structure of optimal measures in this setting.

Contribution

It establishes the existence and uniqueness of Lyapunov maximizing measures for balanced pairs of 2x2 matrices and characterizes them as Sturmian, a novel result in the field.

Findings

Unique Lyapunov maximizing measure exists for each balanced pair.

The measure is always Sturmian.

Provides a structural understanding of maximizing measures for these matrix pairs.

Abstract

We show that every balanced pair (see Definition 1.1) of real $2\times 2$ matrices admits a unique Lyapunov maximizing measure, and the measure is always Sturmian.