Joint spectral radius, Sturmian measures, and the finiteness conjecture

TL;DR

This paper introduces a new ergodic theory-based approach to generating counterexamples to the Lagarias-Wang finiteness conjecture for the joint spectral radius of 2x2 matrices, revealing complex parameter relationships.

Contribution

It develops a novel method using Sturmian measures to produce uncountably many counterexamples, advancing understanding of the conjecture's limitations.

Findings

Identified an open subset of matrix pairs with uncountably many counterexamples.

Proved the relation between parameter t and Sturmian parameter P(t) forms a devil's staircase.

Provided a short proof connecting ergodic theory to the finiteness conjecture.

Abstract

The joint spectral radius of a pair of 2x2 real matrices $(A_0,A_1)\in M_2(\mathbb{R})^2$ is defined to be $r(A_0,A_1)= \limsup_{n\to\infty} \max \{\|A_{i_1}...A_{i_n}\|^{1/n}: i_j\in\{0,1\}\}$, the optimal growth rate of the norm of products of these matrices. The Lagarias-Wang finiteness conjecture, asserting that $r(A_0,A_1)$ is always the nth root of the spectral radius of some length-n product $A_{i_1}...A_{i_n}$, has been refuted by Bousch & Mairesse, with subsequent counterexamples presented by Blondel, Theys & Vladimirov; Kozyakin; Hare, Morris, Sidorov & Theys. In this article we introduce a new approach to generating finiteness counterexamples, and use this to exhibit an open subset of $M_2(\mathbb{R})^2$ with the property that each member $(A_0,A_1)$ of the subset generates uncountably many counterexamples of the form $(A_0, tA_1)$. Our methods employ ergodic theory, in particular the analysis of Sturmian invariant measures; this approach allows a short proof that the relation between the parameter $t$ and the Sturmian parameter $P(t)$ is a devil's staircase.