An explicit counterexample to the Lagarias-Wang finiteness conjecture

TL;DR

This paper provides the first explicit example of a set of matrices that disproves the long-standing Lagarias-Wang finiteness conjecture by explicitly identifying a specific set of 2x2 matrices that lacks the finiteness property.

Contribution

It explicitly constructs a counterexample to the Lagarias-Wang finiteness conjecture, resolving a long-standing open problem in the theory of joint spectral radius.

Findings

Identifies a specific value of alpha where the set fails the finiteness property

Provides the first explicit counterexample to the conjecture

Confirms the existence of non-finite sets of matrices in the 2x2 case

Abstract

The joint spectral radius of a finite set of real $d \times d$ matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the \emph{finiteness property} if there exists a periodic product which achieves this maximal rate of growth. J.C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real $d \times d$ matrices satisfies the finiteness property. However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the finiteness conjecture exist, showing in particular that there exists a family of pairs of $2 \times 2$ matrices which contains a counterexample. Similar results were subsequently given by V.D. Blondel, J. Theys and A.A. Vladimirov and by V.S. Kozyakin, but no explicit counterexample to the finiteness conjecture has so far been given. The purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the Lagarias-Wang finiteness conjecture. Namely, for the set \[ \mathsf{A}_{\alpha_*}:= \{({cc}1&1\\0&1), \alpha_*({cc}1&0\\1&1)\}\] we give an explicit value of \alpha_* \simeq 0.749326546330367557943961948091344672091327370236064317358024...] such that $\mathsf{A}_{\alpha_*}$ does not satisfy the finiteness property.