Extremal sequences of polynomial complexity

TL;DR

This paper investigates the complexity of extremal sequences related to the joint spectral radius of matrix sets, showing that for certain matrices, all extremal sequences have polynomial subword complexity, thus addressing a longstanding conjecture.

Contribution

It constructs specific matrix pairs for which all extremal sequences exhibit polynomial subword complexity, extending previous results on the complexity of extremal sequences.

Findings

Existence of matrix pairs with extremal sequences of polynomial complexity

Counterexamples to the periodic extremal sequence conjecture

Subword complexity grows at least as a polynomial for these sequences

Abstract

The joint spectral radius of a bounded set of $d \times d$ real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called \emph{extremal} if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. Wang asked whether every finite set of matrices admits an extremal sequence which is periodic. This is equivalent to the assertion that every finite set of matrices admits an extremal sequence with bounded subword complexity. Counterexamples were subsequently constructed which have the property that every extremal sequence has at least linear subword complexity. In this paper we extend this result to show that for each integer $p \geq 1$, there exists a pair of square matrices of dimension $2^p(2^{p+1}-1)$ for which every extremal sequence has subword complexity at least $2^{-p^2}n^p$.