Extremal sequences of polynomial complexity (Q2845616)

The authors define the joint spectral radius of a bounded set of \(d\times d\) real matrices as 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 extremal if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of \textit{J. C. Lagarias} and \textit{Y. Wang} [Linear Algebra Appl. 214, 17--42 (1995; Zbl 0818.15007)] 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 the paper, the authors 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 \(2^{ -p^2}n^p\).