Extremal sequences of polynomial complexity

Hare, KG, Morris, ID and Sidorov, N (2013) Extremal sequences of polynomial complexity Mathematical Proceedings of the Cambridge Philosophical Society, 155 (2). pp. 191-205.

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Abstract

The joint spectral radius of a bounded set of d × 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 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 ≥ 1, there exists a pair of square matrices of dimension 2p(2p+1-1) for which every extremal sequence has subword complexity at least 2 p2. © 2013 Cambridge Philosophical Society.

Item Type:	Article
Divisions :	Faculty of Engineering and Physical Sciences > Mathematics
Authors :	Name Email ORCID Hare, KG Morris, ID i.morris@surrey.ac.uk Sidorov, N
Date :	1 September 2013
DOI :	10.1017/S0305004113000157
Depositing User :	Symplectic Elements
Date Deposited :	17 May 2017 12:57
Last Modified :	11 Jun 2019 09:59
URI:	http://epubs.surrey.ac.uk/id/eprint/837383

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