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Extremal sequences of polynomial complexity

Published online by Cambridge University Press:  02 May 2013

KEVIN G. HARE ,
IAN D. MORRIS  and
NIKITA SIDOROV
KEVIN G. HARE
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, CanadaN2L 3G1 e-mail: [email protected]
IAN D. MORRIS
Affiliation:
Department of Mathematics, University of Surrey, Guildford GU2 7XH. e-mail: [email protected]
NIKITA SIDOROV
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M 13 9PL. e-mail: [email protected]
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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−p2np.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 2 , September 2013 , pp. 191 - 205
Copyright
Copyright © Cambridge Philosophical Society 2013 

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