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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: kghare@uwaterloo.ca
IAN D. MORRIS
Affiliation:
Department of Mathematics, University of Surrey, Guildford GU2 7XH. e-mail: i.morris@surrey.ac.uk
NIKITA SIDOROV
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M 13 9PL. e-mail: sidorov@manchester.ac.uk
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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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References

REFERENCES

[1]Allouche, J.–P. and Shallit, J.Automatic Sequences (Cambridge University Press, 2003).CrossRefGoogle Scholar
[2]Barabanov, N. E.On the Lyapunov exponent of discrete inclusions. I, Avtomat. i Telemekh. (1988), no. 2, 40–46.Google Scholar
[3]Blondel, V., Theys, J. and Vladimirov, A. A.An elementary counterexample to the finiteness conjecture. SIAM J. Matrix Anal. Appl. 24 (2003), 963–970.CrossRefGoogle Scholar
[4]Fogg, N. PytheasSubstitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Math. 1794 (Springer-Verlag, Berlin, 2002).CrossRefGoogle Scholar
[5]Gurvits, L.Stability of discrete linear inclusion. Linear Algebra Appl. 231 (1995), 47–85.CrossRefGoogle Scholar
[6]Hare, K. G., Morris, I. D., Sidorov, N. and Theys, J.An explicit counterexample to the Lagarias–Wang finiteness conjecture. Adv. Math. 226 (2011), 4667–4701.CrossRefGoogle Scholar
[7]Horn, R. A. and Johnson, C. R., Topics in Matrix Analysis (Cambridge University Press, 1991).CrossRefGoogle Scholar
[8]Katok, A. and Hasselblatt, B.Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, 1996).Google Scholar
[9]Jungers, R. M. and Blondel, V. D. Is the joint spectral radius of rational matrices reachable by a finite product? In Proceedings of the satellite workshops of DLT2007, Turku Centre for Computer Science, pages 25–37 (2007).Google Scholar
[10]Jungers, R. M. and Blondel, V. D.On the finiteness property for rational matrices. Lin. Alg. Appl. 428 (2008), 2283–2295.CrossRefGoogle Scholar
[11]Jungers, R. M.The joint spectral radius: theory and applications, Lecture Notes in Control and Information Sciences, vol. 385 (Springer-Verlag, 2009).CrossRefGoogle Scholar
[12]Kozyakin, V. S.Structure of extremal trajectories of discrete linear systems and the finiteness conjecture. Automation and Remote Control 68 (2007), Issue 1, 174–209.CrossRefGoogle Scholar
[13]Lagarias, J. and Wang, Y.The finiteness conjecture for the generalized spectral radius of a set of matrices. Linear Algebra Appl. 214 (1995), 17–42.CrossRefGoogle Scholar
[14]Lothaire, M.Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications (Cambridge University Press, 2002).CrossRefGoogle Scholar
[15]Morris, I. D.A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory. Adv. Math. 225 (2010), 3425–3445.CrossRefGoogle Scholar
[16]Morris, I. D. Mather sets for sequences of matrices and applications to the study of joint spectral radii, to appear in Proc. London Math. Soc. http://arxiv.org/abs/1109.4615.Google Scholar
[17]Morris, I. D. and Sidorov, N. On a Devil's staircase associated to the joint spectral radii of a family of pairs of matrices, to appear in J. Eur. Math. Soc. (JEMS), http://arxiv.org/abs/1107.3506.Google Scholar
[18]Peres, Y.A combinatorial application of the maximal ergodic theorem. Bull. London Math. Soc. 20 (1988), no. 3, 248–252.CrossRefGoogle Scholar
[19]Rota, G.-C. and Strang, W. G.A note on the joint spectral radius. Indag. Math. 22 (1960), 379–381.CrossRefGoogle Scholar
[20]Teichner, R. and Margaliot, M.Explicit construction of a Barabanov norm for a class of positive planar discrete-time linear switched systems. Automatica. 48 (2012), Issue 1, 95–101.CrossRefGoogle Scholar
[21]Wirth, F.The generalized spectral radius and extremal norms. Linear Algebra Appl. 342 (2002), 17–40.CrossRefGoogle Scholar