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Joint spectral radius, Sturmian measures and the finiteness conjecture
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Jenkinson, Oliver and Pollicott, Mark (2018) Joint spectral radius, Sturmian measures and the finiteness conjecture. Ergodic Theory and Dynamical Systems, 38 (8). pp. 3062-3100. doi:10.1017/etds.2017.18 ISSN 0143-3857.
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Official URL: https://doi.org/10.1017/etds.2017.18
Abstract
The joint spectral radius of a pair of 2×2 real matrices (A0,A1)∈M2(R)2(A0,A1)∈M2(R)2 is defined to be r(A0,A1)=lim supn→∞max{∥Ai1⋯Ain∥1/n:ij∈{0,1}}r(A0,A1)=lim supn→∞max{‖Ai1⋯Ain‖1/n:ij∈{0,1}} , the optimal growth rate of the norm of products of these matrices. The Lagarias–Wang finiteness conjecture [Lagarias and Wang. The finiteness conjecture for the generalized spectral radius of a set of matrices. Linear Algebra Appl. 214 (1995), 17–42], asserting that r(A0,A1)r(A0,A1) is always the nn th root of the spectral radius of some length- nn product Ai1⋯AinAi1⋯Ain , has been refuted by Bousch and Mairesse [Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture. J. Amer. Math. Soc. 15 (2002), 77–111], with subsequent counterexamples presented by Blondel et al [An elementary counterexample to the finiteness conjecture. SIAM J. Matrix Anal. 24 (2003), 963–970], Kozyakin [A dynamical systems construction of a counterexample to the finiteness conjecture. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (Seville, Spain, December 2005). IEEE, Piscataway, NJ, pp. 2338–2343] and Hare et al [An explicit counterexample to the Lagarias–Wang finiteness conjecture. Adv. Math. 226 (2011), 4667–4701]. In this article, we introduce a new approach to generating finiteness counterexamples, and use this to exhibit an open subset of M2(R)2M2(R)2 with the property that each member (A0,A1)(A0,A1) of the subset generates uncountably many counterexamples of the form (A0,tA1)(A0,tA1) . Our methods employ ergodic theory; in particular, the analysis of Sturmian invariant measures. This approach allows a short proof that the relationship between the parameter tt and the Sturmian parameter P(t)P(t) is a devil’s staircase
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Ergodic theory , Matrices | ||||||||
Journal or Publication Title: | Ergodic Theory and Dynamical Systems | ||||||||
Publisher: | Cambridge University Press | ||||||||
ISSN: | 0143-3857 | ||||||||
Official Date: | December 2018 | ||||||||
Dates: |
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Volume: | 38 | ||||||||
Number: | 8 | ||||||||
Page Range: | pp. 3062-3100 | ||||||||
DOI: | 10.1017/etds.2017.18 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 31 January 2017 | ||||||||
Date of first compliant Open Access: | 2 December 2017 |
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