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Lyapunov-maximising measures for pairs of weighted shift operators

Morris, Ian (2017) Lyapunov-maximising measures for pairs of weighted shift operators Ergodic Theory and Dynamical Systems.

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Motivated by recent investigations of ergodic optimisation for matrix cocycles, we study the measures of maximum top Lyapunov exponent for pairs of bounded weighted shift operators on a separable Hilbert space. We prove that for generic pairs of weighted shift operators the Lyapunovmaximising measure is unique, and show that there exist pairs of operators whose unique Lyapunov-maximising measure takes any prescribed value less than log 2 for its metric entropy. We also show that in contrast to the matrix case, the Lyapunov-maximising measures of pairs of bounded operators are in general not characterised by their supports: we construct explicitly a pair of operators, and a pair of ergodic measures on the 2-shift with identical supports, such that one of the two measures is Lyapunov-maximising for the pair of operators and the other measure is not. Our proofs make use of the Ornstein d-metric to estimate di erences in the top Lyapunov exponent of a pair of weighted shift operators as the underlying measure is varied.

Item Type: Article
Subjects : Mathematics
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Date : 4 May 2017
Identification Number : 10.1017/etds.2017.22
Copyright Disclaimer : © Cambridge University Press, 2017 This article has been published in a revised form in Ergodic Theory and Dynamical Systems This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works.
Related URLs :
Depositing User : Symplectic Elements
Date Deposited : 03 Feb 2017 11:55
Last Modified : 19 Jul 2017 09:44

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