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Ergodic properties of matrix equilibrium states

Morris, ID (2016) Ergodic properties of matrix equilibrium states Ergodic Theory and Dynamical Systems.

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Abstract. Given a nite irreducible set of real d d matrices A1; : : : ;AM and a real parameter s > 0, there exists a unique shift-invariant equilibrium state on f1; : : : ;MgN associated to (A1; : : : ;AM; s). In this article we characterise the ergodic properties of such equilibrium states in terms of the algebraic properties of the semigroup generated by the associated matrices. We completely characterise when the equilibrium state has zero entropy, when it gives distinct Lyapunov exponents to the natural cocycle generated by A1; : : : ;AM, and when it is a Bernoulli measure. We also give a general su cient condition for the equilibrium state to be mixing, and give an example where the equilibrium state is ergodic but not totally ergodic. Connections with a class of measures investigated by S. Kusuoka are explored in an appendix.

Item Type: Article
Subjects : Mathematics
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Morris, ID
Date : 2016
Copyright Disclaimer : This article has will be 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. © Cambridge University Press 2016
Related URLs :
Depositing User : Symplectic Elements
Date Deposited : 25 Oct 2016 07:57
Last Modified : 25 Oct 2016 07:57

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