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Ergodic properties of matrix equilibrium states

Published online by Cambridge University Press:  14 March 2017

IAN D. MORRIS
IAN D. MORRIS*
Affiliation:
Mathematics Department, University of Surrey, Guildford, GU2 7XH, UK email i.morris@surrey.ac.uk
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Abstract

Given a finite irreducible set of real $d\times d$ matrices $A_{1},\ldots ,A_{M}$ and a real parameter $s>0$, there exists a unique shift-invariant equilibrium state on $\{1,\ldots ,M\}^{\mathbb{N}}$ associated to $(A_{1},\ldots ,A_{M},s)$. In this paper we characterize the ergodic properties of such equilibrium states in terms of the algebraic properties of the semigroup generated by the associated matrices. We completely characterize when the equilibrium state has zero entropy, when it gives distinct Lyapunov exponents to the natural cocycle generated by $A_{1},\ldots ,A_{M}$, and when it is a Bernoulli measure. We also give a general sufficient 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 Kusuoka are explored in an appendix.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 6 , September 2018 , pp. 2295 - 2320
Copyright
© Cambridge University Press, 2017 

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