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A necessary and sufficient condition for a matrix equilibrium state to be mixing

Published online by Cambridge University Press:  04 December 2017

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

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Since the 1970s there has been a rich theory of equilibrium states over shift spaces associated to Hölder-continuous real-valued potentials. The construction of equilibrium states associated to matrix-valued potentials is much more recent, with a complete description of such equilibrium states being achieved by Feng and Käenmäki [Equilibrium states of the pressure function for products of matrices. Discrete Contin. Dyn. Syst.30(3) (2011), 699–708]. In a recent article [Ergodic properties of matrix equilibrium states. Ergod. Th. & Dynam. Sys. (2017), to appear] the author investigated the ergodic-theoretic properties of these matrix equilibrium states, attempting in particular to give necessary and sufficient conditions for mixing, positive entropy, and the property of being a Bernoulli measure with respect to the natural partition, in terms of the algebraic properties of the semigroup generated by the matrices. Necessary and sufficient conditions were successfully established for the latter two properties, but only a sufficient condition for mixing was given. The purpose of this note is to complete that investigation by giving a necessary and sufficient condition for a matrix equilibrium state to be mixing.

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 8 , August 2019, pp. 2223 - 2234
Copyright
© Cambridge University Press, 2017 

References

Bárany, B., Käenmäki, A. and Koivusalo, H.. Dimension of self-affine sets for fixed translation vectors. Preprint, 2016, arXiv:1611.09196.Google Scholar
Cao, Y.-L., Feng, D.-J. and Huang, W.. The thermodynamic formalism for sub-additive potentials. Discrete Contin. Dyn. Syst. 20(3) (2008), 639–657.Google Scholar
Falconer, K. and Kempton, T.. Planar self-affine sets with equal Hausdorff, box and affinity dimensions. Ergod. Th. & Dynam. Sys. (2017), to appear.Google Scholar
Feng, D.-J. and Käenmäki, A.. Equilibrium states of the pressure function for products of matrices. Discrete Contin. Dyn. Syst. 30(3) (2011), 699–708.Google Scholar
Hueter, I. and Lalley, S. P.. Falconer’s formula for the Hausdorff dimension of a self-affine set in R 2 . Ergod. Th. & Dynam. Sys. 15(1) (1995), 77–97.Google Scholar
Morris, I. D.. Ergodic properties of matrix equilibrium states. Ergod. Th. & Dynam. Sys. (2017), to appear.Google Scholar
Morris, I. D. and Shmerkin, P.. On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems. Preprint, 2016, arXiv:1602.08789. Trans. Amer. Math. Soc., to appear.Google Scholar
Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990), 268.Google Scholar