Abstract
It is well known that the space of invariant probability measures for transitive sub-shifts of finite type is a Poulsen simplex. In this article we prove that in the non-compact setting, for a large family of transitive countable Markov shifts, the space of invariant sub-probability measures is a Poulsen simplex and that its extreme points are the ergodic invariant probability measures together with the zero measure. In particular, we obtain that the space of invariant probability measures is a Poulsen simplex minus a vertex and the corresponding convex combinations. Our results apply to finite entropy non-locally compact transitive countable Markov shifts and to every locally compact transitive countable Markov shift. In order to prove these results we introduce a topology on the space of measures that generalizes the vague topology to a class of non-locally compact spaces, the topology of convergence on cylinders. We also prove analogous results for suspension flows defined over countable Markov shifts.
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Acknowledgments
We would like to thank Mike Todd for a wealth of relevant and interesting comments on the subject of this article. We would also like to thank the referee for many useful comments and suggestions. This paper was initiated while the second author was visiting the first author at Pontificia Universidad Catolica de Chile. The second author would like to thank the dynamics group at PUC for making his visit very stimulating. He would also like to thank Richard Canary for his invitation to participate in ‘Workshop on Groups, Geometry and Dynamics’ held at Universidad de la Republica, where an important part of this work was prepared.
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G. I. was partially supported by CONICYT PIA ACT172001 and by Proyecto Fondecyt 1190194.
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Iommi, G., Velozo, A. The space of invariant measures for countable Markov shifts. JAMA 143, 461–501 (2021). https://doi.org/10.1007/s11854-021-0159-2
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DOI: https://doi.org/10.1007/s11854-021-0159-2