The space of invariant measures for countable Markov shifts

• DOI: 10.1007/s11854-021-0159-2
• Publication Date: 2021-06-29T11:02:44Z
• Authors: Godofredo Iommi; Anibal Velozo
• Keywords: Simplex; Markov kernel; Second-countable space

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.<br/>Comments welcome; Final version to appear in Journal d'analyse<br/>