Non-compact spaces of invariant measures

Abstract

We study a compactification of the space of invariant probability measures for a transitive countable Markov shift. We prove that it is affine homeomorphic to the Poulsen simplex. Furthermore, we establish that, depending on a combinatorial property of the shift space, the compactification contains either a single new ergodic measure or a dense set of them. As an application of our results, we prove that the space of ergodic probability measures of a transitive countable Markov shift is homeomorphic to $\ell_2$, extending to the non-compact setting a known result for subshifts of finite type. Additionally, we explore implications for thermodynamic formalism, including a version of the dual variational principle for transitive countable Markov shifts with uniformly continuous potentials.