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Non-compact spaces of invariant measures

Godofredo Iommi, Anibal Velozo

TL;DR

The paper tackles the problem of non-compact spaces of invariant measures for transitive countable Markov shifts by introducing a metric compactification that yields a compact invariant-measure space affine homeomorphic to the Poulsen simplex. It then analyzes how the original space embeds densely inside this compactification and unveils a dichotomy: the new ergodic measures added are either a single atomic point $\\delta_{\\bar{\\infty}}$ (when the $\\mathcal{F}$-property holds) or a dense set of extreme points (when it fails). The authors further prove that the space of ergodic measures for the non-compact system is homeomorphic to $\\ell_2$, generalizing known results for subshifts of finite type, and develop a dual variational principle that extends thermodynamic formalism to this setting. Together, these results provide a unified geometric and variational framework for invariant measures on countable Markov shifts and connect compactification methods with equilibrium theory in non-compact dynamics.

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.

Non-compact spaces of invariant measures

TL;DR

The paper tackles the problem of non-compact spaces of invariant measures for transitive countable Markov shifts by introducing a metric compactification that yields a compact invariant-measure space affine homeomorphic to the Poulsen simplex. It then analyzes how the original space embeds densely inside this compactification and unveils a dichotomy: the new ergodic measures added are either a single atomic point (when the -property holds) or a dense set of extreme points (when it fails). The authors further prove that the space of ergodic measures for the non-compact system is homeomorphic to , generalizing known results for subshifts of finite type, and develop a dual variational principle that extends thermodynamic formalism to this setting. Together, these results provide a unified geometric and variational framework for invariant measures on countable Markov shifts and connect compactification methods with equilibrium theory in non-compact dynamics.

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 , 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.

Paper Structure

This paper contains 23 sections, 23 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Compactification of the space of invariant probability measures
  3. The space of ergodic measures
  4. Thermodynamic formalism
  5. Preliminaries
  6. Countable Markov Shifts
  7. The weak$^*$ and the cylinder topologies
  8. Thermodynamic formalism
  9. The $\mathcal{F}-$property.
  10. Finite uniform Rome
  11. Metric compactification of $\Sigma$
  12. A compactification of $\Sigma$
  13. Topology on $\mathcal{M}(\bar{\Sigma},{\bar{\sigma}})$
  14. Finitely additive measures
  15. Affine geometry of $\mathcal{M}(\bar{\Sigma},{\bar{\sigma}})$ and $\mathcal{M}(\Sigma,\sigma)$
  16. ...and 8 more sections

Key Result

theorem 1.1

Let $(\Sigma, \sigma)$ be a transitive countable Markov shift. Then, $\mathcal{M}(\bar{\Sigma},\bar{\sigma})$ is affine homeomorphic to the Poulsen simplex, and $\mathcal{M}(\Sigma,\sigma)$ is a dense subset of $\mathcal{M}(\bar{\Sigma},\bar{\sigma})$.

Theorems & Definitions (58)

  • theorem 1.1
  • theorem 1.2
  • theorem 1.3
  • theorem 1.4
  • remark 2.1
  • definition 2.2
  • definition 2.3
  • remark 2.4
  • theorem 2.5
  • definition 2.6
  • ...and 48 more