On the existence of maximizing measures for irreducible countable Markov shifts: a dynamical proof

Abstract

We prove that if ${\Sigma }_{\mathbf{A}}(\mathbb{N})$ is an irreducible Markov shift space over $\mathbb{N}$ and $f:{\Sigma }_{\mathbf{A}}(\mathbb{N})\to \mathbb{R}$ is coercive with bounded variation then there exists a maximizing probability measure for f, whose support lies on a Markov subshift over a finite alphabet. Furthermore, the support of any maximizing measure is contained in this same compact subshift. To the best of our knowledge, this is the first proof beyond the finitely primitive case on the general irreducible non-compact setting. It's also noteworthy that our technique works for the full shift over positive real sequences.

14 pages

Cited by in corpus (10)

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• Zero temperature limits of Gibbs states for almost-additive potentials
• Gibbs States and Gibbsian Specifications on the space ${\mathbb{R}}^{\mathbb{N}}$
• The Ruelle operator for symmetric $\beta$-shifts
• A Large Deviation Principle for Gibbs States on Markov Shifts at Zero Temperature
• Peierls barrier for countable Markov shifts
• Existence of the zero temperature limit of equilibrium states on topologically transitive countable Markov shifts
• Pressure at infinity on countable Markov shifts
• Maximizing measures for countable alphabet shifts via blur shift spaces