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

Published online by Cambridge University Press:  18 March 2013

RODRIGO BISSACOT  and
RICARDO DOS SANTOS FREIRE JR
RODRIGO BISSACOT
Affiliation:
Department of Applied Mathematics, IME-USP, Brazil email rodrigo.bissacot@gmail.com
RICARDO DOS SANTOS FREIRE JR
Affiliation:
Department of Mathematics, IME-USP, Brazil email rfreire@usp.br
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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} )\rightarrow \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 in the general irreducible non-compact setting. It is also noteworthy that our technique works for the full shift over positive real sequences.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 4 , August 2014 , pp. 1103 - 1115
Copyright
Copyright ©2013 Cambridge University Press 

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