We study Markov shifts over countable (finite or countably infinite) alphabets, i.e. shifts generated by incidence matrices. In particular, we derive necessary and sufficient conditions for the existence of a Gibbs state for a certain class of infinite Markov shifts. We further establish a characterization of the existence, uniqueness and ergodicity of invariant Gibbs states for this class of shifts. Our results generalize the well-known results for finitely irreducible Markov shifts.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm221-3-3,
author = {Andrei E. Ghenciu and Mario Roy},
title = {Gibbs states for non-irreducible countable Markov shifts},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {231-265},
zbl = {1286.37011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm221-3-3}
}
Andrei E. Ghenciu; Mario Roy. Gibbs states for non-irreducible countable Markov shifts. Fundamenta Mathematicae, Tome 220 (2013) pp. 231-265. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm221-3-3/