Gibbs measures on Subshifts
Abstract
The notion of Gibbs Measure is used by many researchers of the communities of Mathematical Physics, Probability, Thermodynamic Formalism, Symbolic Dynamics, and others. A natural question is when these several different notions of Gibbs measure coincide. We study the properties of Gibbs measures for functions with summable variation defined on a subshift . Based on Meyerovitch's work, we prove that if is a subshift of finite type (SFT), then any equilibrium measure is also a Gibbs measure. Although the definition provided by Meyerovitch does not make any mention to conditional expectations, we show that in the case where is a SFT, it is possible to characterize these measures in terms of more familiar notions presented in the literature of Mathematical Physics using DLR equations.
Cite
@article{arxiv.2008.13727,
title = {Gibbs measures on Subshifts},
author = {Bruno Kimura},
journal= {arXiv preprint arXiv:2008.13727},
year = {2020}
}
Comments
Motivated by colleagues' questions, this Master Dissertation proves that, under the suitable hypothesis, the notions of Gibbs measure defined by Capocaccia, Dobrushin-Lanford-Ruelle (DLR), Conformal, and others coincide. Some implications are true for SFT, others in general. A review will be written in the future, including the notion of DLR measure in the setting of groupoids