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 $d-$summable variation defined on a subshift $X$. Based on Meyerovitch's work, we prove that if $X$ 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 $X$ 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.

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

References in corpus (1)

• Gibbs and equilibrium measures for some families of subshifts

Cited by in corpus (1)

• A Dobrushin-Lanford-Ruelle theorem for irreducible sofic shifts