Gibbs and equilibrium measures for some families of subshifts

Abstract

Abstract. For SFTs, any equilibrium measure is Gibbs, as long a f has dsummable variation. This is a theorem of Lanford and Ruelle. Conversely, a theorem of Dobrušin states that for strongly-irreducible subshifts, shiftinvariant Gibbs-measures are equilibrium measures.Here we prove a generalization of the Lanford-Ruelle theorem: for all subshifts, any equilibrium measure for a function with d-summable variation is "topologically Gibbs". This is a relaxed notion which coincides with the usual notion of a Gibbs measure for SFTs.In the second part of the paper, we study Gibbs and equilibrium measures for some interesting families of subshifts: β-shifts, Dyck-shifts and Kalikowtype shifts (defined below). In all of these cases, a Lanford-Ruelle type theorem holds. For each of these families we provide a specific proof of the result.

Gibbs measures on subshifts

“…For a deeper study of some aspects of ergodic theory of equilibrium states, we strongly recommend the reader to see [Kel98]. The definition of a Gibbs measure for subshifts goes back to Schmidt [Sch97], Petersen and Schmidt [PS97], Aaronson and Nakada [AN07], and Meyerovitch [Mey13]. Differently from the usual approach, this definition was provided by using more abstract concepts involving conformal measures, without mentioning conditional expectations.…”

“…In Chapter 5, we begin the study of Gibbs measures for a specific class of functions, the so-called functions with d-summable variation ( [Mey13]) or regular local energy functions ( [Kel98], [Mui11a]). Adopting Meyerovitch's approach, we provide the definitions of a Gibbs measure and of a topological Gibbs measure for such a function f .…”

“…Nós estudamos as propriedades de medidas de Gibbs para funções com variação d-somável definidas em um subshift X. Baseado no trabalho de Meyerovitch [Mey13], provamos que se X é um subshift de tipo finito (STF), então qualquer medida de equilíbrio é também uma medida de Gibbs. Embora a definição fornecida por Meyerovitch não faz qualquer menção à esperanças condicionais, mostramos que no caso em que X é um STF, é possível caracterizar estas medidas em termos de noções mais familiares apresentadas na literatura (por exemplo, [Cap76], [Geo11], [Rue04]).…”

Extender sets and measures of maximal entropy for subshifts

“…Remark In fact the theorem from is more general; it treats equilibrium states for a class of potentials with a property called ‐summable variation, and the statement here for measures of maximal entropy corresponds to the case only.…”

“…Due to our weaker hypothesis, , our proof techniques are different from those used in . In particular, the case of different length treated in Theorem requires some subtle arguments about the ways in which can overlap themselves and each other.…”