Gibbs and Equilibrium Measures for Some Families of Subshifts

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.