Gibbs and equilibrium measures for some families of subshifts

Details Gibbs and equilibrium measures for some families of subshifts Gibbs and equilibrium measures for some families of subshifts

2009-03-08

arxiv.org/abs/0903.1426v1

Mathematics

Dynamical Systems

19 pages

Scientific paper

For SFTs, any equilibrium measure is Gibbs, as long a $f$ has $d$-summable variation. This is a theorem of Lanford and Ruelle. Conversely, a theorem of Dobru{\v{s}}in states that for strongly-irreducible subshifts, shift-invariant 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: $\beta$-shifts, Dyck-shifts and Kalikow-type 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.

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