Tail invariant measures of the Dyck shift

Abstract

We show that the one-sided Dyck shift has a unique tail invariant topologically σ-finite measure (up to scaling). This invariant measure of the one sided Dyck turns out to be a shift-invariant probability. Furthermore, it is one of the two ergodic probabilities obtaining maximal entropy. For the two sided Dyck shift we show that there are exactly three ergodic double-tail invariant probabilities. We show that the two sided Dyck has a double-tail invariant probability, which is also shift invariant, with entropy strictly less than the topological entropy.

Gibbs and equilibrium measures for some families of subshifts

“…The Dyck-shift is a certain subshift whose origin is in the study of formal languages. The Dyck shift is an interesting example to study when attempting to extend a theory for which applies to SFTs, as in [6,8,10,14]. For completeness, we write the definition of the Dyck-shift D = D 2 ⊂ S Z :…”

“…There are 2 ergodic measures of maximal entropy for D, as shown in [8]. In [14] all T-invariant measures for the Dyck-shift were explicitly described. This set of T-invariant measures consists the convex-hull of the two measures of maximal entropy plus a third ergodic shift-invariant measure.…”

“…In section 5 β-shifts are shown to have a unique tail-invariant measure. Section 6 contains a proof of a restricted form of Lanford-Ruelle theorem for the Dyck shift, which is a supplement to the results of [14].…”

An algorithm for generating all CR sequences in the de Bruijn sequences of length 2<I>ⁿ</I> where <I>n</I> is any odd number

“…Following [11] and [12], we define the Dyck language L(D n ) (n ≥ 1) from the viewpoint of symbolic dynamics. We set Σ = {α m , β m : 1 ≤ m ≤ n}.…”

Kreiger Graphs and Fischer Covers vs Dynamical Properties

“…Dyke (or Dyck!) subshifts and certain β-shifts are non-synchronized but half synchronized systems [3].…”