Tail Invariant Measures of the Dyck Shift

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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.