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Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. The Dyck and Motzkin shifts are well-known examples of transitive subshifts over a finite alphabet that are not intrinsically ergodic. We show that the space of continuous functions on any Dyck-Motzkin shift splits into two subsets: one is a dense G_ set with empty interior for which any maximizing measure has zero entropy; the other is contained in the closure of the set of functions having uncountably many, fully supported measures that are Bernoulli. One key ingredient of a proof of this result is the path connectedness of the space of ergodic measures of the Dyck-Motzkin shift.
Shinoda et al. (Fri,) studied this question.