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arXiv · 2406.19828

Ergodic optimization for continuous functions on the Dyck-Motzkin shifts

Mao Shinoda·Hiroki Takahasi·Kenichiro Yamamoto
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Abstract

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 $G_\delta$ 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.

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Mao Shinoda, Hiroki Takahasi, Kenichiro Yamamoto. 2024-06-28. Ergodic optimization for continuous functions on the Dyck-Motzkin shifts. https://arxiv.org/abs/2406.19828

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