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<p>Ergodic optimization aims to describe properties of 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 with non-unique maximal entropy measures. We show that the space of continuous functions on any Dyck-Motzkin shift contains two disjoint subsets: one is a dense <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3486_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(G_\delta \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mi>δ</mi> </msub> </math></EquationSource> </InlineEquation> set with empty interior for which any maximizing measure is not mixing and has zero entropy; the other is a dense set of functions for which there exist uncountably many, fully supported maximizing measures that are Bernoulli. Key ingredients of a proof of this result are the density of closed orbit measures in the space of ergodic measures and the path connectedness of the space of ergodic measures of any Dyck-Motzkin shift.</p>

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Ergodic Optimization for Continuous Functions on the Dyck-Motzkin Shifts

  • 期刊论文

  • 发表日期：

    2025年7月18日

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  • Mao Shinoda,
  • Hiroki Takahasi,
  • Kenichiro Yamamoto

摘要

Ergodic optimization aims to describe properties of 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 with non-unique maximal entropy measures. We show that the space of continuous functions on any Dyck-Motzkin shift contains two disjoint subsets: one is a dense \(G_\delta \) G δ set with empty interior for which any maximizing measure is not mixing and has zero entropy; the other is a dense set of functions for which there exist uncountably many, fully supported maximizing measures that are Bernoulli. Key ingredients of a proof of this result are the density of closed orbit measures in the space of ergodic measures and the path connectedness of the space of ergodic measures of any Dyck-Motzkin shift.