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<p>We prove that there exists an open and dense subset <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3575_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\({\cal U}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">U</mi> </mrow> </math></EquationSource> </InlineEquation> in the space of <i>C</i><sup>2</sup> expanding self-maps of the circle <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3575_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb T}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that the Lyapunov minimizing measures of any <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3575_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(T \in {\cal U}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>T</mi> <mo>∈</mo> <mrow> <mi mathvariant="script">U</mi> </mrow> </math></EquationSource> </InlineEquation> are uniquely supported on a periodic orbit. This answers a conjecture of Jenkinson-Morris in the <i>C</i><sup>2</sup> topology.</p>

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Lyapunov Optimizing Measures and Periodic Measures for C2 Expanding Maps

  • 期刊论文

  • 发表日期：

    2025年9月15日

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  • Wen Huang,
  • Leiye Xu,
  • Dawei Yang

摘要

We prove that there exists an open and dense subset \({\cal U}\) U in the space of C2 expanding self-maps of the circle \({\mathbb T}\) T such that the Lyapunov minimizing measures of any \(T \in {\cal U}\) T ∈ U are uniquely supported on a periodic orbit. This answers a conjecture of Jenkinson-Morris in the C2 topology.