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<p>We study the topological recurrence phenomenon of actions of locally compact abelian groups on compact metric spaces. In the case of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2088_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Z}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> actions, we develop new techniques to analyze Bohr recurrence sets, and show that Bohr recurrence implies recurrence for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2088_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Z}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>-Weyl systems. This family encompasses, for example, all <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2088_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Z}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>-affine nilsystems.</p>

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On recurrence for \(\mathbb {Z}^d\)-Weyl systems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月3日

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  • Sebastián Donoso,
  • Felipe Hernández,
  • Alejandro Maass

摘要

We study the topological recurrence phenomenon of actions of locally compact abelian groups on compact metric spaces. In the case of \(\mathbb {Z}^d\) Z d actions, we develop new techniques to analyze Bohr recurrence sets, and show that Bohr recurrence implies recurrence for \(\mathbb {Z}^d\) Z d -Weyl systems. This family encompasses, for example, all \(\mathbb {Z}^d\) Z d -affine nilsystems.