{"title":"阿贝尔群中的刚性、弱混合和递归","authors":"Ethan Ackelsberg","doi":"10.3934/dcds.2021168","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about <inline-formula><tex-math id=\"M1\">\\begin{document}$ \\mathbb{Z} $\\end{document}</tex-math></inline-formula>-actions extend to this setting:</p><p style='text-indent:20px;'>1. If <inline-formula><tex-math id=\"M2\">\\begin{document}$ (a_n) $\\end{document}</tex-math></inline-formula> is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.</p><p style='text-indent:20px;'>2. There exists a sequence <inline-formula><tex-math id=\"M3\">\\begin{document}$ (r_n) $\\end{document}</tex-math></inline-formula> such that every translate is both a rigidity sequence and a set of recurrence.</p><p style='text-indent:20px;'>The first of these results was shown for <inline-formula><tex-math id=\"M4\">\\begin{document}$ \\mathbb{Z} $\\end{document}</tex-math></inline-formula>-actions by Adams [<xref ref-type=\"bibr\" rid=\"b1\">1</xref>], Fayad and Thouvenot [<xref ref-type=\"bibr\" rid=\"b20\">20</xref>], and Badea and Grivaux [<xref ref-type=\"bibr\" rid=\"b2\">2</xref>]. The latter was established in <inline-formula><tex-math id=\"M5\">\\begin{document}$ \\mathbb{Z} $\\end{document}</tex-math></inline-formula> by Griesmer [<xref ref-type=\"bibr\" rid=\"b23\">23</xref>]. While techniques for handling <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\mathbb{Z} $\\end{document}</tex-math></inline-formula>-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.</p><p style='text-indent:20px;'>As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in <inline-formula><tex-math id=\"M7\">\\begin{document}$ \\mathbb{Z} $\\end{document}</tex-math></inline-formula>, while others exhibit new phenomena.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Rigidity, weak mixing, and recurrence in abelian groups\",\"authors\":\"Ethan Ackelsberg\",\"doi\":\"10.3934/dcds.2021168\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p style='text-indent:20px;'>The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about <inline-formula><tex-math id=\\\"M1\\\">\\\\begin{document}$ \\\\mathbb{Z} $\\\\end{document}</tex-math></inline-formula>-actions extend to this setting:</p><p style='text-indent:20px;'>1. If <inline-formula><tex-math id=\\\"M2\\\">\\\\begin{document}$ (a_n) $\\\\end{document}</tex-math></inline-formula> is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.</p><p style='text-indent:20px;'>2. There exists a sequence <inline-formula><tex-math id=\\\"M3\\\">\\\\begin{document}$ (r_n) $\\\\end{document}</tex-math></inline-formula> such that every translate is both a rigidity sequence and a set of recurrence.</p><p style='text-indent:20px;'>The first of these results was shown for <inline-formula><tex-math id=\\\"M4\\\">\\\\begin{document}$ \\\\mathbb{Z} $\\\\end{document}</tex-math></inline-formula>-actions by Adams [<xref ref-type=\\\"bibr\\\" rid=\\\"b1\\\">1</xref>], Fayad and Thouvenot [<xref ref-type=\\\"bibr\\\" rid=\\\"b20\\\">20</xref>], and Badea and Grivaux [<xref ref-type=\\\"bibr\\\" rid=\\\"b2\\\">2</xref>]. The latter was established in <inline-formula><tex-math id=\\\"M5\\\">\\\\begin{document}$ \\\\mathbb{Z} $\\\\end{document}</tex-math></inline-formula> by Griesmer [<xref ref-type=\\\"bibr\\\" rid=\\\"b23\\\">23</xref>]. While techniques for handling <inline-formula><tex-math id=\\\"M6\\\">\\\\begin{document}$ \\\\mathbb{Z} $\\\\end{document}</tex-math></inline-formula>-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.</p><p style='text-indent:20px;'>As an application of our results, we give several new constructions of rigidity sequences in torsion groups. 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引用次数: 0
摘要
The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about \begin{document}$ \mathbb{Z} $\end{document}-actions extend to this setting:1. If \begin{document}$ (a_n) $\end{document} is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.2. There exists a sequence \begin{document}$ (r_n) $\end{document} such that every translate is both a rigidity sequence and a set of recurrence.The first of these results was shown for \begin{document}$ \mathbb{Z} $\end{document}-actions by Adams [1], Fayad and Thouvenot [20], and Badea and Grivaux [2]. The latter was established in \begin{document}$ \mathbb{Z} $\end{document} by Griesmer [23]. While techniques for handling \begin{document}$ \mathbb{Z} $\end{document}-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in \begin{document}$ \mathbb{Z} $\end{document}, while others exhibit new phenomena.
Rigidity, weak mixing, and recurrence in abelian groups
The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about \begin{document}$ \mathbb{Z} $\end{document}-actions extend to this setting:
1. If \begin{document}$ (a_n) $\end{document} is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.
2. There exists a sequence \begin{document}$ (r_n) $\end{document} such that every translate is both a rigidity sequence and a set of recurrence.
The first of these results was shown for \begin{document}$ \mathbb{Z} $\end{document}-actions by Adams [1], Fayad and Thouvenot [20], and Badea and Grivaux [2]. The latter was established in \begin{document}$ \mathbb{Z} $\end{document} by Griesmer [23]. While techniques for handling \begin{document}$ \mathbb{Z} $\end{document}-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.
As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in \begin{document}$ \mathbb{Z} $\end{document}, while others exhibit new phenomena.