Magdalena Fory's-Krawiec, Jana Hant'akov'a, P. Oprocha
{"title":"On the structure of $ \\alpha $-limit sets of backward trajectories for graph maps","authors":"Magdalena Fory's-Krawiec, Jana Hant'akov'a, P. Oprocha","doi":"10.3934/dcds.2021159","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>In the paper we study what sets can be obtained as <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\alpha $\\end{document}</tex-math></inline-formula>-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\alpha $\\end{document}</tex-math></inline-formula>-limit sets are <inline-formula><tex-math id=\"M4\">\\begin{document}$ \\omega $\\end{document}</tex-math></inline-formula>-limit sets and for all but finitely many points <inline-formula><tex-math id=\"M5\">\\begin{document}$ x $\\end{document}</tex-math></inline-formula>, we can obtain every <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\omega $\\end{document}</tex-math></inline-formula>-limits set as the <inline-formula><tex-math id=\"M7\">\\begin{document}$ \\alpha $\\end{document}</tex-math></inline-formula>-limit set of a backward trajectory starting in <inline-formula><tex-math id=\"M8\">\\begin{document}$ x $\\end{document}</tex-math></inline-formula>. For zero entropy maps, every <inline-formula><tex-math id=\"M9\">\\begin{document}$ \\alpha $\\end{document}</tex-math></inline-formula>-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"27 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Continuous Dynamical Systems - S","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/dcds.2021159","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In the paper we study what sets can be obtained as \begin{document}$ \alpha $\end{document}-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those \begin{document}$ \alpha $\end{document}-limit sets are \begin{document}$ \omega $\end{document}-limit sets and for all but finitely many points \begin{document}$ x $\end{document}, we can obtain every \begin{document}$ \omega $\end{document}-limits set as the \begin{document}$ \alpha $\end{document}-limit set of a backward trajectory starting in \begin{document}$ x $\end{document}. For zero entropy maps, every \begin{document}$ \alpha $\end{document}-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.
In the paper we study what sets can be obtained as \begin{document}$ \alpha $\end{document}-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those \begin{document}$ \alpha $\end{document}-limit sets are \begin{document}$ \omega $\end{document}-limit sets and for all but finitely many points \begin{document}$ x $\end{document}, we can obtain every \begin{document}$ \omega $\end{document}-limits set as the \begin{document}$ \alpha $\end{document}-limit set of a backward trajectory starting in \begin{document}$ x $\end{document}. For zero entropy maps, every \begin{document}$ \alpha $\end{document}-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.