{"title":"相对破碎家庭敏感性","authors":"Zhuo Wei Liu, Tao Yu","doi":"10.1007/s10114-024-3007-3","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>π</i>: (<i>X</i>, <i>T</i>) → (<i>Y</i>, <i>S</i>) be a factor map between two topological dynamical systems, and <span>\\(\\cal{F}\\)</span> a Furstenberg family of ℤ. We introduce the notion of <i>relative broken</i> <span>\\(\\cal{F}\\)</span>-<i>sensitivity</i>. Let <span>\\(\\cal{F}_{s}\\)</span> (resp. <span>\\(\\cal{F}_{\\text{pubd}},\\cal{F}_{\\text{inf}}\\)</span>) be the families consisting of all syndetic subsets (resp. positive upper Banach density subsets, infinite subsets). We show that for a factor map <i>π</i>: (<i>X</i>, <i>T</i>) → (<i>Y</i>, <i>S</i>) between transitive systems, <i>π</i> is relatively broken <span>\\(\\cal{F}\\)</span>-sensitive for <span>\\(\\cal{F}=\\cal{F}_{s}\\)</span> or <span>\\(\\cal{F}_{\\text{pubd}}\\)</span> if and only if there exists a relative sensitive pair which is an <span>\\(\\cal{F}\\)</span>-recurrent point of (<i>R</i><sub><i>π</i></sub>, <i>T</i><sup>(2)</sup>); is relatively broken <span>\\(\\cal{F}_{\\text{inf}}\\)</span>-sensitive if and only if there exists a relative sensitive pair which is not asymptotic. For a factor map <i>π</i>: (<i>X</i>, <i>T</i>) → (<i>Y</i>, <i>S</i>) between minimal systems, we get the structure of relative broken <span>\\(\\cal{F}\\)</span>-sensitivity by the factor map to its maximal equicontinuous factor.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Relative Broken Family Sensitivity\",\"authors\":\"Zhuo Wei Liu, Tao Yu\",\"doi\":\"10.1007/s10114-024-3007-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>π</i>: (<i>X</i>, <i>T</i>) → (<i>Y</i>, <i>S</i>) be a factor map between two topological dynamical systems, and <span>\\\\(\\\\cal{F}\\\\)</span> a Furstenberg family of ℤ. We introduce the notion of <i>relative broken</i> <span>\\\\(\\\\cal{F}\\\\)</span>-<i>sensitivity</i>. Let <span>\\\\(\\\\cal{F}_{s}\\\\)</span> (resp. <span>\\\\(\\\\cal{F}_{\\\\text{pubd}},\\\\cal{F}_{\\\\text{inf}}\\\\)</span>) be the families consisting of all syndetic subsets (resp. positive upper Banach density subsets, infinite subsets). We show that for a factor map <i>π</i>: (<i>X</i>, <i>T</i>) → (<i>Y</i>, <i>S</i>) between transitive systems, <i>π</i> is relatively broken <span>\\\\(\\\\cal{F}\\\\)</span>-sensitive for <span>\\\\(\\\\cal{F}=\\\\cal{F}_{s}\\\\)</span> or <span>\\\\(\\\\cal{F}_{\\\\text{pubd}}\\\\)</span> if and only if there exists a relative sensitive pair which is an <span>\\\\(\\\\cal{F}\\\\)</span>-recurrent point of (<i>R</i><sub><i>π</i></sub>, <i>T</i><sup>(2)</sup>); is relatively broken <span>\\\\(\\\\cal{F}_{\\\\text{inf}}\\\\)</span>-sensitive if and only if there exists a relative sensitive pair which is not asymptotic. For a factor map <i>π</i>: (<i>X</i>, <i>T</i>) → (<i>Y</i>, <i>S</i>) between minimal systems, we get the structure of relative broken <span>\\\\(\\\\cal{F}\\\\)</span>-sensitivity by the factor map to its maximal equicontinuous factor.</p></div>\",\"PeriodicalId\":50893,\"journal\":{\"name\":\"Acta Mathematica Sinica-English Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-07-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Sinica-English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10114-024-3007-3\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-024-3007-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let π: (X, T) → (Y, S) be a factor map between two topological dynamical systems, and \(\cal{F}\) a Furstenberg family of ℤ. We introduce the notion of relative broken\(\cal{F}\)-sensitivity. Let \(\cal{F}_{s}\) (resp. \(\cal{F}_{\text{pubd}},\cal{F}_{\text{inf}}\)) be the families consisting of all syndetic subsets (resp. positive upper Banach density subsets, infinite subsets). We show that for a factor map π: (X, T) → (Y, S) between transitive systems, π is relatively broken \(\cal{F}\)-sensitive for \(\cal{F}=\cal{F}_{s}\) or \(\cal{F}_{\text{pubd}}\) if and only if there exists a relative sensitive pair which is an \(\cal{F}\)-recurrent point of (Rπ, T(2)); is relatively broken \(\cal{F}_{\text{inf}}\)-sensitive if and only if there exists a relative sensitive pair which is not asymptotic. For a factor map π: (X, T) → (Y, S) between minimal systems, we get the structure of relative broken \(\cal{F}\)-sensitivity by the factor map to its maximal equicontinuous factor.
期刊介绍:
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.