{"title":"三维矢量场的熵","authors":"Fei Li, Wanlou Wu","doi":"10.1007/s10884-024-10383-6","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we show that for any <span>\\(C^1\\)</span> three-dimensional vector fields with positive topological entropy, the topological entropy can be approximated by horseshoes. Precisely, for any <span>\\(C^1\\)</span> three-dimensional vector field <i>X</i> with positive topological entropy, there exists a vector field <i>Y</i> arbitrarily close (in the <span>\\(C^1\\)</span> topology) to <i>X</i> exhibiting a horseshoe <span>\\(\\Lambda \\)</span> such that the topological entropy of <i>Y</i> restricted on <span>\\(\\Lambda \\)</span> can arbitrarily approximate the topological entropy of <i>X</i>. This extends a classical result (Katok in Inst Hautes Études Sci Publ Math 51:137–173, 1980) of Katok for <span>\\(C^{1+\\alpha }(\\alpha >0)\\)</span> surface diffeomorphisms and a result (Wu and Liu in Proc Am Math Soc 148(1):223–233, 2020) for <span>\\(C^1\\)</span> surface diffeomorphisms.\n</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Entropy for Three-dimensional Vector Fields\",\"authors\":\"Fei Li, Wanlou Wu\",\"doi\":\"10.1007/s10884-024-10383-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we show that for any <span>\\\\(C^1\\\\)</span> three-dimensional vector fields with positive topological entropy, the topological entropy can be approximated by horseshoes. Precisely, for any <span>\\\\(C^1\\\\)</span> three-dimensional vector field <i>X</i> with positive topological entropy, there exists a vector field <i>Y</i> arbitrarily close (in the <span>\\\\(C^1\\\\)</span> topology) to <i>X</i> exhibiting a horseshoe <span>\\\\(\\\\Lambda \\\\)</span> such that the topological entropy of <i>Y</i> restricted on <span>\\\\(\\\\Lambda \\\\)</span> can arbitrarily approximate the topological entropy of <i>X</i>. This extends a classical result (Katok in Inst Hautes Études Sci Publ Math 51:137–173, 1980) of Katok for <span>\\\\(C^{1+\\\\alpha }(\\\\alpha >0)\\\\)</span> surface diffeomorphisms and a result (Wu and Liu in Proc Am Math Soc 148(1):223–233, 2020) for <span>\\\\(C^1\\\\)</span> surface diffeomorphisms.\\n</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-08-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10884-024-10383-6\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10884-024-10383-6","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
摘要
在本文中，我们证明了对于任何具有正拓扑熵的\(C^1\)三维向量场，拓扑熵都可以用马蹄铁来近似。准确地说，对于任何具有正拓扑熵的\(C^1\)三维向量场X，存在一个与X任意接近（在\(C^1\)拓扑中）的向量场Y，它展示了一个马蹄形\(\Lambda \)，使得Y限制在\(\Lambda \)上的拓扑熵可以任意逼近X的拓扑熵。这扩展了卡托克关于\(C^{1+\alpha }(\alpha >0)\) 曲面差分的经典结果（Katok in Inst Hautes Études Sci Publ Math 51:137-173, 1980）和关于\(C^1\) 曲面差分的结果（Wu and Liu in Proc Am Math Soc 148(1):223-233, 2020）。
In this paper, we show that for any \(C^1\) three-dimensional vector fields with positive topological entropy, the topological entropy can be approximated by horseshoes. Precisely, for any \(C^1\) three-dimensional vector field X with positive topological entropy, there exists a vector field Y arbitrarily close (in the \(C^1\) topology) to X exhibiting a horseshoe \(\Lambda \) such that the topological entropy of Y restricted on \(\Lambda \) can arbitrarily approximate the topological entropy of X. This extends a classical result (Katok in Inst Hautes Études Sci Publ Math 51:137–173, 1980) of Katok for \(C^{1+\alpha }(\alpha >0)\) surface diffeomorphisms and a result (Wu and Liu in Proc Am Math Soc 148(1):223–233, 2020) for \(C^1\) surface diffeomorphisms.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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