{"title":"On Isolated Periodic Points of Diffeomorphisms with Expanding Attractors of Codimension 1","authors":"Marina K. Barinova","doi":"10.1134/S1560354724050022","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we consider an <span>\\(\\Omega\\)</span>-stable 3-diffeomorphism whose chain-recurrent set consists of isolated periodic points and hyperbolic 2-dimensional nontrivial attractors. Nontrivial attractors in this case can only be expanding, orientable or not. The most known example from the class under consideration is the DA-diffeomorphism obtained from the algebraic Anosov diffeomorphism, given on a 3-torus, by Smale’s surgery. Each such attractor has bunches of degree 1 and 2. We estimate the minimum number of isolated periodic points using information about the structure of attractors. Also, we investigate the topological structure of ambient manifolds for diffeomorphisms with k bunches and k isolated periodic points.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"29 5","pages":"794 - 802"},"PeriodicalIF":0.8000,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Regular and Chaotic Dynamics","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1134/S1560354724050022","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we consider an \(\Omega\)-stable 3-diffeomorphism whose chain-recurrent set consists of isolated periodic points and hyperbolic 2-dimensional nontrivial attractors. Nontrivial attractors in this case can only be expanding, orientable or not. The most known example from the class under consideration is the DA-diffeomorphism obtained from the algebraic Anosov diffeomorphism, given on a 3-torus, by Smale’s surgery. Each such attractor has bunches of degree 1 and 2. We estimate the minimum number of isolated periodic points using information about the structure of attractors. Also, we investigate the topological structure of ambient manifolds for diffeomorphisms with k bunches and k isolated periodic points.
在本文中，我们考虑了一个 \(\Omega\)-stable 3-diffeomorphism，它的链循环集由孤立的周期点和双曲的二维非难吸引子组成。在这种情况下，非难吸引子只能是扩展的、可定向的或不可定向的。在我们所研究的这一类吸引子中，最著名的例子是由代数阿诺索夫衍射通过斯马尔手术得到的 DA 衍射。每个这样的吸引子都有阶数为 1 和 2 的束。我们利用吸引子结构的信息来估计孤立周期点的最小数量。此外，我们还研究了具有 k 个束和 k 个孤立周期点的衍射的周围流形的拓扑结构。
期刊介绍：
Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.
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