{"title":"Periodic Generalized Birkhoff Solutions and Farey Intervals for Monotone Recurrence Relations","authors":"","doi":"10.1007/s10884-024-10364-9","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>The aim of this paper is to extend the results associated with periodic orbits from two-dimensions to higher-dimensions. Because of the one-to-one correspondence between solutions for the monotone recurrence relation and orbits for the induced high-dimensional cylinder twist map, we consider the system of solutions for monotone recurrence relations instead. By introducing intersections of type (<em>k</em>, <em>l</em>), we propose the definition of generalized Birkhoff solutions, generalizing the concept of Birkhoff solutions. We show that if there is a (<em>p</em>, <em>q</em>)-periodic solution which is not a generalized Birkhoff solution, then the system has positive topological entropy and the Farey interval of <em>p</em>/<em>q</em> is contained in the rotation set.</p>","PeriodicalId":15624,"journal":{"name":"Journal of Dynamics and Differential Equations","volume":"300 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Dynamics and Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10884-024-10364-9","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The aim of this paper is to extend the results associated with periodic orbits from two-dimensions to higher-dimensions. Because of the one-to-one correspondence between solutions for the monotone recurrence relation and orbits for the induced high-dimensional cylinder twist map, we consider the system of solutions for monotone recurrence relations instead. By introducing intersections of type (k, l), we propose the definition of generalized Birkhoff solutions, generalizing the concept of Birkhoff solutions. We show that if there is a (p, q)-periodic solution which is not a generalized Birkhoff solution, then the system has positive topological entropy and the Farey interval of p/q is contained in the rotation set.
期刊介绍：
Journal of Dynamics and Differential Equations serves as an international forum for the publication of high-quality, peer-reviewed original papers in the field of mathematics, biology, engineering, physics, and other areas of science. The dynamical issues treated in the journal cover all the classical topics, including attractors, bifurcation theory, connection theory, dichotomies, stability theory and transversality, as well as topics in new and emerging areas of the field.
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