{"title":"Winning property of distal set for β-transformations","authors":"Qianqian Yang, Shuailing Wang","doi":"10.1080/14689367.2020.1844154","DOIUrl":null,"url":null,"abstract":"For any , let be the β-transformation dynamical system. For any , define the distal set of a given point y as Yang, Li, et al. [15] proved that the Hausdorff dimension of the distal set of any point is one for any β>1. In this paper, we study the winning property of the distal set of a given point y. We prove that the distal set of a given point y is α-winning for any and , where is a constant. By the definition of winning set, it's obvious that the distal set of a given point y is a dense set.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"36 1","pages":"128 - 141"},"PeriodicalIF":0.5000,"publicationDate":"2020-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/14689367.2020.1844154","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Dynamical Systems-An International Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/14689367.2020.1844154","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
For any , let be the β-transformation dynamical system. For any , define the distal set of a given point y as Yang, Li, et al. [15] proved that the Hausdorff dimension of the distal set of any point is one for any β>1. In this paper, we study the winning property of the distal set of a given point y. We prove that the distal set of a given point y is α-winning for any and , where is a constant. By the definition of winning set, it's obvious that the distal set of a given point y is a dense set.
对于任意，设为β-变换动力系统。对于任意，定义给定点y的远端集为Yang, Li, et al.[15]证明了任意点的远端集的Hausdorff维数对于任意β>1为1。本文研究了给定点y的远端集合的致胜性质，证明了给定点y的远端集合对任意和都是α-致胜的，其中为常数。根据获胜集的定义，可以明显地看出，给定点y的远端集是一个稠密集。
期刊介绍：
Dynamical Systems: An International Journal is a world-leading journal acting as a forum for communication across all branches of modern dynamical systems, and especially as a platform to facilitate interaction between theory and applications. This journal publishes high quality research articles in the theory and applications of dynamical systems, especially (but not exclusively) nonlinear systems. Advances in the following topics are addressed by the journal:
•Differential equations
•Bifurcation theory
•Hamiltonian and Lagrangian dynamics
•Hyperbolic dynamics
•Ergodic theory
•Topological and smooth dynamics
•Random dynamical systems
•Applications in technology, engineering and natural and life sciences
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