{"title":"Relative entropy dimension for countable amenable group actions","authors":"Zubiao Xiao, Zhengyu Yin","doi":"10.1007/s10473-023-0607-4","DOIUrl":null,"url":null,"abstract":"<div><p>We study the topological complexities of relative entropy zero extensions acted upon by countable-infinite amenable groups. First, for a given Følner sequence <span>\\(\\{{F_n}\\}_{n = 0}^{+ \\infty}\\)</span>, we define the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. we also investigate the relations among these. Second, we introduce the notion of a relative dimension set. Moreover, using the method, we discuss the disjointness between the relative entropy zero extensions via the relative dimension sets of two extensions, which says that if the relative dimension sets of two extensions are different, then the extensions are disjoint.</p></div>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"43 6","pages":"2430 - 2448"},"PeriodicalIF":1.2000,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Scientia","FirstCategoryId":"1089","ListUrlMain":"https://link.springer.com/article/10.1007/s10473-023-0607-4","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
We study the topological complexities of relative entropy zero extensions acted upon by countable-infinite amenable groups. First, for a given Følner sequence \(\{{F_n}\}_{n = 0}^{+ \infty}\), we define the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. we also investigate the relations among these. Second, we introduce the notion of a relative dimension set. Moreover, using the method, we discuss the disjointness between the relative entropy zero extensions via the relative dimension sets of two extensions, which says that if the relative dimension sets of two extensions are different, then the extensions are disjoint.
期刊介绍:
Acta Mathematica Scientia was founded by Prof. Li Guoping (Lee Kwok Ping) in April 1981.
The aim of Acta Mathematica Scientia is to present to the specialized readers important new achievements in the areas of mathematical sciences. The journal considers for publication of original research papers in all areas related to the frontier branches of mathematics with other science and technology.