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THE PROXIMAL RELATION,REGIONALLY PROXIMAL RELATION AND BANACH PROXIMAL RELATION FOR amenable group ACTIONS

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Acta Mathematica Scientia 2021年第3期41卷 729-752页

作者： Yuan LIAN Xiaojun HUANG Zhiqiang LI College of Mathematics and Statistics Chongqing UniversityChongqing 401331China Chongqing Key Laboratory of Analytic Mathematics and Applications Chongqing UniversityChongqing 401331China

In this paper,we study the proximal relation,regionally proximal relation and Banach proximal relation of a topological dynamical system for amenable group actions.A useful tool is the support of a topological dynamical system which is used to study the structure of the Banach proximal relation,and we prove that above three relations all coincide on a Banach mean equicontinuous system generated by an amenable group action.

关键词： Proximal relation amenable group mean equicontinuity

Dynamical intricacy and average sample complexity of amenable group actions

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Science China Mathematics 2022年第6期65卷 1247-1266页

作者： Jie Li Siming Tu School of Mathematics and Statistics Jiangsu Normal UniversityXuzhou 221116China School of Mathematics(Zhuhai) Sun Yat-sen UniversityZhuhai 519082China

In 2018,Petersen and Wilson introduced the notion of dynamical intricacy and average sample complexity for dynamical systems of Z-action,based on the past works on the notion of intricacy in the research of brain network and probability *** one wants to take into account underlying system geometry in applications,more general group actions may need to be taken into *** this paper,we consider this notion in the case of amenable group *** show that many basic properties in the Z-action case remain *** also show that their suprema over covers or partitions are equal to the amenable topological entropy and the measure entropy,using the quasitiling technique in the theory of the amenable group.

关键词： intricacy average sample complexity topological entropy amenable group

The Symbolic Extension Theory in Topological Dynamics

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Acta Mathematica Sinica,English Series 2022年第1期38卷 107-136页

作者： Tomasz DOWNAROWICZ Guo Hua ZHANG Faculty of Pure and Applied Mathematics Wroclaw University of Science and TechnologyWybrze˙ze Wyspia′nskiego 2750-370 WroclawPoland School of Mathematical Sciences and Shanghai Center for Mathematical Sciences Fudan UniversityShanghai 200433P.R.China

In this survey we will present the symbolic extension theory in topological dynamics,which was built over the past twenty years.

关键词： Symbolic extension (symbolic)extension entropy function entropy structure superenvelope principal extension asymptotic h-expansiveness amenable group F?lner sequence tiling system quasi-symbolic extension residually finite group comparison property subexponential group