In this paper, a characterization of almost periodicity of topological transformation groups on uniform spaces is given. By searching the appropriate base for uniform structure, it is shown that the topological transf...In this paper, a characterization of almost periodicity of topological transformation groups on uniform spaces is given. By searching the appropriate base for uniform structure, it is shown that the topological transformation group is topologically equivalent to an isometric one if it is uniformly equicontinuous.展开更多
Given a topological dynamical system(X,T)and a T-invariant measureμ,let B denote the Borel σ-algebra on X.This paper proves that(X,B,μ,T)is rigid if and only if(X,T)isμ-A-equicontinuous in the mean for some subseq...Given a topological dynamical system(X,T)and a T-invariant measureμ,let B denote the Borel σ-algebra on X.This paper proves that(X,B,μ,T)is rigid if and only if(X,T)isμ-A-equicontinuous in the mean for some subsequence A of N,and a function f∈L^(2)(μ)is rigid if and only if f is μ-A-equicontinuous in the mean for some subsequence A of N.In particular,this gives a positive answer to Question 4.11 in[1].展开更多
In this paper,we present the concept of Banach-mean equicontinuity and prove that the Banach-,Weyl-and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent.Furthermore,we obtain...In this paper,we present the concept of Banach-mean equicontinuity and prove that the Banach-,Weyl-and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent.Furthermore,we obtain that the topological entropy of a transitive,almost Banach-mean equicontinuous dynamical system of Abelian group action is zero.As an application of our main result,we show that the topological entropy of the Banach-mean equicontinuous system under the action of an Abelian groups is zero.展开更多
Let (E,γ) be a locally convex space and E′ its conjugate space. AE′ be an equicontinuous set on (E,γ). In this paper,we show that for each sequence {f i}A and {x j}E, if {x j} converges to 0 in (E,γ), then we can...Let (E,γ) be a locally convex space and E′ its conjugate space. AE′ be an equicontinuous set on (E,γ). In this paper,we show that for each sequence {f i}A and {x j}E, if {x j} converges to 0 in (E,γ), then we can find a f 0∈E′ and extract subsequences {f n i } and {x n j } such that {f n i } converges to f 0 on {x n j } uniformly. If (E,γ) is metrizable,then we can show that the converse is also valid.展开更多
Let(X,G)be a dynamical system(G-system for short),that is,X is a topological space and G is an infinite topological group continuously acting on X.In the paper,the authors introduce the concepts of Hausdorff sensitivi...Let(X,G)be a dynamical system(G-system for short),that is,X is a topological space and G is an infinite topological group continuously acting on X.In the paper,the authors introduce the concepts of Hausdorff sensitivity,Hausdorff equicontinuity and topological equicontinuity for G-systems and prove that a minimal G-system(X,G)is either topologically equicontinuous or Hausdorff sensitive under the assumption that X is a T_(3)-space and they provide a classification of transitive dynamical systems in terms of equicontinuity pairs.In particular,under the condition that X is a Hausdorff uniform space,they give a dichotomy theorem between Hausdorff sensitivity and Hausdorff equicontinuity for G-systems admitting one transitive point.展开更多
文摘In this paper, a characterization of almost periodicity of topological transformation groups on uniform spaces is given. By searching the appropriate base for uniform structure, it is shown that the topological transformation group is topologically equivalent to an isometric one if it is uniformly equicontinuous.
基金Supported by the National Natural Science Founda-tion of China(11790274 and 11871361)partially supported by Qinglan project of Jiangsu Province。
文摘Given a topological dynamical system(X,T)and a T-invariant measureμ,let B denote the Borel σ-algebra on X.This paper proves that(X,B,μ,T)is rigid if and only if(X,T)isμ-A-equicontinuous in the mean for some subsequence A of N,and a function f∈L^(2)(μ)is rigid if and only if f is μ-A-equicontinuous in the mean for some subsequence A of N.In particular,this gives a positive answer to Question 4.11 in[1].
基金supported by NSF of China(11671057)NSF of Chongqing(cstc2020jcyj-msxmX0694)the Fundamental Research Funds for the Central Universities(2018CDQYST0023).
文摘In this paper,we present the concept of Banach-mean equicontinuity and prove that the Banach-,Weyl-and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent.Furthermore,we obtain that the topological entropy of a transitive,almost Banach-mean equicontinuous dynamical system of Abelian group action is zero.As an application of our main result,we show that the topological entropy of the Banach-mean equicontinuous system under the action of an Abelian groups is zero.
文摘Let (E,γ) be a locally convex space and E′ its conjugate space. AE′ be an equicontinuous set on (E,γ). In this paper,we show that for each sequence {f i}A and {x j}E, if {x j} converges to 0 in (E,γ), then we can find a f 0∈E′ and extract subsequences {f n i } and {x n j } such that {f n i } converges to f 0 on {x n j } uniformly. If (E,γ) is metrizable,then we can show that the converse is also valid.
基金supported by the National Natural Science Foundation of China(Nos.12061043,11661054)。
文摘Let(X,G)be a dynamical system(G-system for short),that is,X is a topological space and G is an infinite topological group continuously acting on X.In the paper,the authors introduce the concepts of Hausdorff sensitivity,Hausdorff equicontinuity and topological equicontinuity for G-systems and prove that a minimal G-system(X,G)is either topologically equicontinuous or Hausdorff sensitive under the assumption that X is a T_(3)-space and they provide a classification of transitive dynamical systems in terms of equicontinuity pairs.In particular,under the condition that X is a Hausdorff uniform space,they give a dichotomy theorem between Hausdorff sensitivity and Hausdorff equicontinuity for G-systems admitting one transitive point.
