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 dynamic...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.展开更多
The regionally proximal relation of order d along arithmetic progressions,namely AP[d]for d 2 N,is introduced and investigated.It turns out that if(X;T)is a topological dynamical system with AP[d]=Δ,then each ergodic...The regionally proximal relation of order d along arithmetic progressions,namely AP[d]for d 2 N,is introduced and investigated.It turns out that if(X;T)is a topological dynamical system with AP[d]=Δ,then each ergodic measure of(X;T)is isomorphic to a d-step pro-nilsystem,and thus(X;T)has zero entropy.Moreover,it is shown that if(X;T)is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic,then AP[d]=RP[d]for each d 2 N.It follows that for a minimal 1-pro-nilsystem,AP[d]=RP[d]for each d 2 N.An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.展开更多
We study several stronger versions of sensitivity for minimal group actions,including nsensitivity,thick n-sensitivity and blockily thick n-sensitivity,and characterize them by the regionally proximal relation.
For a finite discrete topological space X with at least two elements,a nonempty setГ,and a map p■is a generalized shift.In this text for■is bijective}we study proximal relations of transformation semigroups(S,X^(Г...For a finite discrete topological space X with at least two elements,a nonempty setГ,and a map p■is a generalized shift.In this text for■is bijective}we study proximal relations of transformation semigroups(S,X^(Г))and(H,XX^(Г)).Regarding proximal relation we prove:■is infinite■.Moreover,for infinite T,both transformation semigroups(S,X^(Г)and(H,XX^(Г))are regionally proximal,i.e.,■is finite.展开更多
基金supported by NSF of China(11671057)NSF of Chongqing(cstc2020jcyj-msxm X0694)the Fundamental Research Funds for the Central Universities(2018CDQYST0023)。
文摘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.
基金This work was supported by National Natural Science Foundation of China(Grant Nos.11431012,11971455,11571335 and 11371339).
文摘The regionally proximal relation of order d along arithmetic progressions,namely AP[d]for d 2 N,is introduced and investigated.It turns out that if(X;T)is a topological dynamical system with AP[d]=Δ,then each ergodic measure of(X;T)is isomorphic to a d-step pro-nilsystem,and thus(X;T)has zero entropy.Moreover,it is shown that if(X;T)is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic,then AP[d]=RP[d]for each d 2 N.It follows that for a minimal 1-pro-nilsystem,AP[d]=RP[d]for each d 2 N.An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.
基金Supported by NNSF of China(Grant Nos.11771264,11871188)NSF of Guangdong Province(Grant No.2018B030306024)。
文摘We study several stronger versions of sensitivity for minimal group actions,including nsensitivity,thick n-sensitivity and blockily thick n-sensitivity,and characterize them by the regionally proximal relation.
文摘For a finite discrete topological space X with at least two elements,a nonempty setГ,and a map p■is a generalized shift.In this text for■is bijective}we study proximal relations of transformation semigroups(S,X^(Г))and(H,XX^(Г)).Regarding proximal relation we prove:■is infinite■.Moreover,for infinite T,both transformation semigroups(S,X^(Г)and(H,XX^(Г))are regionally proximal,i.e.,■is finite.
