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
A topological dynamical system is n-sensitive,if there is a positive constant such that in each non-empty open subset there are n distinct points whose iterates will be apart from the constant at least for a same mome...A topological dynamical system is n-sensitive,if there is a positive constant such that in each non-empty open subset there are n distinct points whose iterates will be apart from the constant at least for a same moment.The properties of n-sensitivity in minimal systems are investigated.It turns out that a minimal system is n-sensitive if and only if the n-th regionally proximal relation Q_n contains a point whose coordinates are pairwise distinct.Moreover,the structure of a minimal system which is n-sensitive but not(n+1)-sensitive(n≥2)is determined.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
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
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.
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the National Natural Science Foundation of China(Grant Nos.10501042,10531010)the Ministry of Education of China(Grant No.20050358053)
文摘A topological dynamical system is n-sensitive,if there is a positive constant such that in each non-empty open subset there are n distinct points whose iterates will be apart from the constant at least for a same moment.The properties of n-sensitivity in minimal systems are investigated.It turns out that a minimal system is n-sensitive if and only if the n-th regionally proximal relation Q_n contains a point whose coordinates are pairwise distinct.Moreover,the structure of a minimal system which is n-sensitive but not(n+1)-sensitive(n≥2)is determined.
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.
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.
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.
