We study the topological complexities of relative entropy zero extensions acted upon by countable-infinite amenable groups.First,for a given Følner sequence,we define the relative entropy dimensions and the dimen...We study the topological complexities of relative entropy zero extensions acted upon by countable-infinite amenable groups.First,for a given Følner sequence,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.展开更多
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
In this paper,we study the properties of the scaled packing topological pressures for a topological dynamical system(X,G),where G is a countable discrete infinite amenable group.We show that the scaled packing topolog...In this paper,we study the properties of the scaled packing topological pressures for a topological dynamical system(X,G),where G is a countable discrete infinite amenable group.We show that the scaled packing topological pressures can be determined by the scaled Bowen topological pressures.We obtain Billingsley’s Theorem for the scaled packing pressures with a G-action.Then we get a variational principle between the scaled packing pressures and the scaled measure-theoretic upper local pressures.Finally,we give some restrictions on the scaled sequence b,then in the case of the set Xμof generic points,we prove thatP^(P)(X_(μ),{F_(n)},f,b)=h_(μ)(X)+∫_(X)∫dμ,if(F(n))is tempered andμis a G-invariant ergodic Borel probability measure.展开更多
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 netw...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 theory.If one wants to take into account underlying system geometry in applications,more general group actions may need to be taken into consideration.In this paper,we consider this notion in the case of amenable group actions.We show that many basic properties in the Z-action case remain true.We 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.展开更多
In this paper, we define and study polynomial entropy on an arbitrary subset and local measure theoretic polynomial entropy for any Borel probability measure on a compact metric space,and investigate the relation betw...In this paper, we define and study polynomial entropy on an arbitrary subset and local measure theoretic polynomial entropy for any Borel probability measure on a compact metric space,and investigate the relation between local measure-theoretic polynomial entropy of Borel probability measures and polynomial entropy on an arbitrary subset. Also, we establish a variational principle for polynomial entropy on compact subsets in the context of amenable group actions.展开更多
Let G be a countable discrete infinite amenable group which acts continuously on a compact metric space X and let μ be an ergodic G-invariant Borel probability measure on X. For a fixed tempered F?lner sequence {Fn} ...Let G be a countable discrete infinite amenable group which acts continuously on a compact metric space X and let μ be an ergodic G-invariant Borel probability measure on X. For a fixed tempered F?lner sequence {Fn} in G with limn→+∞|Fn|/log n= ∞, we prove the following result:h_top^B(G_μ, {F_n}) = h_μ(X, G),where G_μ is the set of generic points for μ with respect to {F_n} and h_top^B(G_μ, {F_n}) is the Bowen topological entropy(along {F_n}) on G_μ. This generalizes the classical result of Bowen(1973).展开更多
In this paper,we study the Bowen entropy of stable sets in positive entropy G-system of amenable group actions.The lower bound of the Bowen entropy of these sets are estimated.
Several equivalent formulations are given for equivariant coarse embedding into Hilbert space.Using these equivalent definitions,it is proved that for a metric space X and a Hilbert space H with proper and isometric g...Several equivalent formulations are given for equivariant coarse embedding into Hilbert space.Using these equivalent definitions,it is proved that for a metric space X and a Hilbert space H with proper and isometric group actions on both of them,if X is coarsely embeddable into H and the group is amenable,then the coarse embedding can be modified to be equivariant by using the invariant mean property of the amenable group.展开更多
Let G be a locally compact group and let F be a closed subgroup of G × G. Pier introduced the notion of F-amenability which gives a new classification of groups. This concept generalizes the concept of amenabilit...Let G be a locally compact group and let F be a closed subgroup of G × G. Pier introduced the notion of F-amenability which gives a new classification of groups. This concept generalizes the concept of amenability and inner amenability for locally compact groups. In this paper, among other things, we extend some standard results for amenable groups to F-amenable groups and give various characterizations for F-amenable groups. A sequence of characterizations of F-amenable groups is given here by developing the well-known Flner＇s conditions for amenable locally compact groups. Several characterizations of inner amenability are also given.展开更多
Let G be an infinite countable group and A be a finite set.IfΣ?A~G is a strongly irreducible subshift of finite type,we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation■onΣand s...Let G be an infinite countable group and A be a finite set.IfΣ?A~G is a strongly irreducible subshift of finite type,we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation■onΣand show that the reduced C^(*)-algebra C_(r)^(*)(■)of■is a unital simple approximately finite(AF)-dimensional C^(*)-algebra.The shift action G of onΣinduces a canonical automorphism action of G on the C^(*)-algebra C_(r)^(*)(■).We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C^(*)-algebras,and show that,if G is an amenable group,then the noncommutative topological entropy of the canonical automorphism action of G on C_(r)^(*)(■)is equal to the topology entropy of the shift action of G onΣ.We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C^(*)-dynamical system(C_(r)^(*)(■),G).展开更多
基金supported by the NNSF of China (12201120,12171233)the Educational Research Project for Young and Middle-aged Teachers of Fujian Province (JAT200045).
文摘We study the topological complexities of relative entropy zero extensions acted upon by countable-infinite amenable groups.First,for a given Følner sequence,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.
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supported by the NNSF of China(12201120)the visiting fellowships supported by Fujian Alliance of Mathematics while visiting Xiamen University in the winter of 2023。
文摘In this paper,we study the properties of the scaled packing topological pressures for a topological dynamical system(X,G),where G is a countable discrete infinite amenable group.We show that the scaled packing topological pressures can be determined by the scaled Bowen topological pressures.We obtain Billingsley’s Theorem for the scaled packing pressures with a G-action.Then we get a variational principle between the scaled packing pressures and the scaled measure-theoretic upper local pressures.Finally,we give some restrictions on the scaled sequence b,then in the case of the set Xμof generic points,we prove thatP^(P)(X_(μ),{F_(n)},f,b)=h_(μ)(X)+∫_(X)∫dμ,if(F(n))is tempered andμis a G-invariant ergodic Borel probability measure.
基金supported by National Natural Science Foundation of China(Grant No.11701231)supported by National Natural Science Foundation of China(Grant Nos.11801584 and 11871228)+1 种基金National Science Foundation of Jiangsu Province(Grant No.BK20170225)Science Foundation of Jiangsu Normal University(Grant No.17XLR011)。
文摘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 theory.If one wants to take into account underlying system geometry in applications,more general group actions may need to be taken into consideration.In this paper,we consider this notion in the case of amenable group actions.We show that many basic properties in the Z-action case remain true.We 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.
基金supported by Foundation in higher education institutions of He’nan Province,P. R. China(Grant No. 23A110020)National Natural Science Foundation of China (Grant No. 11401363)+4 种基金the Foundation for the Training of Young Key Teachers in Colleges and Universities in He’nan Province,P. R. China (Grant No.2018GGJS134)supported by National Natural Science Foundation of China (Gratn No.11971236)China Postdoctoral Science Foundation (Grant No. 2016M591873)China Postdoctoral Science Special Foundation (Grant No. 2017T100384)funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions。
文摘In this paper, we define and study polynomial entropy on an arbitrary subset and local measure theoretic polynomial entropy for any Borel probability measure on a compact metric space,and investigate the relation between local measure-theoretic polynomial entropy of Borel probability measures and polynomial entropy on an arbitrary subset. Also, we establish a variational principle for polynomial entropy on compact subsets in the context of amenable group actions.
基金supported by National Basic Research Program of China (Grant No. 2013CB834100)National Natural Science Foundation of China (Grant Nos. 11271191 and 11431012)
文摘Let G be a countable discrete infinite amenable group which acts continuously on a compact metric space X and let μ be an ergodic G-invariant Borel probability measure on X. For a fixed tempered F?lner sequence {Fn} in G with limn→+∞|Fn|/log n= ∞, we prove the following result:h_top^B(G_μ, {F_n}) = h_μ(X, G),where G_μ is the set of generic points for μ with respect to {F_n} and h_top^B(G_μ, {F_n}) is the Bowen topological entropy(along {F_n}) on G_μ. This generalizes the classical result of Bowen(1973).
基金Supported by NSFC(No.11861010),also supported by NSFC(No.12171175),also supported by NSFC(No.12261006)NSF of Guangxi Province(No.2018GXNSFFA281008)Project of Guangxi First Class Disciplines of Statistics(No.GJKY-2022-01)。
文摘In this paper,we study the Bowen entropy of stable sets in positive entropy G-system of amenable group actions.The lower bound of the Bowen entropy of these sets are estimated.
基金Supported by National Natural Science Foundation of China(11871342)。
文摘Several equivalent formulations are given for equivariant coarse embedding into Hilbert space.Using these equivalent definitions,it is proved that for a metric space X and a Hilbert space H with proper and isometric group actions on both of them,if X is coarsely embeddable into H and the group is amenable,then the coarse embedding can be modified to be equivariant by using the invariant mean property of the amenable group.
文摘Let G be a locally compact group and let F be a closed subgroup of G × G. Pier introduced the notion of F-amenability which gives a new classification of groups. This concept generalizes the concept of amenability and inner amenability for locally compact groups. In this paper, among other things, we extend some standard results for amenable groups to F-amenable groups and give various characterizations for F-amenable groups. A sequence of characterizations of F-amenable groups is given here by developing the well-known Flner＇s conditions for amenable locally compact groups. Several characterizations of inner amenability are also given.
基金supported by National Science CenterPoland(Grant No.2018/30/M/ST1/00061)+1 种基金the Wroc law University of Science and Technology(Grant No.049U/0052/19)supported by National Natural Science Foundation of China(Grants Nos.11671094,11722103 and 11731003)。
文摘In this survey we will present the symbolic extension theory in topological dynamics,which was built over the past twenty years.
基金supported by National Natural Science Foundation of China(Grant Nos.11771379,11271224 and 11371290)。
文摘Let G be an infinite countable group and A be a finite set.IfΣ?A~G is a strongly irreducible subshift of finite type,we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation■onΣand show that the reduced C^(*)-algebra C_(r)^(*)(■)of■is a unital simple approximately finite(AF)-dimensional C^(*)-algebra.The shift action G of onΣinduces a canonical automorphism action of G on the C^(*)-algebra C_(r)^(*)(■).We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C^(*)-algebras,and show that,if G is an amenable group,then the noncommutative topological entropy of the canonical automorphism action of G on C_(r)^(*)(■)is equal to the topology entropy of the shift action of G onΣ.We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C^(*)-dynamical system(C_(r)^(*)(■),G).
