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
We introduce a new definition of measure-theoretic pressure for ergodic measures of con- tinuous maps on a compact metric space.This definition is similar to those of topological pressure involving spanning sets.As an...We introduce a new definition of measure-theoretic pressure for ergodic measures of con- tinuous maps on a compact metric space.This definition is similar to those of topological pressure involving spanning sets.As an application,for C^(1+α)(α>0)diffeomorphisms of a compact manifold, we study the relationship between the measure-theoretic pressure and the periodic points.展开更多
In this paper,we introduce the concept of measure-theoretic r-entropy of a continuous map on a compact metric space,and get the results as follows:1.Measure-theoretic entropy is the limit of measure-theoretic r-entrop...In this paper,we introduce the concept of measure-theoretic r-entropy of a continuous map on a compact metric space,and get the results as follows:1.Measure-theoretic entropy is the limit of measure-theoretic r-entropy and topological entropy is the limit of topological r-entropy(r → 0);2.Topological r-entropy is more than or equal to the supremum of 4r-entropy in the sense of Feldman's definition,where the measure varies among all the ergodic Borel probability measures.展开更多
In this paper, Brin-Katok local entropy formula and Katok＇s definition of the measuretheoretic entropy using spanning set are established for the random dynamical system over an invertible ergodic system.
We provide an alternative characterization of the invariance entropy for uncertain control systems via covers.We also introduce dimension types of invariance entropies and give a dimension-like characterization of the...We provide an alternative characterization of the invariance entropy for uncertain control systems via covers.We also introduce dimension types of invariance entropies and give a dimension-like characterization of the invariance entropy for uncertain control systems.Moreover,we present four versions of measure-theoretic invariance entropies,give an inverse variational principle of the measure-theoretic Bowen invariance entropy,and obtain four positive variational principles of invariance entropies of subsets with finite elements for uncertain control systems.展开更多
We provide three types of invariance pressure for uncertain control systems,namely,invariance pressure,strong invariance pressure,and invariance feedback pressure.The first two respectively extend the corresponding pr...We provide three types of invariance pressure for uncertain control systems,namely,invariance pressure,strong invariance pressure,and invariance feedback pressure.The first two respectively extend the corresponding pressures for deterministic control systems proposed by Colonius,Cossich,and Santana(2018)and by Nie,Wang,and Huang(2022);and the third generalizes invariance feedback entropy of uncertain control systems presented by Tomar,Rungger,and Zamani(2020),by adding potentials on the control range.Then we prove that(1)an explicit formula for invariance pressure of a controlled invariant set with respect to a potential by the logarithm of the spectral radius of the admissible weighted matrix determined by this potential under some suitable conditions;(2)an explicit formula for pressure of invariant quasi-partitions by maximum mean weight over all irreducible periodic sequences;(3)the invariance feedback pressure of a controlled invariant set is equal to the pressure of an atom partition under some technical assumptions;(4)lower and upper bounds for pressure of invariant quasi-partitions w.r.t.a potential by the logarithm of the spectral radius of the weighted adjacency matrix determined by this potential;(5)a variational principle for strong invariance pressure.展开更多
We investigate the relations between Pesin–Pitskel topological pressure on an arbitrary subset and measure-theoretic pressure of Borel probability measures for finitely generated semigroup actions.Let(X,G)be a system...We investigate the relations between Pesin–Pitskel topological pressure on an arbitrary subset and measure-theoretic pressure of Borel probability measures for finitely generated semigroup actions.Let(X,G)be a system,where X is a compact metric space and G is a finite family of continuous maps on X.Given a continuous function f on X,we define Pesin–Pitskel topological pressure PG(Z,f)for any subset Z■X and measure-theoretical pressure Pμ,G(X,f)for anyμ∈M(X),where M(X)denotes the set of all Borel probability measures on X.For any non-empty compact subset Z of X,we show that PG(Z,f)=sup{Pμ,G(X,f):μ∈M(X),μ(Z)=1}.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
For any C;diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy,a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms o...For any C;diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy,a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent.The main line of our approach to this result is under the setting of topological dynamical systems,which is also applicable to infinite-dimensional C;dynamical systems.展开更多
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Project Supported by National Natural Science Foundation of China
文摘We introduce a new definition of measure-theoretic pressure for ergodic measures of con- tinuous maps on a compact metric space.This definition is similar to those of topological pressure involving spanning sets.As an application,for C^(1+α)(α>0)diffeomorphisms of a compact manifold, we study the relationship between the measure-theoretic pressure and the periodic points.
基金supported by National Natural Science Foundation of China (Grant No. 11071054)the fund of Hebei Normal University of Science and Technology (Grant Nos. ZDJS2009 andCXTD2010-05)
文摘In this paper,we introduce the concept of measure-theoretic r-entropy of a continuous map on a compact metric space,and get the results as follows:1.Measure-theoretic entropy is the limit of measure-theoretic r-entropy and topological entropy is the limit of topological r-entropy(r → 0);2.Topological r-entropy is more than or equal to the supremum of 4r-entropy in the sense of Feldman's definition,where the measure varies among all the ergodic Borel probability measures.
基金the National Natural Science Foundation of China(No.10701032)Natural Science Foundation of Hebei Province(No.A2008000132)
文摘In this paper, Brin-Katok local entropy formula and Katok＇s definition of the measuretheoretic entropy using spanning set are established for the random dynamical system over an invertible ergodic system.
基金supported by National Natural Science Foundation of China(Grant Nos.12171492 and 12201135).
文摘We provide an alternative characterization of the invariance entropy for uncertain control systems via covers.We also introduce dimension types of invariance entropies and give a dimension-like characterization of the invariance entropy for uncertain control systems.Moreover,we present four versions of measure-theoretic invariance entropies,give an inverse variational principle of the measure-theoretic Bowen invariance entropy,and obtain four positive variational principles of invariance entropies of subsets with finite elements for uncertain control systems.
基金Supported by National Natural Science Foundation of China(Grant Nos.12171492,12201135)。
文摘We provide three types of invariance pressure for uncertain control systems,namely,invariance pressure,strong invariance pressure,and invariance feedback pressure.The first two respectively extend the corresponding pressures for deterministic control systems proposed by Colonius,Cossich,and Santana(2018)and by Nie,Wang,and Huang(2022);and the third generalizes invariance feedback entropy of uncertain control systems presented by Tomar,Rungger,and Zamani(2020),by adding potentials on the control range.Then we prove that(1)an explicit formula for invariance pressure of a controlled invariant set with respect to a potential by the logarithm of the spectral radius of the admissible weighted matrix determined by this potential under some suitable conditions;(2)an explicit formula for pressure of invariant quasi-partitions by maximum mean weight over all irreducible periodic sequences;(3)the invariance feedback pressure of a controlled invariant set is equal to the pressure of an atom partition under some technical assumptions;(4)lower and upper bounds for pressure of invariant quasi-partitions w.r.t.a potential by the logarithm of the spectral radius of the weighted adjacency matrix determined by this potential;(5)a variational principle for strong invariance pressure.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11771459,11701584 and 11871228)Guangdong Basic and Applied Basic Research Foundation(Grant No.2019A1515110932)the Natural Science Research Project of Guangdong Province(Grant No.2018KTSCX122)。
文摘We investigate the relations between Pesin–Pitskel topological pressure on an arbitrary subset and measure-theoretic pressure of Borel probability measures for finitely generated semigroup actions.Let(X,G)be a system,where X is a compact metric space and G is a finite family of continuous maps on X.Given a continuous function f on X,we define Pesin–Pitskel topological pressure PG(Z,f)for any subset Z■X and measure-theoretical pressure Pμ,G(X,f)for anyμ∈M(X),where M(X)denotes the set of all Borel probability measures on X.For any non-empty compact subset Z of X,we show that PG(Z,f)=sup{Pμ,G(X,f):μ∈M(X),μ(Z)=1}.
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 Natural Science Foundation of China(Grant No.11701394)supported by National Natural Science Foundation of China(Grant Nos.11971455 and 11731003)supported by National Natural Science Foundation of China(Grant Nos.11671279 and 11541003)。
文摘For any C;diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy,a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent.The main line of our approach to this result is under the setting of topological dynamical systems,which is also applicable to infinite-dimensional C;dynamical systems.
