A Furstenberg family $\mathcal{F}$ is a family, consisting of some subsets of the set of positive integers, which is hereditary upwards, i.e. A ? B and A ∈ $\mathcal{F}$ imply B ∈ $\mathcal{F}$ . For a given system ...A Furstenberg family $\mathcal{F}$ is a family, consisting of some subsets of the set of positive integers, which is hereditary upwards, i.e. A ? B and A ∈ $\mathcal{F}$ imply B ∈ $\mathcal{F}$ . For a given system (i.e., a pair of a complete metric space and a continuous self-map of the space) and for a Furstenberg family $\mathcal{F}$ , the definition of $\mathcal{F}$ -scrambled pairs of points in the space has been given, which brings the well-known scrambled pairs in Li-Yorke sense and the scrambled pairs in distribution sense to be $\mathcal{F}$ -scrambled pairs corresponding respectively to suitable Furstenberg family $\mathcal{F}$ . In the present paper we explore the basic properties of the set of $\mathcal{F}$ -scrambled pairs of a system. The generically $\mathcal{F}$ -chaotic system and the generically strongly $\mathcal{F}$ -chaotic system are defined. A criterion for a generically strongly $\mathcal{F}$ -chaotic system is showed.展开更多
Consider the subshifts induced by constant-length primitive substitutions on two symbols. By investigating the equivalent version for the existence of Li-Yorke scrambled sets and by proving the non-existence of Schwei...Consider the subshifts induced by constant-length primitive substitutions on two symbols. By investigating the equivalent version for the existence of Li-Yorke scrambled sets and by proving the non-existence of Schweizer-Smítal scrambled sets, we completely reveal for this class of subshifts the chaotic behaviors possibly occurring in the sense of Li-Yorke and Schweizer-Smítal.展开更多
In this paper,we observe a special kind of continuous functions on graphs.By estimating the integrals of these functions,we prove that there are no sensitive commutative group actions on graphs.Furthermore,we consider...In this paper,we observe a special kind of continuous functions on graphs.By estimating the integrals of these functions,we prove that there are no sensitive commutative group actions on graphs.Furthermore,we consider a 1-dimensional continuum composed of a spiral curve and a circle and show that there exist sensitive homeomorphisms on it,which answers negatively a question proposed by Kato in 1993.展开更多
基金This work was supported by the National Natural Science Foundation of China (Grant No.10471049)
文摘A Furstenberg family $\mathcal{F}$ is a family, consisting of some subsets of the set of positive integers, which is hereditary upwards, i.e. A ? B and A ∈ $\mathcal{F}$ imply B ∈ $\mathcal{F}$ . For a given system (i.e., a pair of a complete metric space and a continuous self-map of the space) and for a Furstenberg family $\mathcal{F}$ , the definition of $\mathcal{F}$ -scrambled pairs of points in the space has been given, which brings the well-known scrambled pairs in Li-Yorke sense and the scrambled pairs in distribution sense to be $\mathcal{F}$ -scrambled pairs corresponding respectively to suitable Furstenberg family $\mathcal{F}$ . In the present paper we explore the basic properties of the set of $\mathcal{F}$ -scrambled pairs of a system. The generically $\mathcal{F}$ -chaotic system and the generically strongly $\mathcal{F}$ -chaotic system are defined. A criterion for a generically strongly $\mathcal{F}$ -chaotic system is showed.
基金the National Natural Science Foundation of China (Grant No. 10771084)the Education Department Foundation of Jilin Province (Grant No. 200568)the Foundations of Dalian Nationalities University and Jilin Normal University
文摘Consider the subshifts induced by constant-length primitive substitutions on two symbols. By investigating the equivalent version for the existence of Li-Yorke scrambled sets and by proving the non-existence of Schweizer-Smítal scrambled sets, we completely reveal for this class of subshifts the chaotic behaviors possibly occurring in the sense of Li-Yorke and Schweizer-Smítal.
基金the Special Foundation of National Prior Basic Researches of China(Grant No.G1999075108)partially supported by the National Natural Science Foundation of China(Grant No.10501042)
文摘In this paper,we observe a special kind of continuous functions on graphs.By estimating the integrals of these functions,we prove that there are no sensitive commutative group actions on graphs.Furthermore,we consider a 1-dimensional continuum composed of a spiral curve and a circle and show that there exist sensitive homeomorphisms on it,which answers negatively a question proposed by Kato in 1993.
