A Furstenberg family F is a family,consisting of some subsets of the set of positive integers,which is hereditary upwards,i.e.A?B and A∈F imply B∈F.For a given system(i.e.,a pair of a complete metric space and a con...A Furstenberg family F is a family,consisting of some subsets of the set of positive integers,which is hereditary upwards,i.e.A?B and A∈F imply B∈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 F,the definition of 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 F-scrambled pairs corresponding respectively to suitable Furstenberg family F.In the present paper we explore the basic properties of the set of F-scrambled pairs of a system.The generically F-chaotic system and the generically strongly F-chaotic system are defined.A criterion for a generically strongly 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
1 A problem or scrambled setCHAOTIC behavior is a manifestation of complexity of nonlinear dynamical systems. Since theobjects, methods, aims, or emphases of study are distinct, there are some variant definitionsof ch...1 A problem or scrambled setCHAOTIC behavior is a manifestation of complexity of nonlinear dynamical systems. Since theobjects, methods, aims, or emphases of study are distinct, there are some variant definitionsof chaos given by different authors, or given by the same author in his different works. Thefollowing definition mainly stems from Li and Yorke.展开更多
In the past twenty years,great achievements have been made by many researchers in the studies of chaotic behavior and local entropy theory of dynamical systems.Most of the results have been generalized to the relative...In the past twenty years,great achievements have been made by many researchers in the studies of chaotic behavior and local entropy theory of dynamical systems.Most of the results have been generalized to the relative case in the sense of a given factor map.In this survey we offer an overview of these developments.展开更多
In the present paper we show that a tree map is totally transitive iff it is topologically mixing. Using this result, we prove that the tree maps having a chaotic (or scrambled) subset with full Lebesgue measure is de...In the present paper we show that a tree map is totally transitive iff it is topologically mixing. Using this result, we prove that the tree maps having a chaotic (or scrambled) subset with full Lebesgue measure is dense in the space consisting of all topologically mixing (transitive, respectively) maps.展开更多
Recently, Du has given a new strong definition of chaos by using the shift map. In this paper, we give a proof of the main theorem by constructing a dense uncountable invariant subset of the symbol space E2 containing...Recently, Du has given a new strong definition of chaos by using the shift map. In this paper, we give a proof of the main theorem by constructing a dense uncountable invariant subset of the symbol space E2 containing transitive points in a simpler way with the help of a different metric. We also provide two examples, which support this new definition.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 F is a family,consisting of some subsets of the set of positive integers,which is hereditary upwards,i.e.A?B and A∈F imply B∈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 F,the definition of 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 F-scrambled pairs corresponding respectively to suitable Furstenberg family F.In the present paper we explore the basic properties of the set of F-scrambled pairs of a system.The generically F-chaotic system and the generically strongly F-chaotic system are defined.A criterion for a generically strongly 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.
文摘1 A problem or scrambled setCHAOTIC behavior is a manifestation of complexity of nonlinear dynamical systems. Since theobjects, methods, aims, or emphases of study are distinct, there are some variant definitionsof chaos given by different authors, or given by the same author in his different works. Thefollowing definition mainly stems from Li and Yorke.
基金supported by Foundation for the Authors of National Excellent Doctoral Dissertation of China (Grant No.201018)National Natural Science Foundation of China (Grant No. 10801035)Ministry of Education of China (Grant No. 200802461004)
文摘In the past twenty years,great achievements have been made by many researchers in the studies of chaotic behavior and local entropy theory of dynamical systems.Most of the results have been generalized to the relative case in the sense of a given factor map.In this survey we offer an overview of these developments.
文摘In the present paper we show that a tree map is totally transitive iff it is topologically mixing. Using this result, we prove that the tree maps having a chaotic (or scrambled) subset with full Lebesgue measure is dense in the space consisting of all topologically mixing (transitive, respectively) maps.
基金CSIR (project no. F.NO. 8/3 (45)/2005-EMR-I) for their financial support
文摘Recently, Du has given a new strong definition of chaos by using the shift map. In this paper, we give a proof of the main theorem by constructing a dense uncountable invariant subset of the symbol space E2 containing transitive points in a simpler way with the help of a different metric. We also provide two examples, which support this new definition.
