Let <img alt="" src="Edit_6a94976d-35be-4dd4-b74f-d0bf6f497453.png" />be a non-autonomous discrete system and <img alt="" src="Edit_3516e048-3d23-4ae8-81ac-e7e732efbc89...Let <img alt="" src="Edit_6a94976d-35be-4dd4-b74f-d0bf6f497453.png" />be a non-autonomous discrete system and <img alt="" src="Edit_3516e048-3d23-4ae8-81ac-e7e732efbc89.png" /> be a set-valued discrete system induced by it. Where, <img alt="" src="Edit_f67612c1-bbf4-4c21-8b37-7d156ca9502d.png" />is the space formed by all non-empty compact subsets of <em>X</em> endowed with the Hausdorff metric <em>H</em>, <img alt="" src="Edit_cca16788-f64a-47c4-9645-e9c8cf9080fd.png" />is a set-valued mapping sequence induced by <img alt="" src="Edit_5a6d2e7f-3245-4dbd-98ec-dc977e23f3d8.png" />. It is proved that <img alt="" src="Edit_a25ef428-a2ff-46d5-9109-dcc67b57fbec.png" /> is <img alt="" src="Edit_ee8759ba-215c-4088-8590-db9f57eb4a7c.png" />-chaos, then <img alt="" src="Edit_f54b347a-033e-43e2-a3a1-d2fe5ac1f39d.png" />is <img alt="" src="Edit_72a57e59-dc43-4071-b0fe-432e379ddcc9.png" />-chaos. Where <img alt="" src="Edit_97813401-14af-4776-99fe-1e6cd08c3df1.png" />-chaos is denoted to <img alt="" src="Edit_9e2d88b4-7ece-430e-8978-800ff3280799.png" />-sensitive, <img alt="" src="Edit_440b79c1-f679-4571-b14d-6f804f402d75.png" />-sensitive, <img alt="" src="Edit_839b7b55-9961-4d80-b5cb-e7219a0ae871.png" />-transitive, <img alt="" src="Edit_feb0a032-255b-4cbd-b489-6a937c5a287a.png" />-accessible, <img alt="" src="Edit_3ba59c02-6df0-4ae1-8ac0-5c1b620e4a88.png" />-weakly mixing, <img alt="" src="Edit_7362ed03-8686-4cf7-94df-f0933b7abbff.png" />-<em>m</em>-sensitive, infinitely sensitive, or syndetically transitive.展开更多
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
For each sequence of positive real numbers,tending to positive infinity,a Furstenberg family is defined.All these Furstenberg families are compatible with dynamical systems.Then,chaos with respect to such Furstenberg ...For each sequence of positive real numbers,tending to positive infinity,a Furstenberg family is defined.All these Furstenberg families are compatible with dynamical systems.Then,chaos with respect to such Furstenberg families are intently discussed.This greatly improves some classica results of distributional chaos.To confirm the effectiveness of these improvements,the relevant examples are provided finally.展开更多
In the present paper, we define sensitive pairs via Furstenberg families and discuss the relation of three definitions： sensitivity, F -sensitivity and F -sensitive pairs, see Theorem 1. For transitive systems, we gi...In the present paper, we define sensitive pairs via Furstenberg families and discuss the relation of three definitions： sensitivity, F -sensitivity and F -sensitive pairs, see Theorem 1. For transitive systems, we give some sufficient conditions to ensure the existence of F -sensitive pairs. In particular, each non-minimal E system （M system, P system） has positive lower density （ Fs , Fr resp.）-sensitive pairs almost everywhere. Moreover, each non-minimal M system is Fts -sensitive. Finally, by some examples we show that： （1） F -sensitivity can not imply the existence of F -sensitive pairs. That means there exists an F -sensitive system, which has no F -sensitive pairs. （2） There is no immediate relation between the existence of sensitive pairs and Li-Yorke chaos, i.e., there exists a system （X, f ） without Li-Yorke scrambled pairs, which has κ B -sensitive pairs almost everywhere. （3） If the system （G, f ） is sensitive, where G is a finite graph, then it has κ B -sensitive pairs almost everywhere.展开更多
Letπ:(X,T)→(Y,S)be a factor map between two topological dynamical systems,and F_(a) Furstenberg family of Z.We introduce the notion of relative broken F-sensitivity.Let Fs(resp.Fpubd,Finf)be the families consisting ...Letπ:(X,T)→(Y,S)be a factor map between two topological dynamical systems,and F_(a) Furstenberg family of Z.We introduce the notion of relative broken F-sensitivity.Let Fs(resp.Fpubd,Finf)be the families consisting of all syndetic subsets(resp.positive upper Banach density subsets,infinite subsets).We show that for a factor mapπ:(X,T)→(Y,S)between transitive systems,πis relatively broken F-sensitive for F=Fs or Fpubd if and only if there exists a relative sensitive pair which is an F-recurrent point of(R_(π),T^((2)));is relatively broken Finf-sensitive if and only if there exists a relative sensitive pair which is not asymptotic.For a factor mapπ:(X,T)→(Y,S)between minimal systems,we get the structure of relative broken F-sensitivity by the factor map to its maximal equicontinuous factor.展开更多
文摘Let <img alt="" src="Edit_6a94976d-35be-4dd4-b74f-d0bf6f497453.png" />be a non-autonomous discrete system and <img alt="" src="Edit_3516e048-3d23-4ae8-81ac-e7e732efbc89.png" /> be a set-valued discrete system induced by it. Where, <img alt="" src="Edit_f67612c1-bbf4-4c21-8b37-7d156ca9502d.png" />is the space formed by all non-empty compact subsets of <em>X</em> endowed with the Hausdorff metric <em>H</em>, <img alt="" src="Edit_cca16788-f64a-47c4-9645-e9c8cf9080fd.png" />is a set-valued mapping sequence induced by <img alt="" src="Edit_5a6d2e7f-3245-4dbd-98ec-dc977e23f3d8.png" />. It is proved that <img alt="" src="Edit_a25ef428-a2ff-46d5-9109-dcc67b57fbec.png" /> is <img alt="" src="Edit_ee8759ba-215c-4088-8590-db9f57eb4a7c.png" />-chaos, then <img alt="" src="Edit_f54b347a-033e-43e2-a3a1-d2fe5ac1f39d.png" />is <img alt="" src="Edit_72a57e59-dc43-4071-b0fe-432e379ddcc9.png" />-chaos. Where <img alt="" src="Edit_97813401-14af-4776-99fe-1e6cd08c3df1.png" />-chaos is denoted to <img alt="" src="Edit_9e2d88b4-7ece-430e-8978-800ff3280799.png" />-sensitive, <img alt="" src="Edit_440b79c1-f679-4571-b14d-6f804f402d75.png" />-sensitive, <img alt="" src="Edit_839b7b55-9961-4d80-b5cb-e7219a0ae871.png" />-transitive, <img alt="" src="Edit_feb0a032-255b-4cbd-b489-6a937c5a287a.png" />-accessible, <img alt="" src="Edit_3ba59c02-6df0-4ae1-8ac0-5c1b620e4a88.png" />-weakly mixing, <img alt="" src="Edit_7362ed03-8686-4cf7-94df-f0933b7abbff.png" />-<em>m</em>-sensitive, infinitely sensitive, or syndetically transitive.
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supported by National Natural Science Foundation of China(Grant Nos.11071084 and 11026095)Natural Science Foundation of Guangdong Province(Grant No.10451063101006332)+1 种基金the Foundation for Distinguished Young Talents in Higher Education of Guangdong Province(Grant No.2012LYM 0133)Scientific Technology Planning of Guangzhou Education Bureau(Grant No.2012A075)
文摘For each sequence of positive real numbers,tending to positive infinity,a Furstenberg family is defined.All these Furstenberg families are compatible with dynamical systems.Then,chaos with respect to such Furstenberg families are intently discussed.This greatly improves some classica results of distributional chaos.To confirm the effectiveness of these improvements,the relevant examples are provided finally.
基金supported by NSFC (10771079 10871186+5 种基金 11071084 11026095)NSF of Guangdong Province (10451063101006332)supported by NSFC (11001071)Hefei University of Technology (GDBJ2008-024 2010HGXJ0200)
文摘In the present paper, we define sensitive pairs via Furstenberg families and discuss the relation of three definitions： sensitivity, F -sensitivity and F -sensitive pairs, see Theorem 1. For transitive systems, we give some sufficient conditions to ensure the existence of F -sensitive pairs. In particular, each non-minimal E system （M system, P system） has positive lower density （ Fs , Fr resp.）-sensitive pairs almost everywhere. Moreover, each non-minimal M system is Fts -sensitive. Finally, by some examples we show that： （1） F -sensitivity can not imply the existence of F -sensitive pairs. That means there exists an F -sensitive system, which has no F -sensitive pairs. （2） There is no immediate relation between the existence of sensitive pairs and Li-Yorke chaos, i.e., there exists a system （X, f ） without Li-Yorke scrambled pairs, which has κ B -sensitive pairs almost everywhere. （3） If the system （G, f ） is sensitive, where G is a finite graph, then it has κ B -sensitive pairs almost everywhere.
基金Supported by NNSF of China(Grant Nos.12001354,12171298)。
文摘Letπ:(X,T)→(Y,S)be a factor map between two topological dynamical systems,and F_(a) Furstenberg family of Z.We introduce the notion of relative broken F-sensitivity.Let Fs(resp.Fpubd,Finf)be the families consisting of all syndetic subsets(resp.positive upper Banach density subsets,infinite subsets).We show that for a factor mapπ:(X,T)→(Y,S)between transitive systems,πis relatively broken F-sensitive for F=Fs or Fpubd if and only if there exists a relative sensitive pair which is an F-recurrent point of(R_(π),T^((2)));is relatively broken Finf-sensitive if and only if there exists a relative sensitive pair which is not asymptotic.For a factor mapπ:(X,T)→(Y,S)between minimal systems,we get the structure of relative broken F-sensitivity by the factor map to its maximal equicontinuous factor.
