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
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
We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and the...We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.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.11371339,11431012,11401362,11471125)NSF of Guangdong province(Grant No.S2013040014084)
文摘We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.
