强一致收敛下的Li-Yorke混沌和分布混沌

Li-Yorke Chaos and Distributed Chaos under Strongly Uniform Convergence

【目的】研究混沌中序列映射与极限映射的关系。【方法】在超空间上,引入强一致收敛、Li-Yorke混沌、Li-Yorke-δ混沌和分布混沌的定义,然后利用强一致收敛的定义去讨论Li-Yorke混沌、Li-Yorke-δ混沌和分布混沌中的序列映射与极限映射的关系。【结果】若超空间上的序列映射是Li-Yorke混沌(Li-Yorke-δ混沌、分布混沌)且Li-Yorke混沌集(δ混沌集、分布混沌集)的所有交是不可数集,那么超空间上的极限映射就为Li-Yorke混沌(Li-Yorke-δ混沌、分布混沌);若超空间上的序列映射是Li-Yorke混沌且满足两个条件,则超空间上的极限映射是Li-Yorke-δ混沌。【结论】在超空间上,强一致收敛的条件下,序列映射上的混沌与极映射上的混沌具有保持性。

[Purposes]In order to study the relation between the sequence mapping and the limit mapping in chaos.[Methods]The definition of strongly uniform convergence,Li-Yorke chaos,Li-Yorke-δ chaos and distributional chaos are introduced on the hyperspace.Then,the relation between sequence mapping and limit mapping in Li-Yorke chaos,Li-Yorke-δchaos and distributional chaos are discussed by using the definition of strong uniform convergence.[Findings]If the sequence mapping is Li-Yorke chaos（LiYorke-δchaos,distributional chaos）and that all the intersections of the Li-Yorke scrambled set are uncountable sets on the hyperspace,then the limit mapping is Li-Yorke chaos（Li-Yorke-δchaos,distributional chaos）on the hyperspace.If the sequence mapping is Li-Yorke chaos and satisfies two conditions on the hyperspace,then the limit mapping is Li-Yorke-δchaos.[Conclusions]On the hyperspace,the sequence map and limit map in the chaos are preserved under the condition of strong uniform convergence.