研究了一类Li-Yorke混沌系统,该系统没有真子系统是Li-Yorke混沌的,我们称之为混沌极小系统.本文证明混沌极小系统是拓扑传递的,而且该系统每个非空开集都包含一个不可数混乱集.混沌极小系统不一定是极小的,本文构造了一个这样的反例.特别地,我们考察了线段连续自映射,指出该类系统都不是混沌极小的,线段上混沌极小子系统的存在性和该系统有正熵是等价的.
Abstract
We call a dynamical system chaos-minimal if it’s Li-Yorke chaos but has no proper subsystem being Li-Yorke chaos. It is shown that a chaos-minimal system must be topologically transitive and each nonempty open subset contains an uncountable scrambled set. We also construct a chaos-minimal dynamical system which is not minimal. At last, we consider the interval maps and point out an interval map must not be chaos-minimal, but the existence of a chaos-minimal subsystem is equivalent to having positive entropy.
关键词
Li-Yorke 混沌 /
混沌极小 /
拓扑传递 /
线段映射
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Key words
Li-Yorke chaos /
chaos-minimality /
topological transitivity /
interval maps
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参考文献
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脚注
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基金
国家自然科学基金资助项目(11071263); 广东省自然科学基金资助项目
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