含混沌真子系统的Li-Yorke混沌

王肖义, 黄煜

数学学报 ›› 2012, Vol. 55 ›› Issue (4) : 749-756.

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数学学报 ›› 2012, Vol. 55 ›› Issue (4) : 749-756. DOI: 10.12386/A2012sxxb0072
论文

含混沌真子系统的Li-Yorke混沌

    王肖义, 黄煜
作者信息 +

Li-Yorke Chaos System Without a Proper Subsystem Being Li-Yorke Chaos

    Xiao Yi WANG, Yu HUANG
Author information +
文章历史 +

摘要

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

Key words

Li-Yorke chaos / chaos-minimality / topological transitivity / interval maps

引用本文

导出引用
王肖义, 黄煜. 含混沌真子系统的Li-Yorke混沌. 数学学报, 2012, 55(4): 749-756 https://doi.org/10.12386/A2012sxxb0072
Xiao Yi WANG, Yu HUANG. Li-Yorke Chaos System Without a Proper Subsystem Being Li-Yorke Chaos. Acta Mathematica Sinica, Chinese Series, 2012, 55(4): 749-756 https://doi.org/10.12386/A2012sxxb0072

参考文献

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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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