Chaos and indecomposability
Chaos and indecomposability
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混沌与不可分解性
DOI：
10.1016/j.aim.2016.09.012
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Udayan Darji and Hisao Kato
中科院分区：
文献类型：
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作者：
Hisaaki Endo;Seiichi Kamada;Isao Hasegawa;Kokoro Tanaka;Udayan Darji and Hisao Kato
我们利用局部熵理论的最新发展证明了动力系统中的混沌意味着底层空间中复杂结构的存在。早期的证明，如果X是一个弧状连续统，其中允许同胚f具有正拓扑熵，则X包含一个不可分解的子连续统。Barge和Diamond证明了：如果G是有限图，f：G→ G是任意具有正拓扑熵的映射，则逆极限空间lim <$（G，f）包含一个不可分解的连续统.本文证明了：如果X是有限图G的类G连续统，且f：X→ X是任意具有正拓扑熵的映射，则lim←（X，f）包含不可分解连续统.作为推论，我们得到在f是同胚的情况下，X包含一个不可分解的连续统。此外，如果f具有一致正的上熵，则X是不可分解连续统。我们的结果回答了Dupton提出的一些问题，推广了Dupton以及Barge和Diamond的上述工作。我们还引入了一个新的概念，称为锯齿对，试图捕捉的复杂性的动力系统从连续统理论的角度来看，并促进主要结果的证明。
We use recent developments in local entropy theory to prove that chaos in dynamical systems implies the existence of complicated structure in the underlying space. Earlier Mouron proved that if X is an arc-like continuum which admits a homeomorphism f with positive topological entropy, then X contains an indecomposable subcontinuum. Barge and Diamond proved that if G is a finite graph and f: G→ G is any map with positive topological entropy, then the inverse limit space lim←(G, f) contains an indecomposable continuum. In this paper we show that if X is a G-like continuum for some finite graph G and f: X→ X is any map with positive topological entropy, then lim←(X, f) contains an indecomposable continuum. As a corollary, we obtain that in the case that f is a homeomorphism, X contains an indecomposable continuum. Moreover, if f has uniformly positive upper entropy, then X is an indecomposable continuum. Our results answer some questions raised by Mouron and generalize the above mentioned work of Mouron and also that of Barge and Diamond. We also introduce a new concept called zigzag pair which attempts to capture the complexity of a dynamical systems from the continuum theoretic perspective and facilitates the proof of the main result.