The three versions of distributional chaos
The three versions of distributional chaos
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DOI：
10.1016/j.chaos.2004.06.011
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发表时间：
2005-03
影响因子：
7.8
通讯作者：
F. Balibrea;J. Smítal;M. Stefánková
中科院分区：
文献类型：
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作者：
F. Balibrea;J. Smítal;M. Stefánková
分布混沌的概念是由Schweizer和smir[译]引入的。阿米尔。数学。Soc. 344(1994) 737]用于区间的连续映射。然而，事实证明，对于紧致度量空间的连续映射，可以考虑三个相互不等价的分布混沌版本DC1-DC3。在本文中，我们考虑最弱的DC3。我们证明DC3并不意味着李和约克意义上的混乱。我们还证明了DC3对于拓扑共轭不是不变的。换句话说，存在由紧度量空间（M, ρ）的连续映射f生成的上下分布函数Φxyand Φxy*，使得Φxy*(t)>Φxy(t)对于区间内的所有t。然而，f在相同的空间M上，但是有一个度规ρ '生成与ρ相同的拓扑不再是DC3。回想一下，与此相反，DC1或DC2都是拓扑共轭不变的，并且意味着Li和Yorke混沌（参见[chaos， Solitons & fractal 21(2004) 1125]）。
The notion of distributional chaos was introduced by Schweizer and Smı́tal [Trans. Amer. Math. Soc. 344 (1994) 737] for continuous maps of the interval. However, it turns out that, for continuous maps of a compact metric space three mutually nonequivalent versions of distributional chaos, DC1–DC3, can be considered. In this paper we consider the weakest one, DC3. We show that DC3 does not imply chaos in the sense of Li and Yorke. We also show that DC3 is not invariant with respect to topological conjugacy. In other words, there are lower and upper distribution functions Φxyand Φxy*generated by a continuous map f of a compact metric space (M, ρ) such that Φxy*(t)>Φxy(t) for all t in an interval. However, f on the same space M, but with a metric ρ′ generating the same topology as ρ is no more DC3. Recall that, contrary to this, either DC1 or DC2 is topological conjugacy invariant and implies Li and Yorke chaos (cf. [Chaos, Solitons & Fractals 21 (2004) 1125]).