Specification property and distributional chaos almost everywhere
Specification property and distributional chaos almost everywhere
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DOI：
10.1090/s0002-9939-08-09602-0
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发表时间：
2008-06
期刊：
影响因子：
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通讯作者：
P. Oprocha;M. Stefánková
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文献类型：
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作者：
P. Oprocha;M. Stefánková
我们的主要结果表明，作用在紧致度量空间（X，p）上的具有较弱形式的规范性质和一对远点的连续映射f在很强的意义下是分布混沌的.严格地说，在X中存在一个分布置乱稠密集S，它是与康托集同胚的不相交集的并集，使得对于任何两个不同的点u，v S，上分布函数相同地为1，下分布函数在某个ε> 0处为零。作为结果，我们描述了一类映射的情况下，当X是k维立方体Ik的满Lebesgue测度的一个混乱的集合。如果X = I，那么我们甚至可以构造其补集的Hausdorff维数为零的乱集。
Our main result shows that a continuous map f acting on a compact metric space (X, p) with a weaker form of specification property and with a pair of distal points is distributionally chaotic in a very strong sense. Strictly speaking, there is a distributionally scrambled set S dense in X which is the union of disjoint sets homeomorphic to Cantor sets so that, for any two distinct points u, v S, the upper distribution function is identically 1 and the lower distribution function is zero at some £ > 0. As a consequence, we describe a class of maps with a scrambled set of full Lebesgue measure in the case when X is the k-dimensional cube I k . If X = I, then we can even construct scrambled sets whose complements have zero Hausdorff dimension.