Abstract
The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer i such that limi→∞ f r i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension.
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the National Natural Science Foundation of China.
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Xiong, J., Chen, E. Chaos caused by a strong-mixing measure-preserving transformation. Sci. China Ser. A-Math. 40, 253–260 (1997). https://doi.org/10.1007/BF02874517
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DOI: https://doi.org/10.1007/BF02874517