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The absence of mixing in special flows over rearrangements of segments. (English. Russian original) Zbl 0849.28009
Math. Notes 55, No. 6, 648-650 (1994); translation from Mat. Zametki 55, No. 6, 146-149 (1994).
Let $$X= ([0, 1), \mu)$$ be a Lebesgue measurable space, $$F: X\to \mathbb{R}^+$$ a measurable function such that $$\int Fd\mu= 1$$, and $$S$$ an invertible measure-preserving transformation of $$X$$. To $$F$$ and $$S$$ one associates a flow $$(T_t)$$ acting on the space $$M= \{(x, y)\in \mathbb{R}^2|x\in X, 0\leq y\leq F(x)\}$$. It is known [A. V. Kochergin, Dokl. Akad. Nauk SSSR 205, 515-518 (1972; Zbl 0262.28015)] that when $$F$$ has bounded variation, the flow $$(T_t)$$ does not have the mixing property.
In the paper under review one generalizes the mentioned result to the case when $$X$$ is a union of semi-intervals and $$S$$ is a rearrangement of the given set of intervals. More precisely, one defines the so-called ND-approximation of the transformation $$S$$ and proves that a special flow with recurrence function $$F$$ of bounded variation, constructed over a transformation with ND-approximation, does not have mixing property.

##### MSC:
 28D10 One-parameter continuous families of measure-preserving transformations 37A99 Ergodic theory 28D05 Measure-preserving transformations
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##### References:
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