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.

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.

Reviewer: E.Petrisor (Timişoara)

##### MSC:

28D10 | One-parameter continuous families of measure-preserving transformations |

37A99 | Ergodic theory |

28D05 | Measure-preserving transformations |

##### Keywords:

ergodic rearrangements; measure preserving flow; measure-preserving transformation; mixing property
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\textit{V. V. Ryzhikov}, Math. Notes 55, No. 6, 1 (1994; Zbl 0849.28009); translation from Mat. Zametki 55, No. 6, 146--149 (1994)

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##### References:

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