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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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[1] A. N. Kolmogorov, ”On dynamical systems with an integral invariant on the torus,” Dokl. Akad. Nauk SSSR,93, No. 5, 763–766 (1953). · Zbl 0052.31904
[2] D. V. Anosov, ”The additive functional homology equation that is connected with an ergodic rotation of the circle,” Izv. Akad. Nauk SSSR, Ser. Mat.,37, 1259–1274 (1973). · Zbl 0298.28016
[3] A. V. Kochergin, ”The absence of mixing in special flows over a rotation of the circle and in flows on a two-dimensional torus,” Dokl. Akad. Nauk SSSR,205, No. 3, 515–518 (1972).
[4] A. B. Katok, Ya. G. Sinai, and A. M. Stepin, ”The theory of dynamical systems and general transformation groups with invariant measure,” Itogi Nauki i Tekhniki, Ser. Mat. Anal.,3, 129–262 (1975). · Zbl 0399.28011
[5] I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980).
[6] Ya. G. Sinai and K. M. Khanin, ”Mixing of some classes of special flows over rotations of the circle,” Funkts. Anal. Prilozhen.,26, No. 3, 1–21 (1992). · Zbl 0828.20025
[7] V. V. Ryzhikov, ”Joinings and multiple mixing of finite rank actions,” Funkts. Anal. Prilozhen.,27, No. 2, 63–78 (1993). · Zbl 0853.28011
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