Mixing for some classes of special flows over rotations of the circle.

*(English. Russian original)*Zbl 0797.58045
Funct. Anal. Appl. 26, No. 3, 155-169 (1992); translation from Funkts. Anal. Prilozh. 26, No. 3, 1-21 (1992).

In Arnol’d’s paper [V. I. Arnol’d, Funct. Anal. Appl. 25, No. 2, 81–90 (1991); translation from Funkts. Anal. Prilozh. 25, No. 2, 1–12 (1991; Zbl 0732.58001)] it was shown that in general position the phase space of a Hamiltonian system with a multiple-valued Hamiltonian function on the two-dimensional torus can be decomposed into a finite number of cells filled by periodic trajectories and one ergodic component; over this component the phase flow is isomorphic to a special flow over rotation of the circle defined by a function with a finite number of asymmetrical logarithmic singularities. In this connection, the question of mixing for such flows arises. Our purpose is to give a positive answer to this question.

##### MSC:

37A25 | Ergodicity, mixing, rates of mixing |

70G10 | Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics |

37C10 | Dynamics induced by flows and semiflows |

70H05 | Hamilton’s equations |

##### Citations:

Zbl 0732.58001
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\textit{Ya. G. Sinaĭ} and \textit{K. M. Khanin}, Funct. Anal. Appl. 26, No. 3, 155--169 (1992; Zbl 0797.58045); translation from Funkts. Anal. Prilozh. 26, No. 3, 1--21 (1992)

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##### References:

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[7] | K. M. Khanin and Ya. G. Sinai, ?A new proof of M. Hermann’s theorem,? Comm. Math. Phys.,112, 89-101 (1987). · Zbl 0628.58021 |

[8] | Ya. G. Sinai and K. M. Khanin, ?Smoothness of conjugacies of diffeomorphisms of the circle with rotations,? Usp. Mat. Nauk,44, No. 1, 57-82 (1989). |

[9] | A. Ya. Khinchin, Continued Fractions [in Russian], Nauka, Moscow (1978). |

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