Topological and ergodic properties of closed 1-forms with incommensurable periods.

*(English. Russian original)*Zbl 0732.58001
Funct. Anal. Appl. 25, No. 2, 81-90 (1991); translation from Funkts. Anal. Prilozh. 25, No. 2, 1-12 (1991).

Let us consider a closed differential 1-form on a compact manifold. The aim of this paper is to investigate families of hypersurfaces on which the form vanishes in the case in which the periods of the form are incommensurable, but otherwise the form is in general position. It turns out that this problem is equivalent to the problem of hyperplane sections of period noncompact manifolds in Euclidean spaces. We prove that a hyperplane pseudoperiodic section divides a periodic manifold into two unbounded parts (and also arbitrarily many bounded parts).

From this it follows that the Hamiltonian system corresponding to a multi-valued Hamiltonian on a two-dimensional torus decomposes into cells that are filled up by periodic trajectories and an ergodic component of positive measure, on which the phase flow is isomorphic to the special flow over the rotation of the circle (by an angle equal to \(2\pi\) times the ratio of the periods of the form).

From this it follows that the Hamiltonian system corresponding to a multi-valued Hamiltonian on a two-dimensional torus decomposes into cells that are filled up by periodic trajectories and an ergodic component of positive measure, on which the phase flow is isomorphic to the special flow over the rotation of the circle (by an angle equal to \(2\pi\) times the ratio of the periods of the form).

##### MSC:

58A10 | Differential forms in global analysis |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

37A99 | Ergodic theory |

37C10 | Dynamics induced by flows and semiflows |

##### Keywords:

pseudoperiodic topology; Hamiltonian flow; differential 1-form; families of hypersurfaces; hyperplane sections; Hamiltonian system; two- dimensional torus
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\textit{V. I. Arnol'd}, Funct. Anal. Appl. 25, No. 2, 81--90 (1991; Zbl 0732.58001); translation from Funkts. Anal. Prilozh. 25, No. 2, 1--12 (1991)

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##### References:

[1] | A. Denjoy, ”Sur les courbes definies par les equations differentielles a la surface du tore,” J. Math. Pure Appl.,11, 333-375 (1932). · Zbl 0006.30501 |

[2] | A. N. Kolmogorov, ”On dynamical systems with an integral invariant on the torus,” Dokl. Akad. Nauk SSSR,39, 763-766 (1953). · Zbl 0052.31904 |

[3] | A. V. Kochergin, ”Nondegenerate saddles and absence of mixing,” Mat. Zametki,19, 453-468 (1976). |

[4] | A. A. Blokhin, ”Smooth ergodic flows,” Tr. Mosk. Mat. Obshch.,27, 113-128 (1972). |

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