Asymptotic behavior of covering curves on the universal coverings of surfaces.

*(English. Russian original)*Zbl 1030.37029
Proc. Steklov Inst. Math. 238, No. 3, 1-46 (2002); translation from Tr. Mat. Inst. Steklova 238, 5-54 (2002).

To date, a large number of publications have appeared that are devoted to the study of asymptotic properties of the lifts of curves without self-intersections on the universal covering and the “collation” (in a certain sense) of these curves with curves of constant geodesic curvature that have the same asymptotic direction as the curves under investigation. The paper is a survey of the results obtained. Ideas of proofs for the main results and sketches of constructions for key examples on this subject are presented.

The paper’s content is as follows: 1. The Weil and Anosov theorems; 2. Asymptotic direction of special curves; 3. Approximation of a curve by the trajectory of a flow; 4. The limit set of covering curves and trajectories; 5. Deviation of a curve from geodesics; 6. Other aspects of the subject.

For the entire collection see [Zbl 1012.00018].

The paper’s content is as follows: 1. The Weil and Anosov theorems; 2. Asymptotic direction of special curves; 3. Approximation of a curve by the trajectory of a flow; 4. The limit set of covering curves and trajectories; 5. Deviation of a curve from geodesics; 6. Other aspects of the subject.

For the entire collection see [Zbl 1012.00018].

Reviewer: Eugene Ershov (St.Peterburg)

##### MSC:

37E35 | Flows on surfaces |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |

37C10 | Dynamics induced by flows and semiflows |

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

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\textit{D. V. Anosov} and \textit{E. V. Zhuzhoma}, in: Monodromy in problems of algebraic geometry and differential equations. Collected papers. Transl. from the Russian. Moskva: Maik Nauka/Interperiodika. 1--46 (2002; Zbl 1030.37029); translation from Tr. Mat. Inst. Steklova 238, 5--54 (2002)