Asymptotic behavior of covering curves on the universal coverings of surfaces. (English. Russian original) Zbl 1030.37029

Anosov, D. V.; Zhuzhoma, E. V.

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].

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 \(\mathbb{C}\)

Keywords:

flows on surfaces; asymptotic behavior; covering curves; geodesics; Weil conjecture