Geodesic
Asymptotic Behavior of Covering Curves on the Universal Coverings of Surfaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Monodromy in problems of algebraic geometry and differential equations, Tome 238 (2002), pp. 5-54

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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 to the universal covering and the “collation” of these curves (in a certain sense) with lines of constant geodesic curvature that have the same asymptotic direction as the curves under investigation. This paper contains a survey of the results obtained. The ideas of proofs for the main results and the sketches of constructions for important examples on this subject field are presented. 
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     title = {Asymptotic {Behavior} of {Covering} {Curves} on the {Universal} {Coverings} of {Surfaces}},
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D. V. Anosov; E. V. Zhuzhoma. Asymptotic Behavior of Covering Curves on the Universal Coverings of Surfaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Monodromy in problems of algebraic geometry and differential equations, Tome 238 (2002), pp. 5-54. http://geodesic.mathdoc.fr/item/TM_2002_238_a0/