Geodesic
On the Lifts to the Plane of Semileaves of Foliations on the Torus with a~Finite Number of Singularities
Informatics and Automation, Algebra. Topology. Differential equations and their applications, Tome 224 (1999), pp. 28-55

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An example of a semi-infinite non-self-intersecting curve on the torus is constructed having the property such that its lifts to the universal covering plane are at infinite Frechet distance from any lift of any semileaf of any foliation on the torus with a finite number of singular points. Thus for the lifts of the semileaves of such foliations there are fewer possible “types of behavior up to a finite Frechet distance” than for the lifts of arbitrary non-self-intersecting curves. 
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     author = {D. V. Anosov},
     title = {On the {Lifts} to the {Plane} of {Semileaves} of {Foliations} on the {Torus} with {a~Finite} {Number} of {Singularities}},
     journal = {Informatics and Automation},
     pages = {28--55},
     publisher = {mathdoc},
     volume = {224},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_1999_224_a2/}
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D. V. Anosov. On the Lifts to the Plane of Semileaves of Foliations on the Torus with a~Finite Number of Singularities. Informatics and Automation, Algebra. Topology. Differential equations and their applications, Tome 224 (1999), pp. 28-55. http://geodesic.mathdoc.fr/item/TRSPY_1999_224_a2/