Geodesic
On continuity of geodesic frameworks of flows on surfaces
Sbornik. Mathematics, Tome 188 (1997) no. 7, pp. 955-972

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For flows on an orientable closed surface $M_g$ of larger genus (that is, of genus $g\geqslant 2$) a special geodesic distribution (the geodesic framework of the flow) is constructed that consists of geodesics with the same asymptotic directions as the trajectories of the flow and that is a complete topological invariant of the irrational flows on such surfaces. The problem of the dependence of the geodesic framework on a perturbation of the flow (or on the parameter of a family of flows) is considered. It is shown that an irreducible elementary irrational geodesic framework of a flow depends continuously on the perturbation of the flow (which is analogous to the continuous dependence of an irrational Poincare rotation number on a perturbation of a flow). 
@article{SM_1997_188_7_a0,
     author = {S. Kh. Aranson and E. V. Zhuzhoma and V. S. Medvedev},
     title = {On continuity of geodesic frameworks of flows on surfaces},
     journal = {Sbornik. Mathematics},
     pages = {955--972},
     publisher = {mathdoc},
     volume = {188},
     number = {7},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_7_a0/}
}
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S. Kh. Aranson; E. V. Zhuzhoma; V. S. Medvedev. On continuity of geodesic frameworks of flows on surfaces. Sbornik. Mathematics, Tome 188 (1997) no. 7, pp. 955-972. http://geodesic.mathdoc.fr/item/SM_1997_188_7_a0/