Geodesic
Properties of the Absolute That Affect Smoothness of Flows on Closed Surfaces
Matematičeskie zametki, Tome 68 (2000) no. 6, pp. 819-829

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Let $M^2_g$ be a closed orientable surface of genus $g\ge2$, endowed with the structure of a Riemann manifold of constant negative curvature. For the universal covering $\Delta$, there is the notion of absolute, each of whose points determines an asymptotic direction of a bundle of parallel equidirected geodesics. In the paper it is proved that there is a set $U_g$ on the absolute having the cardinality of the continuum and such that if an arbitrary flow on $M^2_g$ has a semitrajectory whose covering has asymptotic direction defined by a point from $U_g$, then this flow is not analytical and has infinitely many stationary points. 
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     author = {S. Kh. Aranson and E. V. Zhuzhoma},
     title = {Properties of the {Absolute} {That} {Affect} {Smoothness} of {Flows} on {Closed} {Surfaces}},
     journal = {Matemati\v{c}eskie zametki},
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S. Kh. Aranson; E. V. Zhuzhoma. Properties of the Absolute That Affect Smoothness of Flows on Closed Surfaces. Matematičeskie zametki, Tome 68 (2000) no. 6, pp. 819-829. http://geodesic.mathdoc.fr/item/MZM_2000_68_6_a1/