Geodesic
Closed cross-sections of irrational flows on surfaces
Sbornik. Mathematics, Tome 197 (2006) no. 2, pp. 173-192

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Simple closed transversals to an irrational flow on a closed surface with negative Euler characteristic are considered. A negative answer is given to A. Katok's conjecture about the existence of simple closed transversals to irrational flows on such surfaces in each free homotopy class containing simple closed curves non-homotopic to zero. Bibliography: 17 titles. 
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S. Kh. Aranson; I. A. Gorelikova; E. V. Zhuzhoma. Closed cross-sections of irrational flows on surfaces. Sbornik. Mathematics, Tome 197 (2006) no. 2, pp. 173-192. http://geodesic.mathdoc.fr/item/SM_2006_197_2_a2/