On continuity of geodesic frameworks of flows on surfaces

S. Kh. Aranson   ; E. V. Zhuzhoma   ; V. S. Medvedev

Sbornik. Mathematics, Tome 188 (1997) no. 7, pp. 955-972

Résumé

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).

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