Flows on Closed Surfaces and Related Geometrical Questions

D. V. Anosov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 20-26.

Résumé

When studying flows (continuous one-parameter groups of transformations) on a surface $M$ (a closed two-dimensional manifold, which in our case is assumed to be different from the sphere and the projective plane), one naturally faces several geometrical questions related to the behavior of trajectories lifted to the universal covering plane $\widetilde {M}$ (for example, the questions of whether the lifted trajectory goes to infinity and if it has a certain asymptotic direction at infinity). The same questions can be posed not only for the flow trajectories but also for leaves of one-dimensional foliations and, in general, for non-self-intersecting (semi-)infinite curves. The properties of curves lifted to $\widetilde M$ that we consider here are such that, if two such curves $\widetilde L$ and $\widetilde L'$ are situated at a finite Frechét distance from each other (in this case, we say that the original curves $L$ and $L'$ are $F$-equivalent on $M$), then these properties of the above curves are identical. Certain (a few) results relate to arbitrary non-self-intersecting $L$; other results only relate to flow trajectories under certain additional constraints (that are usually imposed on the set of equilibrium states). The results of the latter type (which do not hold for arbitrary non-self-intersecting curves $L$) imply that, in general, arbitrary $L$ are not $F$-equivalent to the trajectories of such flows. In this relation, nonorientable foliations occupy a kind of intermediate position.

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