Nonlocal Properties of Analytic Flows on Closed Orientable Surfaces

Nonlocal asymptotic properties of analytic flows on closed orientable hyperbolic surfaces are studied. The asymptotic directions of lifts to the universal covering of semitrajectories of analytic flows with arbitrary sets of fixed points are described. A number of assertions about the properties of analytic flows are proved, in particular, (i) the density of analytic vector fields in the space of vector fields endowed with the $C^r$-topology; (ii) the boundedness of the deviation of the semitrajectories of analytic flows from the geodesics with the same asymptotic direction. The properties of points on the absolute that are reachable and unreachable by lifts of semitrajectories of analytic flows to the universal covering are studied.