Maier's theorems and geodesic laminations of surface flows

Abstract

We prove that a nontrivial recurrent semitrajectory has an arational asymptotic direction for flows on any hyperbolic surface. To prove the arationality we apply the structure of the stabilizer of a point of the circle at infinity. For a quasiminimal set (a closure of a nontrivial recurrent trajectory) of a flow we construct the corresponding geodesic lamination (the so called “geodesic framework”). To describe the geodesic framework of quasiminimal sets of flows on the hyperbolic surface of finite genus, we apply Maier's theorem (which states that if one nontrivial recurrent trajectory belongs to the limit set of another nontrivial recurrent trajectory, then the second nontrivial recurrent trajectory belongs to the limit set of the first one). We get the list of types of trajectories of a quasiminimal set that generalizes Cherry's description in the case of the flow on a compact surface. However, we show that Maier's theorems are not valid for flows on a surface of infinite genus. Geodesic laminations allow us to describe all virtual asymptotic directions of all nontrivial recurrent semitrajectories (formally, we identify the set of directions with some set ORA(Γ) of points of the circle at infinity). We prove that the geodesic framework of the quasiminimal set is determined by any of its asymptotic directions. We consider dynamical properties of some subsets of the set ORA (Γ). Orientable geodesic laminations also allow us to classify the set of all virtual asymptotic directions into two classes ORA₁(Γ), ORA₂(Γ), and we show that different asymptotic directions have different dynamic properties. Namely, we prove that a positive semitrajectory of any arational transitive flow has asymptotic directions from ORA₁(Γ) (resp., from ORA₂(Γ)) if and only if a positive semitrajectory of any arational transitive flow has asymptotic directions from ORA₁(Γ) (resp., from ORA₂(Γ)) if and only if this semitrajectory belongs to the nontrivial recurrent trajectory in both directions (resp., belongs to the α-separatrix of some saddle).