caa a22 4500 477118399 CHVBK 20180405111628.0 cr unu---uuuuu 170330e19961001xx s 000 0 eng 10.1007/BF02254703 doi (NATIONALLICENCE)springer-10.1007/BF02254703 Maier's theorems and geodesic laminations of surface flows [Elektronische Daten] [S. Aranson, E. Zhuzhoma] 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). Plenum Publishing Corporation, 1996 Flow on surface nationallicence recurrent semitrajectory nationallicence geodesic lamination nationallicence Maier's theorem nationallicence asymptotic direction at infinity nationallicence Aranson S. Dept. of Math., Agriculture Academy of Nizhny Novgorod, 97 Gagarin Ave., 603107, Nizhny Novgorod, Russia aut Zhuzhoma E. Dept. of Math. and Mech., University of Nizhny Novgorod, 23 Gagarin Ave., 603600, Nizhny Novgorod, Russia aut Journal of Dynamical and Control Systems Kluwer Academic Publishers-Plenum Publishers 2/4(1996-10-01), 557-582 1079-2724 2:4<557 1996 2 10883 https://doi.org/10.1007/BF02254703 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02254703 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Aranson S. Dept. of Math., Agriculture Academy of Nizhny Novgorod, 97 Gagarin Ave., 603107, Nizhny Novgorod, Russia aut NATIONALLICENCE 700 1- Zhuzhoma E. Dept. of Math. and Mech., University of Nizhny Novgorod, 23 Gagarin Ave., 603600, Nizhny Novgorod, Russia aut NATIONALLICENCE 773 0- Journal of Dynamical and Control Systems Kluwer Academic Publishers-Plenum Publishers 2/4(1996-10-01), 557-582 1079-2724 2:4<557 1996 2 10883 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer