Properties of the Absolute That Affect Smoothness of Flows on Closed Surfaces

S. Kh. Aranson ; E. V. Zhuzhoma

Matematičeskie zametki, Tome 68 (2000) no. 6, pp. 819-829.

Résumé

Let $M^2_g$ be a closed orientable surface of genus $g\ge2$, endowed with the structure of a Riemann manifold of constant negative curvature. For the universal covering $\Delta$, there is the notion of absolute, each of whose points determines an asymptotic direction of a bundle of parallel equidirected geodesics. In the paper it is proved that there is a set $U_g$ on the absolute having the cardinality of the continuum and such that if an arbitrary flow on $M^2_g$ has a semitrajectory whose covering has asymptotic direction defined by a point from $U_g$, then this flow is not analytical and has infinitely many stationary points.

Bibliographie

[1] Eberlein P., O'Neill B., “Visibility manifolds”, Pacific J. Math., 46 (1973), 45–109 | MR | Zbl

[2] Anosov D. V., “Kak mogut ukhodit v beskonechnost krivye na universalnoi nakryvayuschei ploskosti, nakryvayuschie nesamoperesekayuschiesya krivye na zamknutoi poverkhnosti”, Tr. MIAN SSSR, 191, Nauka, M., 1989, 34–44 | MR | Zbl

[3] Anosov D. V., “Flows on closed surfaces and behavior of trajectories lifted to the universal covering plane”, J. of Dyn. and Control Syst., 1:1 (1995), 125–138 | DOI | MR | Zbl

[4] Nikolaev I., Zhuzhoma E., Flows on $2$-dimensional manifolds, Lecture Notes in Math., 1705, 1999 | MR | Zbl

[5] Aranson S. Kh., “O nekotorykh arifmeticheskikh svoistvakh dinamicheskikh sistem na dvumernykh mnogoobraziyakh”, Dokl. AN SSSR, 222:2 (1975), 265–268 | MR | Zbl

[6] Gutierrez C., “Smoothing continuous flows on 2-manifolds and recurrences”, Ergodic Theory Dynamical Systems, 6 (1986), 17–44 | MR | Zbl

[7] Aranson S., Belitsky G., Zhuzhoma E., An Introduction to Qualitive Theory of Dynamical Systems on Surfaces, Amer. Math. Soc. Math. Monogr., Amer. Math. Soc., Providence, 1996 | Zbl

[8] Anosov D. V., “O povedenii traektorii na ploskosti Evklida ili Lobachevskogo, nakryvayuschikh traektorii potokov na zamknutykh poverkhnostyakh, 2”, Izv. AN SSSR. Ser. matem., 52:3 (1988), 451–478 | MR | Zbl

[9] Aranson S., Bronshtein I., Nikolaev I., Zhuzhoma E., “Qualitative theory of foliations on closed surfaces”, J. Math. Sci., 90:3 (1998), 2111–2149 | DOI | MR | Zbl

[10] Aranson S. Kh., Grines V. Z., “O nekotorykh invariantakh dinamicheskikh sistem na dvumernykh mnogoobraziyakh (neobkhodimye i dostatochnye usloviya topologicheskoi ekvivalentnosti tranzitivnykh dinamicheskikh sistem)”, Matem. sb., 90:3 (1973), 372–402 | MR | Zbl

[11] Aranson S., Zhuzhoma E., “Maier's theorems and geodesic laminations of surface flows”, J. of Dyn. and Control Syst., 2:4 (1996), 557–582 | DOI | MR | Zbl

[12] Levitt G., “Foliations and laminations on hyperbolic surfaces”, Topology, 22:2 (1983), 119–135 | DOI | MR | Zbl

[13] Aranson S. Kh., Grines V. Z., “O predstavlenii minimalnykh mnozhestv potokov na dvumernykh mnogoobraziyakh geodezicheskimi liniyami”, Izv. AN SSSR. Ser. matem., 42:1 (1978), 104–129 | MR | Zbl

[14] Whitney H., “Regular families of curves”, Ann. of Math., 34:2 (1933), 244–270 | DOI | MR | Zbl

[15] Arnoux P., Yoccoz J. C., “Construction de difféomorphismes pseudo-Anosov”, C. R. Acad. Sci., 292 (1981), 75–78 | MR | Zbl

[16] Aranson S. Kh., Zhuzhoma E. V., “O strukture kvaziminimalnykh mnozhestv sloenii na poverkhnostyakh”, Matem. sb., 185:8 (1994), 31–62 | Zbl

[17] Aranson S. Kh., Topologichekaya klassifikatsiya sloenii s osobennostyami i gomeomorfizmov s invariantnymi sloeniyami na zamknutykh poverkhnostyakh. 1: Sloeniya, Dep. VINITI No. 6887 V-88, GGU im. N. I. Lobachevskogo, Gorkii, 1988, s. 1–194 ; Топологичекая классификация слоений с особенностями и гомеоморфизмов с инвариантными слоениями на замкнутых поверхностях. 2: Гомеоморфизмы, Деп. ВИНИТИ No. 1043 В-89, ГГУ им. Н. И. Лобачевского, Горький, 1989, с. 1–137 | MR

[18] Maier A. G., “O traektoriyakh na orientiruemykh poverkhnostyakh”, Matem. sb., 12:1 (1943), 71–84 | MR

[19] Anosov D. V., “O povedenii traektorii na ploskosti Evklida ili Lobachevskogo, nakryvayuschikh traektorii potokov na zamknutykh poverkhnostyakh, 1”, Izv. AN SSSR. Ser. matem., 51:1 (1987), 16–43 | MR | Zbl

[20] Myrberg P. J., “Ein Approximationssatz für die Fuchsschen Gruppen”, Acta Math., 57 (1931), 389–409 | DOI | Zbl

[21] Hedlund G., “Two-dimensional manifolds and transivity”, Ann. Math., 37:3 (1936), 534–542 | DOI | MR | Zbl

[22] Nestorovich N. M., Geometricheskie postroeniya na ploskosti Lobachevskogo, Geometriya Lobachevskogo i razvitie ee idei, ed. V. F. Kagan, GITTL, M.–L., 1951