Geodesic flows on closed Riemannian manifolds without focal points

Ya. B. Pesin

Izvestiya. Mathematics , Tome 11 (1977) no. 6, pp. 1195-1228.

Résumé

In this paper it is proved that a geodesic flow on a two-dimensional compact manifold of genus greater than 1 with Riemannian metric without focal points is isomorphic with a Bernoulli flow. This result generalizes to the multidimensional case. The proof is based on establishing some metric properties of flows with nonzero Ljapunov exponents (the $K$-property, etc.), and also the construction of horospheres and leaves on a very wide class of Riemannian manifolds, together with a study of some of their geometric properties. Bibliography: 24 titles.

Bibliographie

[1] Anosov D. V., Geodezicheskie potoki na zamknutykh rimanovykh mnogoobraziyakh otritsatelnoi krivizny, Tr. matem. in-ta im. V. A. Steklova AN SSSR, 90, 1967 | MR

[2] Anosov D. V., Sinai Ya. G., “Nekotorye gladkie ergodicheskie sistemy”, Uspekhi matem. nauk, 22:5 (1967), 107–172 | MR | Zbl

[3] Bylov B. F., Vinograd R. E., Grobman D. M., Nemytskii V. V., Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti, Mir, M., 1966

[4] Gromul D., Klingenberg V., Meier V., Rimanova geometriya v tselom, Mir, M., 1971

[5] Kramli A. B., “Geodezicheskie potoki na kompaktnykh rimanovykh poverkhnostyakh bez fokalnykh tochek”, Stud. Sci. Math. Hung., 8:1,2 (1973), 59–78 | MR

[6] Milnor D., Teoriya Morsa, Mir, M., 1965 | MR

[7] Nitetski Z., Vvedenie v differentsialnuyu dinamiku, Mir, M., 1975 | MR | Zbl

[8] Oseledets V. I., “Multiplikativnaya ergodicheskaya teorema. Kharakteristicheskie pokazateli Lyapunova dinamicheskikh sistem”, Tr. Mosk. Matem. ob-va, 19, 1968, 179–210 | Zbl

[9] Pesin Ya. B., “Ergodicheskie svoistva gladkikh dinamicheskikh sistem s invariantnoi meroi i nenulevymi pokazatelyami Lyapunova”, Dokl. AN SSSR, 226:4 (1976), 774–777 | MR | Zbl

[10] Pesin Ya. B., “Semeistva invariantnykh mnogoobrazii, otvechayuschikh nenulevym kharakteristicheskim pokazatelyam”, Izv. AN SSSR. Ser. matem., 40 (1976), 1332–1379 | MR | Zbl

[11] Pesin Ya. B., “Kharakteristicheskie pokazateli Lyapunova i gladkaya ergodicheskaya teoriya”, Uspekhi matem. nauk, 32:4 (1977), 55–112 | MR | Zbl

[12] Pesin Ya. B., “Opisanie $\pi$-razbieniya diffeomorfizma s invariantnoi meroi”, Matem. zametki, 22:1 (1977), 29–44 | MR | Zbl

[13] Rokhlin V. A., “Lektsii po entropiinoi teorii preobrazovanii s invariantnoi meroi”, Uspekhi matem. nauk, 22:5 (1967), 4–51

[14] Eberlein P., “Geodesic Flow in Certain Manifolds Without Conjugate Points”, Trans. Amer. Math. Soc., 167 (1972), 151–170 | DOI | MR | Zbl

[15] Eberlein P., “Geodesic Flow on Negatively Curved Manifolds, I”, Ann. Math. (2), 95 (1972), 492–510 | DOI | MR | Zbl

[16] Eberlein P., “Geodesic Flow on Negatively Curved Manifolds, II”, Trans. Amer. Math. Soc., 178 (1973), 57–82 | DOI | MR | Zbl

[17] Eberlein P., “When is a Geodesic Flow of Anosov Type? I”, G. Differ. Geometry, 8 (1973), 437–463 | MR | Zbl

[18] Eberlein P., “When is a Geodesic Flow of Anosov Type? II”, G. Differ. Geometry, 8 (1973), 565–577 | MR | Zbl

[19] Ornstein D. S., Weiss B., “Geodesic Flows are Bernoullian”, Isr. J. Math., 14:2 (1973), 184–198 | DOI | MR | Zbl

[20] Rudolf D., “A To-valued step-coding for Ergodic Flows”, Proceeding of the International Conference on Dynamical Systems in Mathematical – Physic Rennes, 1975, 14–21, Sept.

[21] Heintze E., Im Hof H.-Ch., On the Geometry of Horospheres, preprint

[22] Everlein P., O'Neill B., “Visibility manifolds”, Pacific J. Math., 49 (1973), 45–109 | MR

[23] Hopf E., “Clouse Surface Without Conjugate Points”, Proc. Nach. Acad. Sci. USA, 34, no. 2, 1948, 47–51 | MR | Zbl

[24] Eschenburg J., “Horospheres and the Stable on the Geodesic Flow”, Math. Z., 153:3 (1977), 237–251 | DOI | MR | Zbl