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Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 6, Pages 1252–1288 (Mi izv2070)  

This article is cited in 19 scientific papers (total in 19 papers)

Geodesic flows on closed Riemannian manifolds without focal points

Ya. B. Pesin


Abstract: 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.

Full text: PDF file (3893 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:6, 1195–1228

Bibliographic databases:

UDC: 517.9
MSC: Primary 28A65, 58F15, 34C35; Secondary 53C20
Received: 16.09.1976

Citation: Ya. B. Pesin, “Geodesic flows on closed Riemannian manifolds without focal points”, Izv. Akad. Nauk SSSR Ser. Mat., 41:6 (1977), 1252–1288; Math. USSR-Izv., 11:6 (1977), 1195–1228

Citation in format AMSBIB
\Bibitem{Pes77}
\by Ya.~B.~Pesin
\paper Geodesic flows on closed Riemannian manifolds without focal points
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1977
\vol 41
\issue 6
\pages 1252--1288
\mathnet{http://mi.mathnet.ru/izv2070}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=488169}
\zmath{https://zbmath.org/?q=an:0376.58012|0399.58010}
\transl
\jour Math. USSR-Izv.
\yr 1977
\vol 11
\issue 6
\pages 1195--1228
\crossref{https://doi.org/10.1070/IM1977v011n06ABEH001766}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Ya. B. Pesin, “Characteristic Lyapunov exponents and smooth ergodic theory”, Russian Math. Surveys, 32:4 (1977), 55–114  mathnet  crossref  mathscinet  zmath
    2. A. Katok, “Lyapunov exponents, entropy and periodic orbits for diffeomorphisms”, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 51:1 (1980), 137  crossref  mathscinet  zmath
    3. M. Brin, “Bernoulli diffeomorphisms with n − 1 non-zero exponents”, Ergod Th Dynam Sys, 1:1 (1981)  crossref  mathscinet  zmath
    4. Ya. B. Pesin, “Geodesic flows with hyperbolic behaviour of the trajectories and objects connected with them”, Russian Math. Surveys, 36:4 (1981), 1–59  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. W. Ballmann, M. Brin, “On the ergodicity of geodesic flows”, Ergod Th Dynam Sys, 2:3-4 (1982)  crossref  mathscinet  zmath
    6. Keith Burns, “Hyperbolic behaviour of geodesic flows on manifolds with no focal points”, Ergod Th Dynam Sys, 3:1 (1983)  crossref  mathscinet
    7. Donal Hurley, “Note on uniform visibility manifolds”, Math Proc Camb Phil Soc, 98:1 (1985), 73  crossref  mathscinet  zmath
    8. Quo-Shin Chi, “Besicovitch Covering Lemma, Hadamard manifolds, and zero entropy”, J Geom Anal, 1:4 (1991), 373  crossref
    9. Anatole Katok, Keith Burns, “Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dyanmical systems”, Ergod Th Dynam Sys, 14:4 (1994)  crossref  mathscinet  zmath
    10. N. I. Chernov, C. Haskell, “Nonuniformly hyperbolic K-systems are Bernoulli”, Ergod Th Dynam Sys, 16:1 (1996)  crossref  mathscinet  zmath
    11. Rafael Oswaldo Ruggiero, “A note on the divergence of geodesic rays in manifolds without conjugate points”, Geom Dedicata, 134:1 (2008), 131  crossref  mathscinet  zmath  isi
    12. F. Rodriguez Hertz, M.A. Rodriguez Hertz, R. Ures, “Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle”, Invent math, 172:2 (2008), 353  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. Luis Barreira, Claudia Valls, “Parameter dependence of stable manifolds for difference equations”, Nonlinearity, 23:2 (2010), 341  crossref  isi
    14. Luis Barreira, Davor Dragičević, Claudia Valls, “Lyapunov Functions and Cone Families”, J Stat Phys, 2012  crossref
    15. Gang Liao, W.X.iang Sun, “Ergodic measures of geodesic flows on compact Lie groups”, Acta. Math. Sin.-English Ser, 29:9 (2013), 1781  crossref
    16. Antón.J..G. Bento, Cés.M.. Silva, “Nonuniform dichotomic behavior: Lipschitz invariant manifolds for ODEs”, Bulletin des Sciences Mathématiques, 2013  crossref
    17. Antón.J.. G. Bento, Cés.M.. Silva, “Generalized Nonuniform Dichotomies and Local Stable Manifolds”, J Dyn Diff Equat, 2013  crossref
    18. Dan Jane, R.O.. Ruggiero, “Boundary of Anosov dynamics and evolution equations for surfaces”, Math. Nachr, 2014, n/a  crossref
    19. Katrin Gelfert, Barbara Schapira, “Pressures for geodesic flows of rank one manifolds”, Nonlinearity, 27:7 (2014), 1575  crossref
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