Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds
Kaimanovich, Vadim A.
Annales de l'I.H.P. Physique théorique, Tome 52 (1990), p. 361-393 / Harvested from Numdam
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@article{AIHPA_1990__53_4_361_0,
     author = {Kaimanovich, Vadim},
     title = {Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     volume = {52},
     year = {1990},
     pages = {361-393},
     mrnumber = {1096098},
     zbl = {0725.58026},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPA_1990__53_4_361_0}
}

Kaimanovich, Vadim A. Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds. Annales de l'I.H.P. Physique théorique, Tome 52 (1990) pp. 361-393. http://gdmltest.u-ga.fr/item/AIHPA_1990__53_4_361_0/

[A1] A. Ancona, Negatively Curved Manifolds, Elliptic Operators, and the Martin Boundary, Ann. Math., Vol. 125, 1987, pp. 495-536. | MR 890161 | Zbl 0652.31008

[A2] A. Ancona, Positive Harmonic Functions and Hyperbolicity, Springer, Lect. Notes Math., Vol. 1344, 1988, pp. 1-23. | MR 973878 | Zbl 0677.31006

[A3] A. Ancona, Sur les fonctions propres des variétés de Cartan-Hadamard, Comm. Math. Helvetici, Vol. 64, 1989, pp. 62-83. | MR 982562 | Zbl 0689.53027

[Al] A.D. Aleksandrov, A Theorem on Triangles in a Metric Space and Some of Its Applications, Trudy Steklov Mat. Inst., Vol. 38, 1951, pp. 5-23 (in Russian). | MR 49584 | Zbl 0049.39501

[AS] M.T. Anderson and R. Schoen, Positive Harmonic Functions on Complete Manifolds of Negative Curvature, Ann. Math., Vol. 121, 1985, pp. 429-461. | MR 794369 | Zbl 0587.53045

[B1] R. Bowen, Periodic Orbits for Hyperbolic Flows, Am. J. Math., Vol. 94, 1972, pp. 1-30. | MR 298700 | Zbl 0254.58005

[B2] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer, Lect. Notes Math., Vol. 470, 1975. | MR 442989 | Zbl 0308.28010

[BGS] W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, Birkhäuser, Basel, 1985. | MR 823981 | Zbl 0591.53001

[BK] M. Brin and A. Katok, On Local Entropy, Springer, Lect. Notes Math., Vol. 1007, 1983, pp. 30-38. | MR 730261 | Zbl 0533.58020

[BR] R. Bowen and D. Ruelle, The Ergodic Theory of Axiom A Flows, Invent. Math., Vol. 29, 1975, pp. 181-202. | MR 380889 | Zbl 0311.58010

[Bu] D. Ju. Burago, A New Approach to the Computation of the Entropy of a Geodesic Flow and Similar Dynamical Systems, Soviet Math. Dokl., Vol. 37, 1988, pp. 501-504. | MR 947227 | Zbl 0706.58050

[C] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, Orlando, 1984. | MR 768584

[CDP] M. Coornaert, T. Delzant, A. Papadopoulos, Notes sur les groupes hyperboliques de Gromov, Publ. I.R.M.A., Strasbourg, 1989.

[CE] J. Cheeger and D.G. Ebin, Comparison Theorems in Riemannian Geometry, North Holland, Amsterdam, 1975. | MR 458335 | Zbl 0309.53035

[D] E.B. Dynkin, Markov processes and related problems of analysis, London Math. Soc. Lect. Note Ser., Vol. 54, 1982. | MR 678918 | Zbl 0498.60067

[E] K.D. Elworthy, Stochastic differential equations on manifolds, London Math. Soc. Lect. Note Ser., Vol. 70, 1982. | MR 675100 | Zbl 0514.58001

[EO'N] P. Eberlein and B. O'Neill, Visibility manifolds, Pac. J. Math., Vol. 46, 1973, pp. 45-109. | MR 336648 | Zbl 0264.53026

[G] M. Gromov, Hyperbolic groups, M.S.R.I. Publ., Vol. 8, 1987, pp. 75-263. | MR 919829 | Zbl 0634.20015

[GH] E. Ghys, P. De La Harpe éd., Sur les groupes hyperboliques d'après Mikhael Gromov, Birkhäuser, Basel, 1990. | MR 1086648 | Zbl 0731.20025

[H1] U. Hamenstadt, A New Description of the Bowen-Margulis measure, Erg. Th. Dynam. Sys., Vol. 9, 1989, pp. 455-464. | MR 1016663 | Zbl 0722.58029

[H2] U. Hamenstadt, Time Preserving Conjugacies of Geodesic Flows, preprint. | MR 1162399

[H3] U. Hamenstadt, An Explicite Description of Harmonic Measure, preprint. | MR 1076134

[IW] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland/Kodansha, Amsterdam, 1981. | MR 637061 | Zbl 0495.60005

[K] V.A. Kaimanovich, Brownian Motion and Harmonic Functions on Covering Manifolds. An Entropy Approach, Soviet Math. Dokl., Vol. 33, 1986, pp. 812- 816. | MR 852647 | Zbl 0615.60074

[Ki] Y. Kifer, A Lower Bound for Haudorff Dimension of Harmonic Measures on Negatively Curved Manifolds, preprint. | MR 1088826

[KL] Y. Kifer and F. Ledrappier, Hausdorff Dimension of Harmonic Measures on Negatively Curved Manifolds, preprint. | MR 951889

[Ko1] T. Kori, Problème de Neumann sur les espaces harmoniques, Math. Ann., Vol. 224, 1976, pp. 53-76. | MR 435428 | Zbl 0346.31006

[Ko2] T. Kori, La théorie des espaces fonctionnelles a nullité 1 et le problème de Neumann sur les espaces harmonique, Ann. Inst. Fourier Grenoble, Vol. 27, 1977, pp. 45-119. | Numdam | MR 508016 | Zbl 0357.31008

[KV] V.A. Kaimanovich and A.M. Vershik, Random Walks on Discrete Groups: Boundary and Entropy, Ann. Prob., Vol. 11, 1983, pp. 457-490. | MR 704539 | Zbl 0641.60009

[L1] F. Ledrappier, Une relation entre entropie, dimension et exposant pour certaines marches aléatoires, C.R. Acad. Sci. Paris, T. 296, Série I, 1983, pp. 369-372. | MR 699165 | Zbl 0529.60071

[L2] F. Ledrappier, Ergodic properties of Brownian motion on covers of compact negatively curved manifolds, Bol. Soc. Bras. Mat., Vol. 19, 1988, pp. 115-140. | MR 1018929 | Zbl 0685.58036

[L3] F. Ledrappier, Harmonic measures and Bowen-Margulis measures, preprint. | MR 1088820

[LY] F. Ledrappier and L.-S. Young, The Metric Entropy of Diffeomorphisms I, II, Ann. Math., Vol. 122, 1985, pp. 509-574. | MR 819556 | Zbl 0605.58028

[M1] G.A. Margulis, Applications of Ergodic Theory to the Investigation of Manifolds of Negative Curvature, Funct. Anal. Appl., Vol. 3, 1969, pp. 335-336. | MR 257933 | Zbl 0207.20305

[M2] G.A. Margulis, Certain Measures Associated with U-Flows on Compact Manifolds, Funct. Anal. Appl., Vol. 4, 1970, pp. 55-67. | MR 272984 | Zbl 0245.58003

[Man] A. Manning, Topological Entropy for Geodesic Flows, Ann. Math., Vol. 110, 1979, pp. 567-573. | MR 554385 | Zbl 0426.58016

[O] J.-P. Otal, Sur la géométrie symplectique de l'espace des géodésiques d'une variété à courbure négative, preprint.

[P1] S.J. Patterson, The Limit Set of a Fuchsian Group, Acta Math., Vol. 136, 1976, pp. 241-243. | MR 450547 | Zbl 0336.30005

[P2] S.J. Patterson, Lectures on Measures on Limit Sets of Kleinian Groups, London Math. Soc. Lect. Note Ser., Vol. 111, 1987, pp. 281-323. | MR 903855 | Zbl 0611.30036

[Pe] Ya B. Pesin, Geodesic Flows with Hyperbolic Behaviour of the Trajectories and Objects Connected with Them, Russian Math. Surveys, Vol. 36, 4, 1981, pp. 1-59. | MR 629682 | Zbl 0493.58001

[Pr] J.-J. Prat, Etude asymptotique et convergence angulaire du mouvement Brownien sur une variété à courbure négative, C.R. Acad. Sci. Paris, T. 280, Series A, 1975, pp. 1539-1542. | MR 388557 | Zbl 0309.60052

[S1] D. Sullivan, Cycles for the Dynamical Study of Foliated Manifolds and Complex Manifolds, Invent. Math., Vol. 36, 1976, pp. 225-255. | MR 433464 | Zbl 0335.57015

[S2] D. Sullivan, The Density at Infinity of a Discrete Group of Hyperbolic Motions, Publ. Math. I.H.E.S., Vol. 50, 1979, pp. 171-202. | Numdam | MR 556586 | Zbl 0439.30034

[Si] Ya G. Sinai, Gibbs Measures in Ergodic Theory, Russian Math. Surveys, Vol. 27, 4, 1972, pp. 21-69. | MR 399421

[Ya] S.-T. Yau, Harmonic Functions on Complete Riemannian Manifolds, Comm. Pure Appl. Math., Vol. 28, 1975, pp. 201-228. | MR 431040 | Zbl 0291.31002

[Yo] L.-S. Young, Dimension, Entropy and Lyapunov Exponents, Erg. Th. Dynam. Sys., Vol. 2, 1982, pp. 109-129. | MR 684248 | Zbl 0523.58024