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Kajmanovich, Vadim A.

Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds. (English) Zbl 0725.58026

Ann. Inst. Henri Poincaré, Phys. Théor. 53, No. 4, 361-393 (1990).
Let M be a simply connected negatively curved manifold, \(\partial M\) its sphere at infinity. There exists a natural one-to-one correspondence between the invariant measures of the geodesic flow on the unit tangent bundle and the measures on \(\partial^ 2M=(\partial M\times \partial M\setminus diagonal).\) Using this correspondence, the author gives new constructions of the maximal entropy and the harmonic invariant measures of the geodesic flow.
Reviewer: Yu.E.Gliklikh (Voronezh)

Cited in 2 Reviews
Cited in 35 Documents

MSC:

37A99 Ergodic theory
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
58J65 Diffusion processes and stochastic analysis on manifolds

Keywords:

negatively curved manifold; sphere at infinity; geodesic flow
PDFBibTeX XMLCite
\textit{V. A. Kajmanovich}, Ann. Inst. Henri Poincaré, Phys. Théor. 53, No. 4, 361--393 (1990; Zbl 0725.58026)
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References:

[1] A. Ancona, Negatively Curved Manifolds, Elliptic Operators, and the Martin Boundary, Ann. Math., Vol. 125, 1987, pp. 495-536. Zbl0652.31008 MR890161 · Zbl 0652.31008 · doi:10.2307/1971409
[2] A. Ancona, Positive Harmonic Functions and Hyperbolicity, Springer, Lect. Notes Math., Vol. 1344, 1988, pp. 1-23. Zbl0677.31006 MR973878 · Zbl 0677.31006
[3] A. Ancona, Sur les fonctions propres des variétés de Cartan-Hadamard, Comm. Math. Helvetici, Vol. 64, 1989, pp. 62-83. Zbl0689.53027 MR982562 · Zbl 0689.53027 · doi:10.1007/BF02564664
[4] 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). Zbl0049.39501 MR49584 · Zbl 0049.39501
[5] M.T. Anderson and R. Schoen, Positive Harmonic Functions on Complete Manifolds of Negative Curvature, Ann. Math., Vol. 121, 1985, pp. 429-461. Zbl0587.53045 MR794369 · Zbl 0587.53045 · doi:10.2307/1971181
[6] R. Bowen, Periodic Orbits for Hyperbolic Flows, Am. J. Math., Vol. 94, 1972, pp. 1-30. Zbl0254.58005 MR298700 · Zbl 0254.58005 · doi:10.2307/2373590
[7] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer, Lect. Notes Math., Vol. 470, 1975. Zbl0308.28010 MR442989 · Zbl 0308.28010
[8] W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, Birkhäuser, Basel, 1985. Zbl0591.53001 MR823981 · Zbl 0591.53001
[9] M. Brin and A. Katok, On Local Entropy, Springer, Lect. Notes Math., Vol. 1007, 1983, pp. 30-38. Zbl0533.58020 MR730261 · Zbl 0533.58020
[10] R. Bowen and D. Ruelle, The Ergodic Theory of Axiom A Flows, Invent. Math., Vol. 29, 1975, pp. 181-202. Zbl0311.58010 MR380889 · Zbl 0311.58010 · doi:10.1007/BF01389848
[11] 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. Zbl0706.58050 MR947227 · Zbl 0706.58050
[12] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, Orlando, 1984. MR768584 · Zbl 0551.53001
[13] M. Coornaert, T. Delzant, A. Papadopoulos, Notes sur les groupes hyperboliques de Gromov, Publ. I.R.M.A., Strasbourg, 1989. · Zbl 0727.20018
[14] J. Cheeger and D.G. Ebin, Comparison Theorems in Riemannian Geometry, North Holland, Amsterdam, 1975. Zbl0309.53035 MR458335 · Zbl 0309.53035
[15] E.B. Dynkin, Markov processes and related problems of analysis, London Math. Soc. Lect. Note Ser., Vol. 54, 1982. Zbl0498.60067 MR678918 · Zbl 0498.60067 · doi:10.1017/CBO9780511662416
[16] K.D. Elworthy, Stochastic differential equations on manifolds, London Math. Soc. Lect. Note Ser., Vol. 70, 1982. Zbl0514.58001 MR675100 · Zbl 0514.58001
[17] P. Eberlein and B. O’Neill, Visibility manifolds, Pac. J. Math., Vol. 46, 1973, pp. 45-109. Zbl0264.53026 MR336648 · Zbl 0264.53026 · doi:10.2140/pjm.1973.46.45
[18] M. Gromov, Hyperbolic groups, M.S.R.I. Publ., Vol. 8, 1987, pp. 75-263. Zbl0634.20015 MR919829 · Zbl 0634.20015
[19] E. Ghys, P. De La Harpe éd., Sur les groupes hyperboliques d’après Mikhael Gromov, Birkhäuser, Basel, 1990. Zbl0731.20025 MR1086648 · Zbl 0731.20025
[20] U. Hamenstadt, A New Description of the Bowen-Margulis measure, Erg. Th. Dynam. Sys., Vol. 9, 1989, pp. 455-464. Zbl0722.58029 MR1016663 · Zbl 0722.58029 · doi:10.1017/S0143385700005095
[21] U. Hamenstadt, Time Preserving Conjugacies of Geodesic Flows, preprint. MR1162399
[22] U. Hamenstadt, An Explicite Description of Harmonic Measure, preprint. MR1076134
[23] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland/Kodansha, Amsterdam, 1981. Zbl0495.60005 MR637061 · Zbl 0495.60005
[24] V.A. Kaimanovich, Brownian Motion and Harmonic Functions on Covering Manifolds. An Entropy Approach, Soviet Math. Dokl., Vol. 33, 1986, pp. 812- 816. Zbl0615.60074 MR852647 · Zbl 0615.60074
[25] Y. Kifer, A Lower Bound for Haudorff Dimension of Harmonic Measures on Negatively Curved Manifolds, preprint. MR1088826
[26] Y. Kifer and F. Ledrappier, Hausdorff Dimension of Harmonic Measures on Negatively Curved Manifolds, preprint. MR951889 · Zbl 0702.58080
[27] T. Kori, Problème de Neumann sur les espaces harmoniques, Math. Ann., Vol. 224, 1976, pp. 53-76. Zbl0346.31006 MR435428 · Zbl 0346.31006 · doi:10.1007/BF01420288
[28] 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. Zbl0357.31008 MR508016 · Zbl 0357.31008 · doi:10.5802/aif.672
[29] V.A. Kaimanovich and A.M. Vershik, Random Walks on Discrete Groups: Boundary and Entropy, Ann. Prob., Vol. 11, 1983, pp. 457-490. Zbl0641.60009 MR704539 · Zbl 0641.60009 · doi:10.1214/aop/1176993497
[30] 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. Zbl0529.60071 MR699165 · Zbl 0529.60071
[31] F. Ledrappier, Ergodic properties of Brownian motion on covers of compact negatively curved manifolds, Bol. Soc. Bras. Mat., Vol. 19, 1988, pp. 115-140. Zbl0685.58036 MR1018929 · Zbl 0685.58036 · doi:10.1007/BF02584822
[32] F. Ledrappier, Harmonic measures and Bowen-Margulis measures, preprint. MR1088820 · Zbl 0728.53029
[33] F. Ledrappier and L.-S. Young, The Metric Entropy of Diffeomorphisms I, II, Ann. Math., Vol. 122, 1985, pp. 509-574. Zbl0605.58028 MR819556 · Zbl 0605.58028 · doi:10.2307/1971328
[34] G.A. Margulis, Applications of Ergodic Theory to the Investigation of Manifolds of Negative Curvature, Funct. Anal. Appl., Vol. 3, 1969, pp. 335-336. Zbl0207.20305 MR257933 · Zbl 0207.20305 · doi:10.1007/BF01076325
[35] G.A. Margulis, Certain Measures Associated with U-Flows on Compact Manifolds, Funct. Anal. Appl., Vol. 4, 1970, pp. 55-67. Zbl0245.58003 MR272984 · Zbl 0245.58003 · doi:10.1007/BF01075620
[36] A. Manning, Topological Entropy for Geodesic Flows, Ann. Math., Vol. 110, 1979, pp. 567-573. Zbl0426.58016 MR554385 · Zbl 0426.58016 · doi:10.2307/1971239
[37] J.-P. Otal, Sur la géométrie symplectique de l’espace des géodésiques d’une variété à courbure négative, preprint. · Zbl 0777.53042 · doi:10.4171/RMI/130
[38] S.J. Patterson, The Limit Set of a Fuchsian Group, Acta Math., Vol. 136, 1976, pp. 241-243. Zbl0336.30005 MR450547 · Zbl 0336.30005 · doi:10.1007/BF02392046
[39] S.J. Patterson, Lectures on Measures on Limit Sets of Kleinian Groups, London Math. Soc. Lect. Note Ser., Vol. 111, 1987, pp. 281-323. Zbl0611.30036 MR903855 · Zbl 0611.30036
[40] 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. Zbl0493.58001 MR629682 · Zbl 0493.58001 · doi:10.1070/RM1981v036n04ABEH002636
[41] 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. Zbl0309.60052 MR388557 · Zbl 0309.60052
[42] D. Sullivan, Cycles for the Dynamical Study of Foliated Manifolds and Complex Manifolds, Invent. Math., Vol. 36, 1976, pp. 225-255. Zbl0335.57015 MR433464 · Zbl 0335.57015 · doi:10.1007/BF01390011
[43] 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. Zbl0439.30034 MR556586 · Zbl 0439.30034 · doi:10.1007/BF02684773
[44] Ya G. Sinai, Gibbs Measures in Ergodic Theory, Russian Math. Surveys, Vol. 27, 4, 1972, pp. 21-69. MR399421 · Zbl 0246.28008
[45] S.-T. Yau, Harmonic Functions on Complete Riemannian Manifolds, Comm. Pure Appl. Math., Vol. 28, 1975, pp. 201-228. Zbl0291.31002 MR431040 · Zbl 0291.31002 · doi:10.1002/cpa.3160280203
[46] L.-S. Young, Dimension, Entropy and Lyapunov Exponents, Erg. Th. Dynam. Sys., Vol. 2, 1982, pp. 109-129. Zbl0523.58024 MR684248 · Zbl 0523.58024 · doi:10.1017/S0143385700009615
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