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Ergodic geometry for non-elementary rank one manifolds
1. | Institut für Algebra und Geometrie, Karlsruhe Institute of Technology (KIT), Englerstr. 2, 76 131 Karlsruhe, Germany |
2. | LMPT, UMR 6083, Faculté des Sciences et Techniques, Parc de Grandmont, 37 200 Tours, France |
References:
[1] |
J. Aaronson and M. Denker, The Poincaré series of $\mathbb C\setminus \mathbb Z$, Ergodic Theory Dynam. Systems, 19 (1999), 1-20.
doi: 10.1017/S0143385799126592. |
[2] |
W. Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144.
doi: 10.1007/BF01456836. |
[3] |
W. Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597-609.
doi: 10.2307/1971331. |
[4] |
W. Ballmann, Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar, Birkhäuser Verlag, Basel, 1995, With an appendix by Misha Brin.
doi: 10.1007/978-3-0348-9240-7. |
[5] |
W. Ballmann, M. Brin and P. Eberlein, Structure of manifolds of nonpositive curvature. I, Ann. of Math. (2), 122 (1985), 171-203.
doi: 10.2307/1971373. |
[6] |
W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, vol. 61 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1985.
doi: 10.1007/978-1-4684-9159-3. |
[7] |
V. Bangert and V. Schroeder, Existence of flat tori in analytic manifolds of nonpositive curvature, Ann. Sci. École Norm. Sup. (4), 24 (1991), 605-634. |
[8] |
M. Bourdon, Structure conforme au bord et flot géodésique d'un $CAT(-1)$-espace, Enseign. Math. (2), 41 (1995), 63-102. |
[9] |
K. Burns, Hyperbolic behaviour of geodesic flows on manifolds with no focal points, Ergodic Theory Dynam. Systems, 3 (1983), 1-12.
doi: 10.1017/S0143385700001796. |
[10] |
K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Études Sci. Publ. Math., 35-59. |
[11] |
M. Coornaert and A. Papadopoulos, Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d'isométries des arbres, Trans. Amer. Math. Soc., 343 (1994), 883-898.
doi: 10.2307/2154747. |
[12] |
F. Dal'bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124.
doi: 10.1007/BF02803518. |
[13] |
P. Eberlein, Surfaces of nonpositive curvature, Mem. Amer. Math. Soc., 20 (1979), x+90.
doi: 10.1090/memo/0218. |
[14] |
E. Hopf, Ergodentheorie, Springer, 1937.
doi: 10.1007/978-3-642-86630-2. |
[15] |
E. Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc., 77 (1971), 863-877.
doi: 10.1090/S0002-9904-1971-12799-4. |
[16] |
V. A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. Reine Angew. Math., 455 (1994), 57-103.
doi: 10.1515/crll.1994.455.57. |
[17] |
G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782.
doi: 10.1007/s000390050025. |
[18] |
G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds, Ann. of Math. (2), 148 (1998), 291-314.
doi: 10.2307/120995. |
[19] |
U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1985, With a supplement by Antoine Brunel.
doi: 10.1515/9783110844641. |
[20] |
G. Link, Asymptotic geometry and growth of conjugacy classes of nonpositively curved manifolds, Ann. Global Anal. Geom., 31 (2007), 37-57.
doi: 10.1007/s10455-006-9016-x. |
[21] |
G. Link, M. Peigné and J.-C. Picaud, Sur les surfaces non-compactes de rang un, Enseign. Math. (2), 52 (2006), 3-36. |
[22] |
P. J. Nicholls, The Ergodic Theory of Discrete Groups, vol. 143 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1989.
doi: 10.1017/CBO9780511600678. |
[23] |
J.-P. Otal and M. Peigné, Principe variationnel et groupes kleiniens, Duke Math. J., 125 (2004), 15-44.
doi: 10.1215/S0012-7094-04-12512-6. |
[24] |
S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.
doi: 10.1007/BF02392046. |
[25] |
T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), vi+96. |
[26] |
T. Roblin, Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative, Israel J. Math., 147 (2005), 333-357.
doi: 10.1007/BF02785371. |
[27] |
V. Schroeder, Existence of immersed tori in manifolds of nonpositive curvature, J. Reine Angew. Math., 390 (1988), 32-46.
doi: 10.1515/crll.1988.390.32. |
[28] |
V. Schroeder, Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature, Manuscripta Math., 64 (1989), 77-105.
doi: 10.1007/BF01182086. |
[29] |
V. Schroeder, Codimension one tori in manifolds of nonpositive curvature, Geom. Dedicata, 33 (1990), 251-263.
doi: 10.1007/BF00181332. |
[30] |
D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 171-202. |
[31] |
M. E. Taylor, Measure Theory and Integration, vol. 76 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/gsm/076. |
[32] |
M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Co., New York, 1975, Reprinting of the 1959 original. |
[33] |
C. Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., 348 (1996), 4965-5005.
doi: 10.1090/S0002-9947-96-01614-5. |
show all references
References:
[1] |
J. Aaronson and M. Denker, The Poincaré series of $\mathbb C\setminus \mathbb Z$, Ergodic Theory Dynam. Systems, 19 (1999), 1-20.
doi: 10.1017/S0143385799126592. |
[2] |
W. Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144.
doi: 10.1007/BF01456836. |
[3] |
W. Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597-609.
doi: 10.2307/1971331. |
[4] |
W. Ballmann, Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar, Birkhäuser Verlag, Basel, 1995, With an appendix by Misha Brin.
doi: 10.1007/978-3-0348-9240-7. |
[5] |
W. Ballmann, M. Brin and P. Eberlein, Structure of manifolds of nonpositive curvature. I, Ann. of Math. (2), 122 (1985), 171-203.
doi: 10.2307/1971373. |
[6] |
W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, vol. 61 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 1985.
doi: 10.1007/978-1-4684-9159-3. |
[7] |
V. Bangert and V. Schroeder, Existence of flat tori in analytic manifolds of nonpositive curvature, Ann. Sci. École Norm. Sup. (4), 24 (1991), 605-634. |
[8] |
M. Bourdon, Structure conforme au bord et flot géodésique d'un $CAT(-1)$-espace, Enseign. Math. (2), 41 (1995), 63-102. |
[9] |
K. Burns, Hyperbolic behaviour of geodesic flows on manifolds with no focal points, Ergodic Theory Dynam. Systems, 3 (1983), 1-12.
doi: 10.1017/S0143385700001796. |
[10] |
K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Études Sci. Publ. Math., 35-59. |
[11] |
M. Coornaert and A. Papadopoulos, Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d'isométries des arbres, Trans. Amer. Math. Soc., 343 (1994), 883-898.
doi: 10.2307/2154747. |
[12] |
F. Dal'bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124.
doi: 10.1007/BF02803518. |
[13] |
P. Eberlein, Surfaces of nonpositive curvature, Mem. Amer. Math. Soc., 20 (1979), x+90.
doi: 10.1090/memo/0218. |
[14] |
E. Hopf, Ergodentheorie, Springer, 1937.
doi: 10.1007/978-3-642-86630-2. |
[15] |
E. Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc., 77 (1971), 863-877.
doi: 10.1090/S0002-9904-1971-12799-4. |
[16] |
V. A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. Reine Angew. Math., 455 (1994), 57-103.
doi: 10.1515/crll.1994.455.57. |
[17] |
G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782.
doi: 10.1007/s000390050025. |
[18] |
G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds, Ann. of Math. (2), 148 (1998), 291-314.
doi: 10.2307/120995. |
[19] |
U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1985, With a supplement by Antoine Brunel.
doi: 10.1515/9783110844641. |
[20] |
G. Link, Asymptotic geometry and growth of conjugacy classes of nonpositively curved manifolds, Ann. Global Anal. Geom., 31 (2007), 37-57.
doi: 10.1007/s10455-006-9016-x. |
[21] |
G. Link, M. Peigné and J.-C. Picaud, Sur les surfaces non-compactes de rang un, Enseign. Math. (2), 52 (2006), 3-36. |
[22] |
P. J. Nicholls, The Ergodic Theory of Discrete Groups, vol. 143 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1989.
doi: 10.1017/CBO9780511600678. |
[23] |
J.-P. Otal and M. Peigné, Principe variationnel et groupes kleiniens, Duke Math. J., 125 (2004), 15-44.
doi: 10.1215/S0012-7094-04-12512-6. |
[24] |
S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.
doi: 10.1007/BF02392046. |
[25] |
T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), vi+96. |
[26] |
T. Roblin, Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative, Israel J. Math., 147 (2005), 333-357.
doi: 10.1007/BF02785371. |
[27] |
V. Schroeder, Existence of immersed tori in manifolds of nonpositive curvature, J. Reine Angew. Math., 390 (1988), 32-46.
doi: 10.1515/crll.1988.390.32. |
[28] |
V. Schroeder, Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature, Manuscripta Math., 64 (1989), 77-105.
doi: 10.1007/BF01182086. |
[29] |
V. Schroeder, Codimension one tori in manifolds of nonpositive curvature, Geom. Dedicata, 33 (1990), 251-263.
doi: 10.1007/BF00181332. |
[30] |
D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 171-202. |
[31] |
M. E. Taylor, Measure Theory and Integration, vol. 76 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/gsm/076. |
[32] |
M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Co., New York, 1975, Reprinting of the 1959 original. |
[33] |
C. Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., 348 (1996), 4965-5005.
doi: 10.1090/S0002-9947-96-01614-5. |
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