On the spectral theory and dynamics of asymptotically hyperbolic manifolds

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1. Michael T. Anderson, Geometric aspects of the AdS/CFT correspondence, AdS/CFT correspondence: Einstein metrics and their conformal boundaries 8 (2005), 1-31, Eur. Math. Soc., Zürich Zbl1071.81553MR2160865
2. Michael T. Anderson, Topics in conformally compact Einstein metrics, Perspectives in Riemannian geometry 40 (2006), 1-26, Amer. Math. Soc., Providence, RI Zbl1110.53031MR2237104
3. D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967) Zbl0163.43604MR224110
4. James Arthur, The trace formula and Hecke operators, Number theory, trace formulas and discrete groups (Oslo, 1987) (1989), 11-27, Academic Press, Boston, MA Zbl0671.10026MR993309
5. Eric Bahuaud, An intrinsic characterization of asymptotically hyperbolic metrics, (2007) Zbl1163.53025MR2710594
6. Eric Bahuaud, Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics, Pacific J. Math. 239 (2009), 231-249 Zbl1163.53025MR2457230
7. Luis Barreira, Yakov B. Pesin, Lyapunov exponents and smooth ergodic theory, 23 (2002), American Mathematical Society, Providence, RI Zbl1195.37002MR1862379
8. Pierre H. Bérard, On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z. 155 (1977), 249-276 Zbl0341.35052MR455055
9. R. L. Bishop, B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49 Zbl0191.52002MR251664
10. J. Bolton, Conditions under which a geodesic flow is Anosov, Math. Ann. 240 (1979), 103-113 Zbl0382.58017MR524660
11. M. Born, W. Heisenberg, P. Jordan, Zur Quantenmechanik II, Zeitschrift für Physik 35 (1925), 557-616 
12. David Borthwick, Scattering theory for conformally compact metrics with variable curvature at infinity, J. Funct. Anal. 184 (2001), 313-376 Zbl1006.58019MR1851001
13. David Borthwick, Upper and lower bounds on resonances for manifolds hyperbolic near infinity, Comm. Partial Differential Equations 33 (2008), 1507-1539 Zbl1168.58012MR2450168
14. David Borthwick, Chris Judge, Peter A. Perry, Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces, Comment. Math. Helv. 80 (2005), 483-515 Zbl1079.58023MR2165200
15. David Borthwick, Peter Perry, Scattering poles for asymptotically hyperbolic manifolds, Trans. Amer. Math. Soc. 354 (2002), 1215-1231 (electronic) Zbl1009.58021MR1867379
16. David Borthwick, Peter Perry, Inverse scattering results for manifolds hyperbolic near infinity, (2009) Zbl1229.58024
17. Rufus Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 1-30 Zbl0254.58005MR298700
18. Rufus Bowen, Maximizing entropy for a hyperbolic flow, Math. Systems Theory 7 (1974), 300-303 Zbl0303.58014MR385928
19. Richard D. Canary, Yair N. Minsky, Edward C. Taylor, Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds, J. Geom. Anal. 9 (1999), 17-40 Zbl0957.57012MR1760718
20. Manfredo Perdigão do Carmo, Riemannian geometry, (1992), Birkhäuser Boston Inc., Boston, MA Zbl0752.53001MR1138207
21. Sun-Yung A. Chang, Matthew J. Gursky, Paul Yang, Conformal invariants associated to a measure, Proc. Natl. Acad. Sci. USA 103 (2006), 2535-2540 Zbl1160.53356MR2203156
22. Sun-Yung A. Chang, Jie Qing, Paul Yang, Some progress in conformal geometry, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007) Zbl1133.53031MR2366900
23. Su Shing Chen, Anthony Manning, The convergence of zeta functions for certain geodesic flows depends on their pressure, Math. Z. 176 (1981), 379-382 Zbl0437.58016MR610218
24. P. Dirac, The quantum theory of the electron, Proc. R. Soc. London Series A, (1928), 610-624 Zbl54.0973.01
25. J. J. Duistermaat, V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39-79 Zbl0307.35071MR405514
26. Patrick Eberlein, Geodesic flows on negatively curved manifolds. I, Ann. of Math. (2) 95 (1972), 492-510 Zbl0217.47304MR310926
27. Patrick Eberlein, Geodesic flows on negatively curved manifolds. II, Trans. Amer. Math. Soc. 178 (1973), 57-82 Zbl0264.53027MR314084
28. Patrick Eberlein, When is a geodesic flow of Anosov type? I,II, J. Differential Geometry 8 (1973), 437-463; ibid. 8 (1973), 565–577 Zbl0295.58009MR380891
29. Patrick Eberlein, Ursula Hamenstädt, Viktor Schroeder, Manifolds of nonpositive curvature, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990) 54 (1993), 179-227, Amer. Math. Soc., Providence, RI Zbl0811.53038MR1216622
30. Patrick Eberlein, B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45-109 Zbl0264.53026MR336648
31. P. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 47 (1935), 777-780 Zbl0012.04201
32. Charles Fefferman, C. Robin Graham, Conformal invariants, Astérisque (1985), 95-116 Zbl0602.53007MR837196
33. Charles Fefferman, C. Robin Graham, $Q$-curvature and Poincaré metrics, Math. Res. Lett. 9 (2002), 139-151 Zbl1016.53031MR1909634
34. Ernesto Franco, Flows with unique equilibrium states, Amer. J. Math. 99 (1977), 486-514 Zbl0368.54014MR442193
35. A. Freire, R. Mañé, On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math. 69 (1982), 375-392 Zbl0476.58019MR679763
36. Ramesh Gangolli, Garth Warner, On Selberg’s trace formula, J. Math. Soc. Japan 27 (1975), 328-343 Zbl0325.22014MR399354
37. Ramesh Gangolli, Garth Warner, Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one, Nagoya Math. J. 78 (1980), 1-44 MR571435
38. C. Robin Graham, Volume and area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School “Geometry and Physics” (Srní, 1999) (2000), 31-42 Zbl0984.53020MR1758076
39. C. Robin Graham, Ralph Jenne, Lionel J. Mason, George A. J. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. (2) 46 (1992), 557-565 Zbl0726.53010MR1190438
40. C. Robin Graham, Maciej Zworski, Scattering matrix in conformal geometry, Invent. Math. 152 (2003), 89-118 Zbl1030.58022MR1965361
41. Colin Guillarmou, Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J. 129 (2005), 1-37 Zbl1099.58011MR2153454
42. Colin Guillarmou, Generalized Krein formula, determinants, and Selberg zeta function in even dimension, Amer. J. Math. 131 (2009), 1359-1417 Zbl1207.58023MR2555844
43. Colin Guillarmou, Frédéric Naud, Wave 0-trace and length spectrum on convex co-compact hyperbolic manifolds, Comm. Anal. Geom. 14 (2006), 945-967 Zbl1127.58028MR2287151
44. Victor Guillemin, Wave-trace invariants and a theorem of Zelditch, Internat. Math. Res. Notices (1993), 303-308 Zbl0798.58073MR1253645
45. Victor Guillemin, Wave-trace invariants, Duke Math. J. 83 (1996), 287-352 Zbl0858.58051MR1390650
46. Laurent Guillopé, Sur la distribution des longueurs des géodésiques fermées d’une surface compacte à bord totalement géodésique, Duke Math. J. 53 (1986), 827-848 Zbl0611.53042MR860674
47. Laurent Guillopé, Maciej Zworski, Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity, Asymptotic Anal. 11 (1995), 1-22 Zbl0859.58028MR1344252
48. Laurent Guillopé, Maciej Zworski, Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Anal. 129 (1995), 364-389 Zbl0841.58063MR1327183
49. Laurent Guillopé, Maciej Zworski, The wave trace for Riemann surfaces, Geom. Funct. Anal. 9 (1999), 1156-1168 Zbl0947.58022MR1736931
50. M. C. Gutzwiller, Periodic orbits and classical quantization conditions, J. Math. Phys. 12 (1971), 343-358 
51. Dennis A. Hejhal, The Selberg trace formula for congruence subgroups, Bull. Amer. Math. Soc. 81 (1975), 752-755 Zbl0304.10018MR371818
52. Dennis A. Hejhal, The Selberg trace formula for $\mathrm{PSL}(2,\mathbf{R})$. Vol. 1 and 2, 548 and 1001 (1976 and 1983), Springer-Verlag, Berlin Zbl0543.10020
53. Lars Hörmander, The analysis of linear partial differential operators. I, (2003), Springer-Verlag, Berlin Zbl1028.35001MR1996773
54. Dmitry Jakobson, Iosif Polterovich, John A. Toth, A lower bound for the remainder in Weyl’s law on negatively curved surfaces, Int. Math. Res. Not. IMRN (2008) Zbl1161.58010MR2418855
55. Mark S. Joshi, Antônio Sá Barreto, Inverse scattering on asymptotically hyperbolic manifolds, Acta Math. 184 (2000), 41-86 Zbl1142.58309MR1756569
56. Mark S. Joshi, Antônio Sá Barreto, The wave group on asymptotically hyperbolic manifolds, J. Funct. Anal. 184 (2001), 291-312 Zbl0997.58010MR1851000
57. A. Karnaukh, Spectral count on compact negatively curved surfaces, (1996) MR2695000
58. Anatole Katok, Boris Hasselblatt, Introduction to the modern theory of dynamical systems, 54 (1995), Cambridge University Press, Cambridge Zbl0878.58019MR1326374
59. Wilhelm Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2) 99 (1974), 1-13 Zbl0272.53025MR377980
60. S. P. Lalley, The “prime number theorem” for the periodic orbits of a Bernoulli flow, Amer. Math. Monthly 95 (1988), 385-398 Zbl0645.28013MR937528
61. Peter D. Lax, Ralph S. Phillips, Translation representation for automorphic solutions of the wave equation in non-Euclidean spaces. I, Comm. Pure Appl. Math. 37 (1984), 303-328 Zbl0544.10024MR739923
62. John M. Lee, The spectrum of an asymptotically hyperbolic Einstein manifold, Comm. Anal. Geom. 3 (1995), 253-271 Zbl0934.58029MR1362652
63. John M. Lee, Fredholm operators and Einstein metrics on conformally compact manifolds, Mem. Amer. Math. Soc. 183 (2006) Zbl1112.53002MR2252687
64. Anthony Manning, Topological entropy for geodesic flows, Ann. of Math. (2) 110 (1979), 567-573 Zbl0426.58016MR554385
65. Anthony Manning, private correspondence, (2008) 
66. Rafe Mazzeo, The Hodge cohomology of a conformally compact metric, J. Differential Geom. 28 (1988), 309-339 Zbl0656.53042MR961517
67. Rafe Mazzeo, Frank Pacard, Maskit combinations of Poincaré-Einstein metrics, Adv. Math. 204 (2006), 379-412 Zbl1097.53029MR2249618
68. Rafe R. Mazzeo, Richard B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), 260-310 Zbl0636.58034MR916753
69. Richard B. Melrose, The Atiyah-Patodi-Singer index theorem, 4 (1993), A K Peters Ltd., Wellesley, MA Zbl0796.58050MR1348401
70. Werner Müller, Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math. 109 (1992), 265-305 Zbl0772.58063MR1172692
71. Frédéric Naud, Classical and quantum lifetimes on some non-compact Riemann surfaces, J. Phys. A 38 (2005), 10721-10729 Zbl1082.81026MR2197679
72. William Parry, Mark Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. of Math. (2) 118 (1983), 573-591 Zbl0537.58038MR727704
73. S. J. Patterson, Lectures on measures on limit sets of Kleinian groups, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) 111 (1987), 281-323, Cambridge Univ. Press, Cambridge Zbl0611.30036MR903855
74. S. J. Patterson, Peter A. Perry, Divisor of the Selberg zeta function for Kleinian groups in even dimensions, Duke Math. J. 326 (2001), 321-390 Zbl1012.11083MR1813434
75. Peter A. Perry, The Laplace operator on a hyperbolic manifold. I. Spectral and scattering theory, J. Funct. Anal. 75 (1987), 161-187 Zbl0631.58030MR911204
76. Peter A. Perry, Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume, Geom. Funct. Anal. 11 (2001), 132-141 Zbl0986.11059MR1829645
77. Peter A. Perry, A Poisson summation formula and lower bounds for resonances in hyperbolic manifolds, Int. Math. Res. Not. (2003), 1837-1851 Zbl1035.58020MR1988782
78. R. S. Phillips, P. Sarnak, The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups, Acta Math. 155 (1985), 173-241 Zbl0611.30037MR806414
79. Ralph Phillips, The spectrum of the Laplacian for domains in hyperbolic space and limit sets of Kleinian groups, Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983) 65 (1984), 521-525, Birkhäuser, Basel Zbl0559.10017MR820548
80. Ralph Phillips, Zeév Rudnick, The circle problem in the hyperbolic plane, J. Funct. Anal. 121 (1994), 78-116 Zbl0812.11035MR1270589
81. Steven Rosenberg, The Laplacian on a Riemannian manifold, 31 (1997), Cambridge University Press, Cambridge Zbl0868.58074MR1462892
82. Julie Rowlett, Dynamics of asymptotically hyperbolic manifolds, Pacific J. Math. 242 (2009), 377-397 Zbl1198.37036MR2546718
83. Michael Rubinstein, Peter Sarnak, Chebyshev’s bias, Experiment. Math. 3 (1994), 173-197 Zbl0823.11050MR1329368
84. Antônio Sá Barreto, Radiation fields, scattering, and inverse scattering on asymptotically hyperbolic manifolds., Duke Math. J. 129 (2005), 407-480 Zbl1154.58310MR2169870
85. E. Schrödinger, An undulatory theory of the mechanics of atoms and molecules, Phys. Rev. 28 (1926), 1049-1070 Zbl52.0965.07
86. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47-87 Zbl0072.08201MR88511
87. Yuguang Shi, Gang Tian, Rigidity of asymptotically hyperbolic manifolds, Comm. Math. Phys. 259 (2005), 545-559 Zbl1092.53033MR2174416
88. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817 Zbl0202.55202MR228014
89. Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. (1979), 171-202 Zbl0439.30034MR556586
90. Peter Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math. 97 (1975), 937-971 Zbl0318.28007MR390180
91. Chengbo Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc. 348 (1996), 4965-5005 Zbl0864.58047MR1348871
92. Steven Zelditch, On the rate of quantum ergodicity. I. Upper bounds, Comm. Math. Phys. 160 (1994), 81-92 Zbl0788.58043MR1262192
93. Steven Zelditch, Wave invariants at elliptic closed geodesics, Geom. Funct. Anal. 7 (1997), 145-213 Zbl0876.58010MR1437476
94. Steven Zelditch, Wave invariants for non-degenerate closed geodesics, Geom. Funct. Anal. 8 (1998), 179-217 Zbl0908.58022MR1601862