Cet article est une présentation rapide de la théorie spectrale et de la dynamique des variétés asymptotiquement hyperboliques à volume infini. Nous commençons par leur géométrie et quelques exemples, nous poursuivons en rappelant leur théorie spectrale, puis continuons sur des développements récents de leur dynamique. Nous concluons par une discussion des résultats qui démontrent un rapport entre leurs mécaniques quantiques et classiques et enfin, nous offrons quelques idées et conjectures.
We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures.
Keywords: Asymptotically hyperbolic, conformally compact, wave trace, negative curvature, resonances, length spectrum, topological entropy, dynamics, geodesic flow, prime orbit theorem, quantum and classical mechanics
Mot clés : variété asymptotiquement hyperbolique, conformement compact, courbures negatives, spectre des longeurs géodesiques, flot géodesique, dynamique, formule de trace dynamique, entropie topologique, mécanique quantique et classique
@article{AIF_2010__60_7_2461_0, author = {Rowlett, Julie}, title = {On the spectral theory and dynamics of asymptotically hyperbolic manifolds}, journal = {Annales de l'Institut Fourier}, pages = {2461--2492}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {60}, number = {7}, year = {2010}, doi = {10.5802/aif.2615}, zbl = {1252.37025}, mrnumber = {2849270}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2615/} }
TY - JOUR AU - Rowlett, Julie TI - On the spectral theory and dynamics of asymptotically hyperbolic manifolds JO - Annales de l'Institut Fourier PY - 2010 SP - 2461 EP - 2492 VL - 60 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2615/ DO - 10.5802/aif.2615 LA - en ID - AIF_2010__60_7_2461_0 ER -
%0 Journal Article %A Rowlett, Julie %T On the spectral theory and dynamics of asymptotically hyperbolic manifolds %J Annales de l'Institut Fourier %D 2010 %P 2461-2492 %V 60 %N 7 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2615/ %R 10.5802/aif.2615 %G en %F AIF_2010__60_7_2461_0
Rowlett, Julie. On the spectral theory and dynamics of asymptotically hyperbolic manifolds. Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2461-2492. doi : 10.5802/aif.2615. http://www.numdam.org/articles/10.5802/aif.2615/
[1] Geometric aspects of the AdS/CFT correspondence, AdS/CFT correspondence: Einstein metrics and their conformal boundaries (IRMA Lect. Math. Theor. Phys.), Volume 8, Eur. Math. Soc., Zürich, 2005, pp. 1-31 | DOI | MR | Zbl
[2] Topics in conformally compact Einstein metrics, Perspectives in Riemannian geometry (CRM Proc. Lecture Notes), Volume 40, Amer. Math. Soc., Providence, RI, 2006, pp. 1-26 | MR | Zbl
[3] Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., Volume 90 (1967), pp. 209 | MR | Zbl
[4] The trace formula and Hecke operators, Number theory, trace formulas and discrete groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 11-27 | MR | Zbl
[5] An intrinsic characterization of asymptotically hyperbolic metrics, University of Washington (2007) (Ph. D. Thesis) | MR
[6] Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics, Pacific J. Math., Volume 239 (2009) no. 2, pp. 231-249 | DOI | MR | Zbl
[7] Lyapunov exponents and smooth ergodic theory, University Lecture Series, 23, American Mathematical Society, Providence, RI, 2002 | MR | Zbl
[8] On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., Volume 155 (1977) no. 3, pp. 249-276 | DOI | MR | Zbl
[9] Manifolds of negative curvature, Trans. Amer. Math. Soc., Volume 145 (1969), pp. 1-49 | DOI | MR | Zbl
[10] Conditions under which a geodesic flow is Anosov, Math. Ann., Volume 240 (1979) no. 2, pp. 103-113 | DOI | MR | Zbl
[11] Zur Quantenmechanik II, Zeitschrift für Physik, Volume 35 (1925), pp. 557-616 | DOI
[12] Scattering theory for conformally compact metrics with variable curvature at infinity, J. Funct. Anal., Volume 184 (2001) no. 2, pp. 313-376 | DOI | MR | Zbl
[13] Upper and lower bounds on resonances for manifolds hyperbolic near infinity, Comm. Partial Differential Equations, Volume 33 (2008) no. 7-9, pp. 1507-1539 | DOI | MR | Zbl
[14] Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces, Comment. Math. Helv., Volume 80 (2005) no. 3, pp. 483-515 | DOI | MR | Zbl
[15] Scattering poles for asymptotically hyperbolic manifolds, Trans. Amer. Math. Soc., Volume 354 (2002) no. 3, p. 1215-1231 (electronic) | DOI | MR | Zbl
[16] Inverse scattering results for manifolds hyperbolic near infinity, 2009 (arXiv:0906.0542v2)
[17] Periodic orbits for hyperbolic flows, Amer. J. Math., Volume 94 (1972), pp. 1-30 | DOI | MR | Zbl
[18] Maximizing entropy for a hyperbolic flow, Math. Systems Theory, Volume 7 (1974) no. 4, pp. 300-303 | MR | Zbl
[19] Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds, J. Geom. Anal., Volume 9 (1999) no. 1, pp. 17-40 | MR | Zbl
[20] Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992 (Translated from the second Portuguese edition by Francis Flaherty) | MR | Zbl
[21] Conformal invariants associated to a measure, Proc. Natl. Acad. Sci. USA, Volume 103 (2006) no. 8, pp. 2535-2540 | DOI | MR | Zbl
[22] Some progress in conformal geometry, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 3 (2007), pp. Paper 122, 17 | DOI | MR | Zbl
[23] The convergence of zeta functions for certain geodesic flows depends on their pressure, Math. Z., Volume 176 (1981) no. 3, pp. 379-382 | DOI | MR | Zbl
[24] The quantum theory of the electron, Proc. R. Soc. London Series A, (1928) no. 778, pp. 610-624 (Containing Papers of a Mathematical and Physical Character 117) | DOI
[25] The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., Volume 29 (1975) no. 1, pp. 39-79 | DOI | MR | Zbl
[26] Geodesic flows on negatively curved manifolds. I, Ann. of Math. (2), Volume 95 (1972), pp. 492-510 | DOI | MR | Zbl
[27] Geodesic flows on negatively curved manifolds. II, Trans. Amer. Math. Soc., Volume 178 (1973), pp. 57-82 | DOI | MR | Zbl
[28] When is a geodesic flow of Anosov type? I,II, J. Differential Geometry, Volume 8 (1973), p. 437-463; ibid. 8 (1973), 565–577 | MR | Zbl
[29] Manifolds of nonpositive curvature, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990) (Proc. Sympos. Pure Math.), Volume 54, Amer. Math. Soc., Providence, RI, 1993, pp. 179-227 | MR
[30] Visibility manifolds, Pacific J. Math., Volume 46 (1973), pp. 45-109 | MR | Zbl
[31] Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev., Volume 47 (1935), pp. 777-780 | DOI | Zbl
[32] Conformal invariants, Astérisque (1985) no. Numero Hors Serie, pp. 95-116 The mathematical heritage of Élie Cartan (Lyon, 1984) | Numdam | MR | Zbl
[33] -curvature and Poincaré metrics, Math. Res. Lett., Volume 9 (2002) no. 2-3, pp. 139-151 | MR | Zbl
[34] Flows with unique equilibrium states, Amer. J. Math., Volume 99 (1977) no. 3, pp. 486-514 | DOI | MR | Zbl
[35] On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math., Volume 69 (1982) no. 3, pp. 375-392 | DOI | MR | Zbl
[36] On Selberg’s trace formula, J. Math. Soc. Japan, Volume 27 (1975), pp. 328-343 | DOI | MR | Zbl
[37] Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one, Nagoya Math. J., Volume 78 (1980), pp. 1-44 http://projecteuclid.org/getRecord?id=euclid.nmj/1118786087 | MR
[38] Volume and area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School “Geometry and Physics” (Srní, 1999) (2000) no. 63, pp. 31-42 | MR | Zbl
[39] Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. (2), Volume 46 (1992) no. 3, pp. 557-565 | DOI | MR | Zbl
[40] Scattering matrix in conformal geometry, Invent. Math., Volume 152 (2003) no. 1, pp. 89-118 | DOI | MR | Zbl
[41] Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J., Volume 129 (2005) no. 1, pp. 1-37 | DOI | MR | Zbl
[42] Generalized Krein formula, determinants, and Selberg zeta function in even dimension, Amer. J. Math., Volume 131 (2009) no. 5, pp. 1359-1417 | DOI | MR | Zbl
[43] Wave 0-trace and length spectrum on convex co-compact hyperbolic manifolds, Comm. Anal. Geom., Volume 14 (2006) no. 5, pp. 945-967 http://projecteuclid.org/getRecord?id=euclid.cag/1175790874 | MR | Zbl
[44] Wave-trace invariants and a theorem of Zelditch, Internat. Math. Res. Notices (1993) no. 12, pp. 303-308 | DOI | MR | Zbl
[45] Wave-trace invariants, Duke Math. J., Volume 83 (1996) no. 2, pp. 287-352 | DOI | MR | Zbl
[46] Sur la distribution des longueurs des géodésiques fermées d’une surface compacte à bord totalement géodésique, Duke Math. J., Volume 53 (1986) no. 3, pp. 827-848 | DOI | MR | Zbl
[47] Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity, Asymptotic Anal., Volume 11 (1995) no. 1, pp. 1-22 | MR | Zbl
[48] Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Anal., Volume 129 (1995) no. 2, pp. 364-389 | DOI | MR | Zbl
[49] The wave trace for Riemann surfaces, Geom. Funct. Anal., Volume 9 (1999) no. 6, pp. 1156-1168 | DOI | MR | Zbl
[50] Periodic orbits and classical quantization conditions, J. Math. Phys., Volume 12 (1971) no. 3, pp. 343-358 | DOI
[51] The Selberg trace formula for congruence subgroups, Bull. Amer. Math. Soc., Volume 81 (1975), pp. 752-755 | DOI | MR | Zbl
[52] The Selberg trace formula for . Vol. 1 and 2, Lecture Notes in Mathematics, 548 and 1001, Springer-Verlag, Berlin, 1976 and 1983 | Zbl
[53] The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003 Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)] | MR | Zbl
[54] A lower bound for the remainder in Weyl’s law on negatively curved surfaces, Int. Math. Res. Not. IMRN (2008) no. 2, pp. Art. ID rnm142, 38 | MR | Zbl
[55] Inverse scattering on asymptotically hyperbolic manifolds, Acta Math., Volume 184 (2000) no. 1, pp. 41-86 | DOI | MR | Zbl
[56] The wave group on asymptotically hyperbolic manifolds, J. Funct. Anal., Volume 184 (2001) no. 2, pp. 291-312 | DOI | MR | Zbl
[57] Spectral count on compact negatively curved surfaces, Princeton University (1996) (Ph. D. Thesis) | MR
[58] Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995 (With a supplementary chapter by Katok and Leonardo Mendoza) | MR | Zbl
[59] Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), Volume 99 (1974), pp. 1-13 | DOI | MR | Zbl
[60] The “prime number theorem” for the periodic orbits of a Bernoulli flow, Amer. Math. Monthly, Volume 95 (1988) no. 5, pp. 385-398 | DOI | MR | Zbl
[61] Translation representation for automorphic solutions of the wave equation in non-Euclidean spaces. I, Comm. Pure Appl. Math., Volume 37 (1984) no. 3, pp. 303-328 | DOI | MR | Zbl
[62] The spectrum of an asymptotically hyperbolic Einstein manifold, Comm. Anal. Geom., Volume 3 (1995) no. 1-2, pp. 253-271 | MR | Zbl
[63] Fredholm operators and Einstein metrics on conformally compact manifolds, Mem. Amer. Math. Soc., Volume 183 (2006) no. 864, pp. vi+83 | MR | Zbl
[64] Topological entropy for geodesic flows, Ann. of Math. (2), Volume 110 (1979) no. 3, pp. 567-573 | DOI | MR | Zbl
[65] private correspondence, 2008
[66] The Hodge cohomology of a conformally compact metric, J. Differential Geom., Volume 28 (1988) no. 2, pp. 309-339 http://projecteuclid.org/getRecord?id=euclid.jdg/1214442281 | MR | Zbl
[67] Maskit combinations of Poincaré-Einstein metrics, Adv. Math., Volume 204 (2006) no. 2, pp. 379-412 | DOI | MR | Zbl
[68] Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., Volume 75 (1987) no. 2, pp. 260-310 | DOI | MR | Zbl
[69] The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, 4, A K Peters Ltd., Wellesley, MA, 1993 | MR | Zbl
[70] Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math., Volume 109 (1992) no. 2, pp. 265-305 | DOI | MR | Zbl
[71] Classical and quantum lifetimes on some non-compact Riemann surfaces, J. Phys. A, Volume 38 (2005) no. 49, pp. 10721-10729 | DOI | MR | Zbl
[72] An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. of Math. (2), Volume 118 (1983) no. 3, pp. 573-591 | DOI | MR | Zbl
[73] Lectures on measures on limit sets of Kleinian groups, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) (London Math. Soc. Lecture Note Ser.), Volume 111, Cambridge Univ. Press, Cambridge, 1987, pp. 281-323 | MR | Zbl
[74] Divisor of the Selberg zeta function for Kleinian groups in even dimensions, Duke Math. J., Volume 326 (2001), pp. 321-390 (with an appendix by C. Epstein) | MR | Zbl
[75] The Laplace operator on a hyperbolic manifold. I. Spectral and scattering theory, J. Funct. Anal., Volume 75 (1987) no. 1, pp. 161-187 | DOI | MR | Zbl
[76] Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume, Geom. Funct. Anal., Volume 11 (2001) no. 1, pp. 132-141 | DOI | MR | Zbl
[77] A Poisson summation formula and lower bounds for resonances in hyperbolic manifolds, Int. Math. Res. Not. (2003) no. 34, pp. 1837-1851 | DOI | MR | Zbl
[78] The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups, Acta Math., Volume 155 (1985) no. 3-4, pp. 173-241 | DOI | MR | Zbl
[79] 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) (Internat. Schriftenreihe Numer. Math.), Volume 65, Birkhäuser, Basel, 1984, pp. 521-525 | MR | Zbl
[80] The circle problem in the hyperbolic plane, J. Funct. Anal., Volume 121 (1994) no. 1, pp. 78-116 | DOI | MR | Zbl
[81] The Laplacian on a Riemannian manifold, London Mathematical Society Student Texts, 31, Cambridge University Press, Cambridge, 1997 (An introduction to analysis on manifolds) | DOI | MR | Zbl
[82] Dynamics of asymptotically hyperbolic manifolds, Pacific J. Math., Volume 242 (2009) no. 2, pp. 377-397 | DOI | MR | Zbl
[83] Chebyshev’s bias, Experiment. Math., Volume 3 (1994) no. 3, pp. 173-197 http://projecteuclid.org/getRecord?id=euclid.em/1048515870 | MR | Zbl
[84] Radiation fields, scattering, and inverse scattering on asymptotically hyperbolic manifolds., Duke Math. J., Volume 129 (2005) no. 3, pp. 407-480 | DOI | MR | Zbl
[85] An undulatory theory of the mechanics of atoms and molecules, Phys. Rev., Volume 28 (1926) no. 6, pp. 1049-1070 | DOI
[86] Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.), Volume 20 (1956), pp. 47-87 | MR | Zbl
[87] Rigidity of asymptotically hyperbolic manifolds, Comm. Math. Phys., Volume 259 (2005) no. 3, pp. 545-559 | DOI | MR | Zbl
[88] Differentiable dynamical systems, Bull. Amer. Math. Soc., Volume 73 (1967), pp. 747-817 | DOI | MR | Zbl
[89] The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. (1979) no. 50, pp. 171-202 | DOI | Numdam | MR | Zbl
[90] A variational principle for the pressure of continuous transformations, Amer. J. Math., Volume 97 (1975) no. 4, pp. 937-971 | DOI | MR | Zbl
[91] The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., Volume 348 (1996) no. 12, pp. 4965-5005 | DOI | MR | Zbl
[92] On the rate of quantum ergodicity. I. Upper bounds, Comm. Math. Phys., Volume 160 (1994) no. 1, pp. 81-92 http://projecteuclid.org/getRecord?id=euclid.cmp/1104269516 | DOI | MR | Zbl
[93] Wave invariants at elliptic closed geodesics, Geom. Funct. Anal., Volume 7 (1997) no. 1, pp. 145-213 | DOI | MR | Zbl
[94] Wave invariants for non-degenerate closed geodesics, Geom. Funct. Anal., Volume 8 (1998) no. 1, pp. 179-217 | DOI | MR | Zbl
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