Egorov theorems and equidistribution of eigenfunctions for the quantized sawtooth and Baker maps
De Bièvre, S. ; Degli Esposti, M.
Annales de l'I.H.P. Physique théorique, Tome 69 (1998), p. 1-30 / Harvested from Numdam
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@article{AIHPA_1998__69_1_1_0,
     author = {De Bi\`evre, Stephan and Degli Esposti, Mirko},
     title = {Egorov theorems and equidistribution of eigenfunctions for the quantized sawtooth and Baker maps},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     volume = {69},
     year = {1998},
     pages = {1-30},
     mrnumber = {1635807},
     zbl = {0922.58074},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPA_1998__69_1_1_0}
}

De Bièvre, S.; Degli Esposti, M. Egorov theorems and equidistribution of eigenfunctions for the quantized sawtooth and Baker maps. Annales de l'I.H.P. Physique théorique, Tome 69 (1998) pp. 1-30. http://gdmltest.u-ga.fr/item/AIHPA_1998__69_1_1_0/

[1] V.I. Arnold, A. Avez, Ergodic Problems in Classical Mechanics, Benjamin, New York, 1968. | MR 232910 | Zbl 0167.22901

[2] N.L. Balazs, A. Voros, The Quantized Baker's Transformation, Annals of Physics, Vol. 190, 1989, 1-31. | MR 994045 | Zbl 0664.58045

[3] A. Bouzouina, S. De Bièvre, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Commun. Math. Phys., Vol. 178, 1996, 83-105. | MR 1387942 | Zbl 0876.58041

[4] M. Basilio De Matos and A.M. Ozerio De Almeida, Quantization of Anosov maps, Annals of Physics, Vol. 237, 1, 1995, 46-65. | MR 1309729 | Zbl 0812.58041

[5] J. Chazarain, Construction de la paramétrix du problème mixte hyperbolique pour l'équation d'ondes, C. R. Acad. Sci. Paris, Vol. 276, 1973, 1213-1215. | MR 320536 | Zbl 0253.35058

[6] N. Chernoff, Ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the two-torus, J. Stat. Phys., Vol. 69, 1992, 111-134. | MR 1184772 | Zbl 0925.58072

[7] Y. Colin De Verdière, Ergodicité et functions propres du Laplacien, Commun. Math. Phys., Vol. 102, 1985, 497-502. | MR 818831 | Zbl 0592.58050

[8] M. Combescure, D. Robert, Distribution of matrix elements and level spacings for classical chaotic systems, Ann. Inst. H. Poincaré, Vol. 61, 4, 1994, 443-483. | Numdam | MR 1311538 | Zbl 0833.58018

[9] S. De Bièvre, M. Degli Esposti and R. Giachetti,Quantization of a class of piecewise affine transformations on the torus, Commun. Math. Phys., Vol. 176, 1995, 73-94. | MR 1372818 | Zbl 0840.58019

[10] M. Degli Esposti, Quantization of the orientation preserving automorphisms of the torus, Ann. Inst. Henri Poincaré, Vol. 58, 1993, 323-341. | Numdam | MR 1222946 | Zbl 0777.58017

[11] M. Degli Esposti, S. Isola and S. Graffi, Classical limit of the quantized hyperbolic toral automorphisms, Commun. Math. Phys., Vol. 167, 1995, 471-507. | MR 1316757 | Zbl 0822.58022

[12] R.E. Edwards, Fourier Series. A modern Introduction. Volume 1, Graduate Text in Mathematics, Vol. 64, 1979. | Zbl 0424.42001

[13] M. Farris, Egorov's theorem on a manifold with diffractive boundary, Comm. P. D. E., Vol. 6, 6, 1981, 651-687. | MR 617785 | Zbl 0474.58019

[14] P. Gerard, E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. Jour., Vol. 71, 1993, 559-607. | MR 1233448 | Zbl 0788.35103

[15] J.H. Hannay, M.V. Berry, Quantization of linear maps on a torus - Fresnel diffraction by a periodic grating, Physica D, Vol. 1, 1980, 267-291. | MR 602111

[16] J.H. Hannay, J.P. Keating, A.M. Ozorio De Almeida, Optical realization of the baker's transformation, Nonlinearity, Vol. 7, 1994, 1327-1342 | MR 1294545 | Zbl 0803.58025

[17] B. Helffer, A.MARTINEZ, D. Robert, Ergodicité et limite semi-classique, Commun. Math. Phys., Vol. 131, 1985, 493-520.

[18] J. Keating, Ph. D. thesis University of Bristol, 1989.

[19] J. Keating, The cat maps: quantum mechanics and classical motion, Nonlinearity, Vol. 4, 1991, 309-341. | MR 1107009 | Zbl 0726.58037

[20] A. Laskshminarayan, Semiclassical theory of the Sawtooth map, Phys. Lett. A, Vol. 192, 1994, 345-354. | MR 1291311 | Zbl 0960.81518

[21] C. Liverani, Decay of correlations, Annals of Mathematics, Vol. 142, 1995, 239-301. | MR 1343323 | Zbl 0871.58059

[22] C. Liverani, M.P. Wojtkowski, Ergodicity in Hamiltonian systems, to appear in Dyn. Rep., Vol. 4. | MR 1346498

[23] P. O'Connor, R. Heller, S. Tomsovic, Semiclassical dynamics in the strongly chaotic regime: breaking the log-time barrier, Physica D, Vol. 55, 1992, 340-357. | MR 1156735 | Zbl 0753.58029

[24] D. Robert, Autour de l'approximation semi-classique, Birkhaüser Boston, 1987. | MR 897108 | Zbl 0621.35001

[25] P. Sarnak, Arithmetic quantum chaos, Tel Aviv Lectures, 1993.

[26] M. Saraceno, Classical structures in the quantized Baker transformation, Annals of Physics, Vol. 199, 1990, 37-60. | MR 1048673 | Zbl 0724.58059

[27] M. Saraceno, A. Voros, Towards a semiclassical theory of the quantum baker's map, Physica D, Vol. 79, 1994, 206-268. | MR 1306462 | Zbl 0888.58036

[28] M. Saraceno, R.O. Vallejos, The quantized D-transformation, (preprint 1995). | MR 1393556

[29] S. Vaienti, Ergodic properties of the discontinuous sawtooth map, J. Stat. Phys., Vol. 67, 1992, 251. | MR 1159464 | Zbl 0892.58049

[30] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. Jour., 55, 1987, 919-941. | MR 916129 | Zbl 0643.58029

[3 1 ] S. Zelditch, Quantum transition amplitudes for ergodic and completely integrable systems, Journ. Funct. Ana., Vol. 94, 1990, 415-436. | MR 1081652 | Zbl 0721.58051

[32] S. Zelditch, Quantum ergodicity of C* dynamical systems, Commun. Math. Phys., Vol. 177, 1996, 507-528. | MR 1384146 | Zbl 0856.58019

[33] S. Zelditch, Index and dynamics of quantized contact transformations, Ann. Inst. Fourier, Vol. 47, 1997, 305-363. | Numdam | MR 1437187 | Zbl 0865.47018

[34] S. Zelditch, M. Zworski, Ergodicity of eigenfunctions for ergodic billiards, Commun. Math. Phys., Vol. 175, 1996, 3, 673-682. | MR 1372814 | Zbl 0840.58048