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Spectral analysis of the transfer operator for the Lorentz gas
1. | Department of Mathematics and Computer Science, Fairfield University, Fairfield CT 06824, United States |
2. | Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003, United States |
References:
[1] |
V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. |
[2] |
V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $\C^\infty$ foliations, in "Algebraic and Topological Dynamics" (eds. Sergiy Kolyada, Yuri Manin and Tom Ward), Contemporary Mathematics, 385, Amer. Math. Society, Providence, RI, (2005), 123-135. |
[3] |
V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Annales de l'Institute Henri Poincaré Analyse Non Linéare, 26 (2009), 1453-1481.
doi: 10.1016/j.anihpc.2009.01.001. |
[4] |
V. Baladi and S. Gouëzel, Banach spaces for piecewise cone-hyperbolic maps, J. Modern Dynam., 4 (2010), 91-137. |
[5] |
V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier (Grenoble), 57 (2007), 127-154.
doi: 10.5802/aif.2253. |
[6] |
V. Baladi and L.-S.Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys., 156 (1993), 355-385.
doi: 10.1007/BF02098487. |
[7] |
L. Bunimovich, Y. G. Sinaĭ and N. Chernov, Markov partitions for two-dimensional hyperbolic billiards, Russian Math. Surveys, 45 (1990), 105-152.
doi: 10.1070/RM1990v045n03ABEH002355. |
[8] |
L. Bunimovich, Y. G. Sinaĭ and N. Chernov, Statistical properties of two-dimensional hyperbolic billiards, Russian Math. Surveys 46 (1991), 47-106.
doi: 10.1070/RM1991v046n04ABEH002827. |
[9] |
M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.
doi: 10.1088/0951-7715/15/6/309. |
[10] |
J. Buzzi, Absolutely continuous invariant probability measures for arbitrary expanding piecewise $\mathbbR$-analytic mappings of the plane, Ergod. Th. and Dynam. Sys., 20 (2000), 697-708.
doi: 10.1017/S0143385700000377. |
[11] |
J. Buzzi and G. Keller, Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps, Ergod. Th. and Dynam. Sys., 21 (2001), 689-716.
doi: 10.1017/S0143385701001341. |
[12] |
J.-R. Chazottes and S. Gouëzel, On almost-sure versions of classical theorems for dynamical systems, Probability Theory and Related Fields, 138 (2007), 195-234.
doi: 10.1007/s00440-006-0021-6. |
[13] |
N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys., 94 (1999), 513-556.
doi: 10.1023/A:1004581304939. |
[14] |
N. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys., 122 (2006), 1061-1094.
doi: 10.1007/s10955-006-9036-8. |
[15] |
N. Chernov and R. Markarian, "Chaotic Billiards," Mathematical Surveys and Monographs, 127, AMS, Providence, RI, 2006. |
[16] |
N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls, in "Hard Ball Systems and the Lorentz Gas" (ed. D. Szasz), Enclyclopaedia of Mathematical Sciences, 101, Springer, Berlin, (2000), 89-120. |
[17] |
A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications," Second edition, Applications of Mathematics, 38, Springer-Verlag, New York, 1998. |
[18] |
M. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc., 360 (2008), 4777-4814.
doi: 10.1090/S0002-9947-08-04464-4. |
[19] |
M. Demers, Functional norms for Young towers, Ergod. Th. Dynam. Sys., 30 (2010), 1371-1398.
doi: 10.1017/S0143385709000534. |
[20] |
W. Doeblin and R. Fortet, Sur des chaînes à liaisons complètes, Bull. Soc. Math. France, 65 (1937), 132-148. |
[21] |
S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergod. Th. and Dynam. Sys., 26 (2006), 189-217. |
[22] |
S. Gouëzel, Almost sure invariance principle for dynamical systems by spectral methods, Ann. Prob., 38 (2010), 1639-1671.
doi: 10.1214/10-AOP525. |
[23] |
H. Hennion and L. Hevré, "Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness," Lectures Notes in Mathematics, 1766, Springer-Verlag, Berlin, 2001. |
[24] |
C. T. Ionescu-Tulcea and G. Marinescu, Théorie ergodique pour des classes d'opérations non complètement continues, Ann. of Math. (2), 52 (1950), 140-147. |
[25] |
T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenchaften, 132, Springer-Verlag, Berlin-New York, 1976. |
[26] |
G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193.
doi: 10.1007/BF01240219. |
[27] |
G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Annali della Scuola Normale Superiore di Pisa, Scienze Fisiche e Matematiche (4), 28 (1999), 141-152. |
[28] |
A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488.
doi: 10.1090/S0002-9947-1973-0335758-1. |
[29] |
C. Liverani, Invariant measures and their properties. A functional analytic point of view, in "Dynamical Systems," Part II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Normale Superiore, Pisa, (2003), 185-237. |
[30] |
C. Liverani, Fredholm determinants, Anosov maps and Ruelle resonances, Discrete and Continuous Dynamical Systems, 13 (2005), 1203-1215.
doi: 10.3934/dcds.2005.13.1203. |
[31] |
I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., 260 (2005), 131-146.
doi: 10.1007/s00220-005-1407-5. |
[32] |
I. Melbourne and M. Nicol, Large deviations for nonuniformly hyperbolic systems, Trans. Amer. Math. Soc., 360 (2008), 6661-6676.
doi: 10.1090/S0002-9947-08-04520-0. |
[33] |
S. V. Nagaev, Some limit theorems for stationary Markov chains, (Russian), Teor. Veroyatnost. i Primenen, 2 (1957), 389-416. |
[34] |
W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Annals of Math. (2), 118 (1983), 573-591.
doi: 10.2307/2006982. |
[35] |
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268 pp. |
[36] |
L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems, Ergod. Th. and Dynam. Systems, 28 (2008), 587-612.
doi: 10.1017/S0143385707000478. |
[37] |
D. Ruelle, Locating resonances for Axiom A dynamical systems, J. Stat. Phys., 44 (1986), 281-292.
doi: 10.1007/BF01011300. |
[38] |
D. Ruelle, Resonances for Axiom $A$ flows, J. Differential Geom., 25 (1987), 99-116. |
[39] |
H. H. Rugh, The correlation spectrum for hyperbolic analytic maps, Nonlinearity, 5 (1992), 1237-1263.
doi: 10.1088/0951-7715/5/6/003. |
[40] |
H. H. Rugh, Fredholm determinants for real-analytic hyperbolic diffeomorphisms of surfaces, "XIth International Congress of Mathematical Physics" (Paris, 1994), Internat. Press, Cambridge, MA, (1995), 297-303. |
[41] |
H. H. Rugh, Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems, Ergod. Th. and Dynam. Sys., 16 (1996), 805-819.
doi: 10.1017/S0143385700009111. |
[42] |
B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248.
doi: 10.1007/BF02773219. |
[43] |
M. Tsujii, Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane, Comm. Math. Phys., 208 (2000), 605-622.
doi: 10.1007/s002200050003. |
[44] |
M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps, Invent. Math., 143 (2001), 349-373.
doi: 10.1007/PL00005797. |
[45] |
L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Math. (2), 147 (1998), 585-650.
doi: 10.2307/120960. |
show all references
References:
[1] |
V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. |
[2] |
V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $\C^\infty$ foliations, in "Algebraic and Topological Dynamics" (eds. Sergiy Kolyada, Yuri Manin and Tom Ward), Contemporary Mathematics, 385, Amer. Math. Society, Providence, RI, (2005), 123-135. |
[3] |
V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Annales de l'Institute Henri Poincaré Analyse Non Linéare, 26 (2009), 1453-1481.
doi: 10.1016/j.anihpc.2009.01.001. |
[4] |
V. Baladi and S. Gouëzel, Banach spaces for piecewise cone-hyperbolic maps, J. Modern Dynam., 4 (2010), 91-137. |
[5] |
V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier (Grenoble), 57 (2007), 127-154.
doi: 10.5802/aif.2253. |
[6] |
V. Baladi and L.-S.Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys., 156 (1993), 355-385.
doi: 10.1007/BF02098487. |
[7] |
L. Bunimovich, Y. G. Sinaĭ and N. Chernov, Markov partitions for two-dimensional hyperbolic billiards, Russian Math. Surveys, 45 (1990), 105-152.
doi: 10.1070/RM1990v045n03ABEH002355. |
[8] |
L. Bunimovich, Y. G. Sinaĭ and N. Chernov, Statistical properties of two-dimensional hyperbolic billiards, Russian Math. Surveys 46 (1991), 47-106.
doi: 10.1070/RM1991v046n04ABEH002827. |
[9] |
M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.
doi: 10.1088/0951-7715/15/6/309. |
[10] |
J. Buzzi, Absolutely continuous invariant probability measures for arbitrary expanding piecewise $\mathbbR$-analytic mappings of the plane, Ergod. Th. and Dynam. Sys., 20 (2000), 697-708.
doi: 10.1017/S0143385700000377. |
[11] |
J. Buzzi and G. Keller, Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps, Ergod. Th. and Dynam. Sys., 21 (2001), 689-716.
doi: 10.1017/S0143385701001341. |
[12] |
J.-R. Chazottes and S. Gouëzel, On almost-sure versions of classical theorems for dynamical systems, Probability Theory and Related Fields, 138 (2007), 195-234.
doi: 10.1007/s00440-006-0021-6. |
[13] |
N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys., 94 (1999), 513-556.
doi: 10.1023/A:1004581304939. |
[14] |
N. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys., 122 (2006), 1061-1094.
doi: 10.1007/s10955-006-9036-8. |
[15] |
N. Chernov and R. Markarian, "Chaotic Billiards," Mathematical Surveys and Monographs, 127, AMS, Providence, RI, 2006. |
[16] |
N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls, in "Hard Ball Systems and the Lorentz Gas" (ed. D. Szasz), Enclyclopaedia of Mathematical Sciences, 101, Springer, Berlin, (2000), 89-120. |
[17] |
A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications," Second edition, Applications of Mathematics, 38, Springer-Verlag, New York, 1998. |
[18] |
M. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc., 360 (2008), 4777-4814.
doi: 10.1090/S0002-9947-08-04464-4. |
[19] |
M. Demers, Functional norms for Young towers, Ergod. Th. Dynam. Sys., 30 (2010), 1371-1398.
doi: 10.1017/S0143385709000534. |
[20] |
W. Doeblin and R. Fortet, Sur des chaînes à liaisons complètes, Bull. Soc. Math. France, 65 (1937), 132-148. |
[21] |
S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergod. Th. and Dynam. Sys., 26 (2006), 189-217. |
[22] |
S. Gouëzel, Almost sure invariance principle for dynamical systems by spectral methods, Ann. Prob., 38 (2010), 1639-1671.
doi: 10.1214/10-AOP525. |
[23] |
H. Hennion and L. Hevré, "Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness," Lectures Notes in Mathematics, 1766, Springer-Verlag, Berlin, 2001. |
[24] |
C. T. Ionescu-Tulcea and G. Marinescu, Théorie ergodique pour des classes d'opérations non complètement continues, Ann. of Math. (2), 52 (1950), 140-147. |
[25] |
T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenchaften, 132, Springer-Verlag, Berlin-New York, 1976. |
[26] |
G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193.
doi: 10.1007/BF01240219. |
[27] |
G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Annali della Scuola Normale Superiore di Pisa, Scienze Fisiche e Matematiche (4), 28 (1999), 141-152. |
[28] |
A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488.
doi: 10.1090/S0002-9947-1973-0335758-1. |
[29] |
C. Liverani, Invariant measures and their properties. A functional analytic point of view, in "Dynamical Systems," Part II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Normale Superiore, Pisa, (2003), 185-237. |
[30] |
C. Liverani, Fredholm determinants, Anosov maps and Ruelle resonances, Discrete and Continuous Dynamical Systems, 13 (2005), 1203-1215.
doi: 10.3934/dcds.2005.13.1203. |
[31] |
I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., 260 (2005), 131-146.
doi: 10.1007/s00220-005-1407-5. |
[32] |
I. Melbourne and M. Nicol, Large deviations for nonuniformly hyperbolic systems, Trans. Amer. Math. Soc., 360 (2008), 6661-6676.
doi: 10.1090/S0002-9947-08-04520-0. |
[33] |
S. V. Nagaev, Some limit theorems for stationary Markov chains, (Russian), Teor. Veroyatnost. i Primenen, 2 (1957), 389-416. |
[34] |
W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Annals of Math. (2), 118 (1983), 573-591.
doi: 10.2307/2006982. |
[35] |
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268 pp. |
[36] |
L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems, Ergod. Th. and Dynam. Systems, 28 (2008), 587-612.
doi: 10.1017/S0143385707000478. |
[37] |
D. Ruelle, Locating resonances for Axiom A dynamical systems, J. Stat. Phys., 44 (1986), 281-292.
doi: 10.1007/BF01011300. |
[38] |
D. Ruelle, Resonances for Axiom $A$ flows, J. Differential Geom., 25 (1987), 99-116. |
[39] |
H. H. Rugh, The correlation spectrum for hyperbolic analytic maps, Nonlinearity, 5 (1992), 1237-1263.
doi: 10.1088/0951-7715/5/6/003. |
[40] |
H. H. Rugh, Fredholm determinants for real-analytic hyperbolic diffeomorphisms of surfaces, "XIth International Congress of Mathematical Physics" (Paris, 1994), Internat. Press, Cambridge, MA, (1995), 297-303. |
[41] |
H. H. Rugh, Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems, Ergod. Th. and Dynam. Sys., 16 (1996), 805-819.
doi: 10.1017/S0143385700009111. |
[42] |
B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248.
doi: 10.1007/BF02773219. |
[43] |
M. Tsujii, Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane, Comm. Math. Phys., 208 (2000), 605-622.
doi: 10.1007/s002200050003. |
[44] |
M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps, Invent. Math., 143 (2001), 349-373.
doi: 10.1007/PL00005797. |
[45] |
L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Math. (2), 147 (1998), 585-650.
doi: 10.2307/120960. |
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