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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 (2000).
|
[2] |
V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $\C^\infty$ foliations,, in, 385 (2005), 123.
|
[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.
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.
|
[5] |
V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms,, Ann. Inst. Fourier (Grenoble), 57 (2007), 127.
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.
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.
doi: 10.1070/RM1990v045n03ABEH002355. |
[8] |
L. Bunimovich, Y. G. Sinaĭ and N. Chernov, Statistical properties of two-dimensional hyperbolic billiards,, Russian Math. Surveys {\bf 46} (1991), 46 (1991), 47.
doi: 10.1070/RM1991v046n04ABEH002827. |
[9] |
M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps,, Nonlinearity, 15 (2002), 1905.
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.
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.
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.
doi: 10.1007/s00440-006-0021-6. |
[13] |
N. Chernov, Decay of correlations and dispersing billiards,, J. Stat. Phys., 94 (1999), 513.
doi: 10.1023/A:1004581304939. |
[14] |
N. Chernov, Advanced statistical properties of dispersing billiards,, J. Stat. Phys., 122 (2006), 1061.
doi: 10.1007/s10955-006-9036-8. |
[15] |
N. Chernov and R. Markarian, "Chaotic Billiards,", Mathematical Surveys and Monographs, 127 (2006).
|
[16] |
N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls,, in, 101 (2000), 89.
|
[17] |
A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Second edition, 38 (1998).
|
[18] |
M. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps,, Trans. Amer. Math. Soc., 360 (2008), 4777.
doi: 10.1090/S0002-9947-08-04464-4. |
[19] |
M. Demers, Functional norms for Young towers,, Ergod. Th. Dynam. Sys., 30 (2010), 1371.
doi: 10.1017/S0143385709000534. |
[20] |
W. Doeblin and R. Fortet, Sur des chaînes à liaisons complètes,, Bull. Soc. Math. France, 65 (1937), 132.
|
[21] |
S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems,, Ergod. Th. and Dynam. Sys., 26 (2006), 189.
|
[22] |
S. Gouëzel, Almost sure invariance principle for dynamical systems by spectral methods,, Ann. Prob., 38 (2010), 1639.
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 (1766).
|
[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.
|
[25] |
T. Kato, "Perturbation Theory for Linear Operators,", Second edition, 132 (1976).
|
[26] |
G. Keller, On the rate of convergence to equilibrium in one-dimensional systems,, Comm. Math. Phys., 96 (1984), 181.
doi: 10.1007/BF01240219. |
[27] |
G. Keller and C. Liverani, Stability of the spectrum for transfer operators,, Annali della Scuola Normale Superiore di Pisa, 28 (1999), 141.
|
[28] |
A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481.
doi: 10.1090/S0002-9947-1973-0335758-1. |
[29] |
C. Liverani, Invariant measures and their properties. A functional analytic point of view,, in, (2003), 185.
|
[30] |
C. Liverani, Fredholm determinants, Anosov maps and Ruelle resonances,, Discrete and Continuous Dynamical Systems, 13 (2005), 1203.
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.
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.
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.
|
[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.
doi: 10.2307/2006982. |
[35] |
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics,, Astérisque, 187-188 (1990), 187.
|
[36] |
L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems,, Ergod. Th. and Dynam. Systems, 28 (2008), 587.
doi: 10.1017/S0143385707000478. |
[37] |
D. Ruelle, Locating resonances for Axiom A dynamical systems,, J. Stat. Phys., 44 (1986), 281.
doi: 10.1007/BF01011300. |
[38] |
D. Ruelle, Resonances for Axiom $A$ flows,, J. Differential Geom., 25 (1987), 99.
|
[39] |
H. H. Rugh, The correlation spectrum for hyperbolic analytic maps,, Nonlinearity, 5 (1992), 1237.
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, (1995), 297.
|
[41] |
H. H. Rugh, Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems,, Ergod. Th. and Dynam. Sys., 16 (1996), 805.
doi: 10.1017/S0143385700009111. |
[42] |
B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps,, Israel J. Math., 116 (2000), 223.
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.
doi: 10.1007/s002200050003. |
[44] |
M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps,, Invent. Math., 143 (2001), 349.
doi: 10.1007/PL00005797. |
[45] |
L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity,, Annals of Math. (2), 147 (1998), 585.
doi: 10.2307/120960. |
show all references
References:
[1] |
V. Baladi, "Positive Transfer Operators and Decay of Correlations,", Advanced Series in Nonlinear Dynamics, 16 (2000).
|
[2] |
V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $\C^\infty$ foliations,, in, 385 (2005), 123.
|
[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.
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.
|
[5] |
V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms,, Ann. Inst. Fourier (Grenoble), 57 (2007), 127.
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.
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.
doi: 10.1070/RM1990v045n03ABEH002355. |
[8] |
L. Bunimovich, Y. G. Sinaĭ and N. Chernov, Statistical properties of two-dimensional hyperbolic billiards,, Russian Math. Surveys {\bf 46} (1991), 46 (1991), 47.
doi: 10.1070/RM1991v046n04ABEH002827. |
[9] |
M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps,, Nonlinearity, 15 (2002), 1905.
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.
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.
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.
doi: 10.1007/s00440-006-0021-6. |
[13] |
N. Chernov, Decay of correlations and dispersing billiards,, J. Stat. Phys., 94 (1999), 513.
doi: 10.1023/A:1004581304939. |
[14] |
N. Chernov, Advanced statistical properties of dispersing billiards,, J. Stat. Phys., 122 (2006), 1061.
doi: 10.1007/s10955-006-9036-8. |
[15] |
N. Chernov and R. Markarian, "Chaotic Billiards,", Mathematical Surveys and Monographs, 127 (2006).
|
[16] |
N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls,, in, 101 (2000), 89.
|
[17] |
A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Second edition, 38 (1998).
|
[18] |
M. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps,, Trans. Amer. Math. Soc., 360 (2008), 4777.
doi: 10.1090/S0002-9947-08-04464-4. |
[19] |
M. Demers, Functional norms for Young towers,, Ergod. Th. Dynam. Sys., 30 (2010), 1371.
doi: 10.1017/S0143385709000534. |
[20] |
W. Doeblin and R. Fortet, Sur des chaînes à liaisons complètes,, Bull. Soc. Math. France, 65 (1937), 132.
|
[21] |
S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems,, Ergod. Th. and Dynam. Sys., 26 (2006), 189.
|
[22] |
S. Gouëzel, Almost sure invariance principle for dynamical systems by spectral methods,, Ann. Prob., 38 (2010), 1639.
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 (1766).
|
[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.
|
[25] |
T. Kato, "Perturbation Theory for Linear Operators,", Second edition, 132 (1976).
|
[26] |
G. Keller, On the rate of convergence to equilibrium in one-dimensional systems,, Comm. Math. Phys., 96 (1984), 181.
doi: 10.1007/BF01240219. |
[27] |
G. Keller and C. Liverani, Stability of the spectrum for transfer operators,, Annali della Scuola Normale Superiore di Pisa, 28 (1999), 141.
|
[28] |
A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481.
doi: 10.1090/S0002-9947-1973-0335758-1. |
[29] |
C. Liverani, Invariant measures and their properties. A functional analytic point of view,, in, (2003), 185.
|
[30] |
C. Liverani, Fredholm determinants, Anosov maps and Ruelle resonances,, Discrete and Continuous Dynamical Systems, 13 (2005), 1203.
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.
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.
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.
|
[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.
doi: 10.2307/2006982. |
[35] |
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics,, Astérisque, 187-188 (1990), 187.
|
[36] |
L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems,, Ergod. Th. and Dynam. Systems, 28 (2008), 587.
doi: 10.1017/S0143385707000478. |
[37] |
D. Ruelle, Locating resonances for Axiom A dynamical systems,, J. Stat. Phys., 44 (1986), 281.
doi: 10.1007/BF01011300. |
[38] |
D. Ruelle, Resonances for Axiom $A$ flows,, J. Differential Geom., 25 (1987), 99.
|
[39] |
H. H. Rugh, The correlation spectrum for hyperbolic analytic maps,, Nonlinearity, 5 (1992), 1237.
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, (1995), 297.
|
[41] |
H. H. Rugh, Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems,, Ergod. Th. and Dynam. Sys., 16 (1996), 805.
doi: 10.1017/S0143385700009111. |
[42] |
B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps,, Israel J. Math., 116 (2000), 223.
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.
doi: 10.1007/s002200050003. |
[44] |
M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps,, Invent. Math., 143 (2001), 349.
doi: 10.1007/PL00005797. |
[45] |
L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity,, Annals of Math. (2), 147 (1998), 585.
doi: 10.2307/120960. |
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