# American Institute of Mathematical Sciences

October  2011, 5(4): 665-709. doi: 10.3934/jmd.2011.5.665

## 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

Received  July 2011 Revised  February 2012 Published  March 2012

We study the billiard map associated with both the finite- and infinite-horizon Lorentz gases having smooth scatterers with strictly positive curvature. We introduce generalized function spaces (Banach spaces of distributions) on which the transfer operator is quasicompact. The mixing properties of the billiard map then imply the existence of a spectral gap and related statistical properties such as exponential decay of correlations and the Central Limit Theorem. Finer statistical properties of the map such as the identification of Ruelle resonances, large deviation estimates and an almost-sure invariance principle follow immediately once the spectral picture is established.
Citation: Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] V. Baladi and S. Gouëzel, Banach spaces for piecewise cone-hyperbolic maps, J. Modern Dynam., 4 (2010), 91-137.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys., 94 (1999), 513-556. doi: 10.1023/A:1004581304939.  Google Scholar [14] N. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys., 122 (2006), 1061-1094. doi: 10.1007/s10955-006-9036-8.  Google Scholar [15] N. Chernov and R. Markarian, "Chaotic Billiards," Mathematical Surveys and Monographs, 127, AMS, Providence, RI, 2006.  Google Scholar [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.  Google Scholar [17] A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications," Second edition, Applications of Mathematics, 38, Springer-Verlag, New York, 1998.  Google Scholar [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.  Google Scholar [19] M. Demers, Functional norms for Young towers, Ergod. Th. Dynam. Sys., 30 (2010), 1371-1398. doi: 10.1017/S0143385709000534.  Google Scholar [20] W. Doeblin and R. Fortet, Sur des chaînes à liaisons complètes, Bull. Soc. Math. France, 65 (1937), 132-148.  Google Scholar [21] S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergod. Th. and Dynam. Sys., 26 (2006), 189-217.  Google Scholar [22] S. Gouëzel, Almost sure invariance principle for dynamical systems by spectral methods, Ann. Prob., 38 (2010), 1639-1671. doi: 10.1214/10-AOP525.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenchaften, 132, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] S. V. Nagaev, Some limit theorems for stationary Markov chains, (Russian), Teor. Veroyatnost. i Primenen, 2 (1957), 389-416.  Google Scholar [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.  Google Scholar [35] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268 pp.  Google Scholar [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.  Google Scholar [37] D. Ruelle, Locating resonances for Axiom A dynamical systems, J. Stat. Phys., 44 (1986), 281-292. doi: 10.1007/BF01011300.  Google Scholar [38] D. Ruelle, Resonances for Axiom $A$ flows, J. Differential Geom., 25 (1987), 99-116.  Google Scholar [39] H. H. Rugh, The correlation spectrum for hyperbolic analytic maps, Nonlinearity, 5 (1992), 1237-1263. doi: 10.1088/0951-7715/5/6/003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [42] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248. doi: 10.1007/BF02773219.  Google Scholar [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.  Google Scholar [44] M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps, Invent. Math., 143 (2001), 349-373. doi: 10.1007/PL00005797.  Google Scholar [45] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Math. (2), 147 (1998), 585-650. doi: 10.2307/120960.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] V. Baladi and S. Gouëzel, Banach spaces for piecewise cone-hyperbolic maps, J. Modern Dynam., 4 (2010), 91-137.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys., 94 (1999), 513-556. doi: 10.1023/A:1004581304939.  Google Scholar [14] N. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys., 122 (2006), 1061-1094. doi: 10.1007/s10955-006-9036-8.  Google Scholar [15] N. Chernov and R. Markarian, "Chaotic Billiards," Mathematical Surveys and Monographs, 127, AMS, Providence, RI, 2006.  Google Scholar [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.  Google Scholar [17] A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications," Second edition, Applications of Mathematics, 38, Springer-Verlag, New York, 1998.  Google Scholar [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.  Google Scholar [19] M. Demers, Functional norms for Young towers, Ergod. Th. Dynam. Sys., 30 (2010), 1371-1398. doi: 10.1017/S0143385709000534.  Google Scholar [20] W. Doeblin and R. Fortet, Sur des chaînes à liaisons complètes, Bull. Soc. Math. France, 65 (1937), 132-148.  Google Scholar [21] S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergod. Th. and Dynam. Sys., 26 (2006), 189-217.  Google Scholar [22] S. Gouëzel, Almost sure invariance principle for dynamical systems by spectral methods, Ann. Prob., 38 (2010), 1639-1671. doi: 10.1214/10-AOP525.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenchaften, 132, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] S. V. Nagaev, Some limit theorems for stationary Markov chains, (Russian), Teor. Veroyatnost. i Primenen, 2 (1957), 389-416.  Google Scholar [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.  Google Scholar [35] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268 pp.  Google Scholar [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.  Google Scholar [37] D. Ruelle, Locating resonances for Axiom A dynamical systems, J. Stat. Phys., 44 (1986), 281-292. doi: 10.1007/BF01011300.  Google Scholar [38] D. Ruelle, Resonances for Axiom $A$ flows, J. Differential Geom., 25 (1987), 99-116.  Google Scholar [39] H. H. Rugh, The correlation spectrum for hyperbolic analytic maps, Nonlinearity, 5 (1992), 1237-1263. doi: 10.1088/0951-7715/5/6/003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [42] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248. doi: 10.1007/BF02773219.  Google Scholar [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.  Google Scholar [44] M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps, Invent. Math., 143 (2001), 349-373. doi: 10.1007/PL00005797.  Google Scholar [45] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Math. (2), 147 (1998), 585-650. doi: 10.2307/120960.  Google Scholar
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