Spectral analysis of the transfer operator for the Lorentz gas

Mark F. Demers; Hong-Kun Zhang

doi:10.3934/jmd.2011.5.665

Previous Article
Ziggurats and rotation numbers
JMD Home
This Issue
Next Article
Planetary Birkhoff normal forms

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

Spectral analysis of the transfer operator for the Lorentz gas

Mark F. Demers ^1, and Hong-Kun Zhang ^2,

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

Abstract
Related Papers

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.

Keywords: spectral gap, limit theorems., transfer operator, Dispersing billiards.

Mathematics Subject Classification: Primary: 37D50; Secondary: 37A25, 37C3.

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 (2000). Google Scholar
[2]	V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $\C^\infty$ foliations,, in, 385 (2005), 123. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. 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. doi: 10.1007/s00440-006-0021-6. Google Scholar
[13]	N. Chernov, Decay of correlations and dispersing billiards,, J. Stat. Phys., 94 (1999), 513. doi: 10.1023/A:1004581304939. Google Scholar
[14]	N. Chernov, Advanced statistical properties of dispersing billiards,, J. Stat. Phys., 122 (2006), 1061. doi: 10.1007/s10955-006-9036-8. Google Scholar
[15]	N. Chernov and R. Markarian, "Chaotic Billiards,", Mathematical Surveys and Monographs, 127 (2006). Google Scholar
[16]	N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls,, in, 101 (2000), 89. Google Scholar
[17]	A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Second edition, 38 (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. 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. 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. Google Scholar
[21]	S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems,, Ergod. Th. and Dynam. Sys., 26 (2006), 189. Google Scholar
[22]	S. Gouëzel, Almost sure invariance principle for dynamical systems by spectral methods,, Ann. Prob., 38 (2010), 1639. 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 (1766). 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. Google Scholar
[25]	T. Kato, "Perturbation Theory for Linear Operators,", Second edition, 132 (1976). Google Scholar
[26]	G. Keller, On the rate of convergence to equilibrium in one-dimensional systems,, Comm. Math. Phys., 96 (1984), 181. 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, 28 (1999), 141. 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. 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, (2003), 185. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	S. V. Nagaev, Some limit theorems for stationary Markov chains, (Russian),, Teor. Veroyatnost. i Primenen, 2 (1957), 389. 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. 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), 187. 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. doi: 10.1017/S0143385707000478. Google Scholar
[37]	D. Ruelle, Locating resonances for Axiom A dynamical systems,, J. Stat. Phys., 44 (1986), 281. doi: 10.1007/BF01011300. Google Scholar
[38]	D. Ruelle, Resonances for Axiom $A$ flows,, J. Differential Geom., 25 (1987), 99. Google Scholar
[39]	H. H. Rugh, The correlation spectrum for hyperbolic analytic maps,, Nonlinearity, 5 (1992), 1237. 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, (1995), 297. 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. doi: 10.1017/S0143385700009111. Google Scholar
[42]	B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps,, Israel J. Math., 116 (2000), 223. 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. doi: 10.1007/s002200050003. Google Scholar
[44]	M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps,, Invent. Math., 143 (2001), 349. 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. doi: 10.2307/120960. Google Scholar

show all references

References:

[1]	V. Baladi, "Positive Transfer Operators and Decay of Correlations,", Advanced Series in Nonlinear Dynamics, 16 (2000). Google Scholar
[2]	V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $\C^\infty$ foliations,, in, 385 (2005), 123. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. 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. doi: 10.1007/s00440-006-0021-6. Google Scholar
[13]	N. Chernov, Decay of correlations and dispersing billiards,, J. Stat. Phys., 94 (1999), 513. doi: 10.1023/A:1004581304939. Google Scholar
[14]	N. Chernov, Advanced statistical properties of dispersing billiards,, J. Stat. Phys., 122 (2006), 1061. doi: 10.1007/s10955-006-9036-8. Google Scholar
[15]	N. Chernov and R. Markarian, "Chaotic Billiards,", Mathematical Surveys and Monographs, 127 (2006). Google Scholar
[16]	N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls,, in, 101 (2000), 89. Google Scholar
[17]	A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Second edition, 38 (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. 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. 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. Google Scholar
[21]	S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems,, Ergod. Th. and Dynam. Sys., 26 (2006), 189. Google Scholar
[22]	S. Gouëzel, Almost sure invariance principle for dynamical systems by spectral methods,, Ann. Prob., 38 (2010), 1639. 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 (1766). 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. Google Scholar
[25]	T. Kato, "Perturbation Theory for Linear Operators,", Second edition, 132 (1976). Google Scholar
[26]	G. Keller, On the rate of convergence to equilibrium in one-dimensional systems,, Comm. Math. Phys., 96 (1984), 181. 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, 28 (1999), 141. 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. 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, (2003), 185. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	S. V. Nagaev, Some limit theorems for stationary Markov chains, (Russian),, Teor. Veroyatnost. i Primenen, 2 (1957), 389. 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. 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), 187. 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. doi: 10.1017/S0143385707000478. Google Scholar
[37]	D. Ruelle, Locating resonances for Axiom A dynamical systems,, J. Stat. Phys., 44 (1986), 281. doi: 10.1007/BF01011300. Google Scholar
[38]	D. Ruelle, Resonances for Axiom $A$ flows,, J. Differential Geom., 25 (1987), 99. Google Scholar
[39]	H. H. Rugh, The correlation spectrum for hyperbolic analytic maps,, Nonlinearity, 5 (1992), 1237. 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, (1995), 297. 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. doi: 10.1017/S0143385700009111. Google Scholar
[42]	B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps,, Israel J. Math., 116 (2000), 223. 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. doi: 10.1007/s002200050003. Google Scholar
[44]	M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps,, Invent. Math., 143 (2001), 349. 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. doi: 10.2307/120960. Google Scholar

[1]	Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
[2]	Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453
[3]	Jory Griffin, Jens Marklof. Limit theorems for skew translations. Journal of Modern Dynamics, 2014, 8 (2) : 177-189. doi: 10.3934/jmd.2014.8.177
[4]	Bruno Colbois, Alexandre Girouard. The spectral gap of graphs and Steklov eigenvalues on surfaces. Electronic Research Announcements, 2014, 21: 19-27. doi: 10.3934/era.2014.21.19
[5]	O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110
[6]	Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012
[7]	Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205
[8]	Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
[9]	Soña Pavlíková, Daniel Ševčovič. On construction of upper and lower bounds for the HOMO-LUMO spectral gap. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 53-69. doi: 10.3934/naco.2019005
[10]	Krzysztof Frączek, Ronggang Shi, Corinna Ulcigrai. Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions. Journal of Modern Dynamics, 2018, 12: 55-122. doi: 10.3934/jmd.2018004
[11]	Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235
[12]	Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure & Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241
[13]	Gershon Wolansky. Limit theorems for optimal mass transportation and applications to networks. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 365-374. doi: 10.3934/dcds.2011.30.365
[14]	Martin Fraas, David Krejčiřík, Yehuda Pinchover. On some strong ratio limit theorems for heat kernels. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 495-509. doi: 10.3934/dcds.2010.28.495
[15]	Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[16]	Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48
[17]	Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835
[18]	Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025
[19]	Arnaud Debussche, Sylvain De Moor, Julien Vovelle. Diffusion limit for the radiative transfer equation perturbed by a Wiener process. Kinetic & Related Models, 2015, 8 (3) : 467-492. doi: 10.3934/krm.2015.8.467
[20]	Laurent Amour, Jérémy Faupin. Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials. Inverse Problems & Imaging, 2013, 7 (4) : 1115-1122. doi: 10.3934/ipi.2013.7.1115

2018 Impact Factor: 0.295

American Institute of Mathematical Sciences

Spectral analysis of the transfer operator for the Lorentz gas

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Spectral analysis of the transfer operator for the Lorentz gas

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors