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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

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
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show all references

References:
[1]

Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.  Google Scholar

[2]

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]

Annales de l'Institute Henri Poincaré Analyse Non Linéare, 26 (2009), 1453-1481. doi: 10.1016/j.anihpc.2009.01.001.  Google Scholar

[4]

J. Modern Dynam., 4 (2010), 91-137.  Google Scholar

[5]

Ann. Inst. Fourier (Grenoble), 57 (2007), 127-154. doi: 10.5802/aif.2253.  Google Scholar

[6]

Comm. Math. Phys., 156 (1993), 355-385. doi: 10.1007/BF02098487.  Google Scholar

[7]

Russian Math. Surveys, 45 (1990), 105-152. doi: 10.1070/RM1990v045n03ABEH002355.  Google Scholar

[8]

Russian Math. Surveys 46 (1991), 47-106. doi: 10.1070/RM1991v046n04ABEH002827.  Google Scholar

[9]

Nonlinearity, 15 (2002), 1905-1973. doi: 10.1088/0951-7715/15/6/309.  Google Scholar

[10]

Ergod. Th. and Dynam. Sys., 20 (2000), 697-708. doi: 10.1017/S0143385700000377.  Google Scholar

[11]

Ergod. Th. and Dynam. Sys., 21 (2001), 689-716. doi: 10.1017/S0143385701001341.  Google Scholar

[12]

Probability Theory and Related Fields, 138 (2007), 195-234. doi: 10.1007/s00440-006-0021-6.  Google Scholar

[13]

J. Stat. Phys., 94 (1999), 513-556. doi: 10.1023/A:1004581304939.  Google Scholar

[14]

J. Stat. Phys., 122 (2006), 1061-1094. doi: 10.1007/s10955-006-9036-8.  Google Scholar

[15]

Mathematical Surveys and Monographs, 127, AMS, Providence, RI, 2006.  Google Scholar

[16]

in "Hard Ball Systems and the Lorentz Gas" (ed. D. Szasz), Enclyclopaedia of Mathematical Sciences, 101, Springer, Berlin, (2000), 89-120.  Google Scholar

[17]

Second edition, Applications of Mathematics, 38, Springer-Verlag, New York, 1998.  Google Scholar

[18]

Trans. Amer. Math. Soc., 360 (2008), 4777-4814. doi: 10.1090/S0002-9947-08-04464-4.  Google Scholar

[19]

Ergod. Th. Dynam. Sys., 30 (2010), 1371-1398. doi: 10.1017/S0143385709000534.  Google Scholar

[20]

Bull. Soc. Math. France, 65 (1937), 132-148.  Google Scholar

[21]

Ergod. Th. and Dynam. Sys., 26 (2006), 189-217.  Google Scholar

[22]

Ann. Prob., 38 (2010), 1639-1671. doi: 10.1214/10-AOP525.  Google Scholar

[23]

Lectures Notes in Mathematics, 1766, Springer-Verlag, Berlin, 2001.  Google Scholar

[24]

Ann. of Math. (2), 52 (1950), 140-147.  Google Scholar

[25]

Second edition, Grundlehren der Mathematischen Wissenchaften, 132, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[26]

Comm. Math. Phys., 96 (1984), 181-193. doi: 10.1007/BF01240219.  Google Scholar

[27]

Annali della Scuola Normale Superiore di Pisa, Scienze Fisiche e Matematiche (4), 28 (1999), 141-152.  Google Scholar

[28]

Trans. Amer. Math. Soc., 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1.  Google Scholar

[29]

in "Dynamical Systems," Part II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Normale Superiore, Pisa, (2003), 185-237.  Google Scholar

[30]

Discrete and Continuous Dynamical Systems, 13 (2005), 1203-1215. doi: 10.3934/dcds.2005.13.1203.  Google Scholar

[31]

Commun. Math. Phys., 260 (2005), 131-146. doi: 10.1007/s00220-005-1407-5.  Google Scholar

[32]

Trans. Amer. Math. Soc., 360 (2008), 6661-6676. doi: 10.1090/S0002-9947-08-04520-0.  Google Scholar

[33]

Teor. Veroyatnost. i Primenen, 2 (1957), 389-416.  Google Scholar

[34]

Annals of Math. (2), 118 (1983), 573-591. doi: 10.2307/2006982.  Google Scholar

[35]

Astérisque, 187-188 (1990), 268 pp.  Google Scholar

[36]

Ergod. Th. and Dynam. Systems, 28 (2008), 587-612. doi: 10.1017/S0143385707000478.  Google Scholar

[37]

J. Stat. Phys., 44 (1986), 281-292. doi: 10.1007/BF01011300.  Google Scholar

[38]

J. Differential Geom., 25 (1987), 99-116.  Google Scholar

[39]

Nonlinearity, 5 (1992), 1237-1263. doi: 10.1088/0951-7715/5/6/003.  Google Scholar

[40]

"XIth International Congress of Mathematical Physics" (Paris, 1994), Internat. Press, Cambridge, MA, (1995), 297-303.  Google Scholar

[41]

Ergod. Th. and Dynam. Sys., 16 (1996), 805-819. doi: 10.1017/S0143385700009111.  Google Scholar

[42]

Israel J. Math., 116 (2000), 223-248. doi: 10.1007/BF02773219.  Google Scholar

[43]

Comm. Math. Phys., 208 (2000), 605-622. doi: 10.1007/s002200050003.  Google Scholar

[44]

Invent. Math., 143 (2001), 349-373. doi: 10.1007/PL00005797.  Google Scholar

[45]

Annals of Math. (2), 147 (1998), 585-650. doi: 10.2307/120960.  Google Scholar

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