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Erschienen in:
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2019 | OriginalPaper | Buchkapitel

Markov Approximations and Statistical Properties of Billiards

verfasst von : Domokos Szász

Erschienen in: The Abel Prize 2013-2017

Verlag: Springer International Publishing

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Abstract

Markov partitions designed by Sinai (Funct Anal Appl 2:245–253, 1968) and Bowen (Am J Math 92:725–747, 1970) proved to be an efficient tool for describing statistical properties of uniformly hyperbolic systems. For hyperbolic systems with singularities, in particular for hyperbolic billiards the construction of a Markov partition by Bunimovich and Sinai (Commun Math Phys 78:247–280, 1980) was a delicate and hard task. Therefore later more and more flexible and simple variants of Markov partitions appeared: Markov sieves (Bunimovich–Chernov–Sinai, Russ Math Surv 45(3):105–152, 1990), Markov towers (Young, Ann Math (2) 147(3):585–650, 1998), standard pairs (Dolgopyat). This remarkable evolution of Sinai’s original idea is surveyed in this paper.

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Vorheriges Kapitel Further Developments of Sinai’s Ideas: The Boltzmann–Sinai Hypothesis
Nächstes Kapitel List of Publications for Yakov G. Sinai
Fußnoten
1
Actually the Lorentz gas suggested by H. A. Lorentz in [50] is also a billiard in a space with infinite invariant measure, cf. Sect. 3 below.
 
2
See, for instance, D. Dolgopyat: Introduction to averaging. Lecture notes, Institut Henri Poincaré, http://​www2.​math.​umd.​edu/​~dolgop/​IANotes.​pdf (2005).
 
Literatur
1.
R.L. Adler and B. Weiss. Entropy, a complete metric invariant for automorphisms of the torus. Proc. Nat. Acad. Sci. U.S.A., 57:1573–1576, 1967.MathSciNetMATHCrossRef
2.
V. Baladi, M.F. Demers, and C. Liverani. Exponential decay of correlations for finite horizon Sinai billiard flows. Invent. Math., 211(1):39–177, 2018.MathSciNetMATHCrossRef
3.
P. Bálint, N. Chernov, and D. Dolgopyat. Limit theorems for dispersing billiards with cusps. Comm. Math. Phys., 308(2):479–510, 2011.MathSciNetMATHCrossRef
4.
P. Bálint, N. Chernov, D. Szász, and I.P. Tóth. Geometry of multi-dimensional dispersing billiards. Astérisque, (286):119–150, 2003.MathSciNetMATH
5.
6.
P. Bálint, T. Gilbert, P. Nándori, D. Szász, Domokos, and I.P. Tóth. On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas. J. Stat. Phys. 166(3–4), 903–925 (2017).MathSciNetMATHCrossRef
7.
P. Bálint, P. Nándori, D. Szász, and I.P. Tóth. Equidistribution for standard pairs in planar dispersing billiard flows. Ann. Henri Poincaré 19(4), 979–1042 (2018).MathSciNetMATHCrossRef
8.
P. Bálint and I.P. Tóth. Exponential decay of correlations in multi-dimensional dispersing billiards. Ann. Henri Poincaré, 9(7):1309–1369, 2008.MathSciNetMATHCrossRef
9.
K.R. Berg. On the conjugacy problem for K-systems. PhD thesis, University of Minnesota, 1967.
10.
11.
G.D. Birkhoff. Proof of the ergodic theorem. Proc. Natl. Acad. Sci. USA, 17:656–660, 1931.MATHCrossRef
12.
P. M. Bleher. Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon. J. Statist. Phys. 66(1–2), 315–373 (1992).MathSciNetMATHCrossRef
13.
14.
R. Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, volume 470 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2008.
15.
L. Bunimovich. Billiards. In H. Holden, R. Piene, editors, The Abel Prize 2013–2017, pages 1–11. Springer, 2019.
16.
17.
L.A. Bunimovich and Ya.G. Sinai. Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys., 78:479–497, 1981.MathSciNetMATHCrossRef
18.
L.A. Bunimovich and Ya.G. Sinai. Erratum [16]. Comm. Math. Phys., 107(2):357–358, 1986.
19.
L.A. Bunimovich, Ya.G. Sinai, and N.I. Chernov. Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv., 46(4), 1991.MathSciNetMATHCrossRef
20.
L.A. Bunimovich, Ya.G. Sinai, and N.I. Chernov. Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv., 45(3):105–152, 1990.MathSciNetMATHCrossRef
21.
22.
23.
24.
N. Chernov and R. Markarian. Anosov maps with rectangular holes. Nonergodic cases. Bol. Soc. Brasil. Mat. (N.S.), 28(2):315–342, 1997.
25.
N. Chernov and R. Markarian. Chaotic Billiards, volume 127 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2006.
26.
N. Chernov, R. Markarian, and S. Troubetzkoy. Invariant measures for Anosov maps with small holes. Ergodic Theory Dynam. Systems, 20(4):1007–1044, 2000.MathSciNetMATHCrossRef
27.
N. Chernov and L.S. Young. Decay of correlations for Lorentz gases and hard balls. In Hard Ball Systems and the Lorentz Gas, volume 101 of Encyclopaedia Math. Sci., pages 89–120. Springer, Berlin, 2000.
28.
N. Chernov, H.-K. Zhang, and P. Zhang. Electrical current in Sinai billiards under general small forces. J. Stat. Phys., 153(6):1065–1083, 2013.MathSciNetMATHCrossRef
29.
30.
N.I. Chernov and D.I. Dolgopyat. Anomalous current in periodic Lorentz gases with infinite horizon. Russ. Math. Surv., 64(4):651–699, 2009.MathSciNetMATHCrossRef
31.
N.I. Chernov, G.L. Eyink, J.L. Lebowitz, and Ya.G. Sinai. Derivation of ohm’s law in a deterministic mechanical model. Phys. Rev. Lett., 70:2209–2212, Apr 1993.CrossRef
32.
N.I. Chernov, G.L. Eyink, J.L. Lebowitz, and Ya.G. Sinai. Steady-state electrical conduction in the periodic Lorentz gas. Comm. Math. Phys., 154(3):569–601, 1993.MathSciNetMATHCrossRef
33.
V. Climenhaga, D. Dolgopyat, and Y. Pesin. Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps. Comm. Math. Phys., 346(2):553–602, 2016.MathSciNetMATHCrossRef
34.
J.-P. Conze. Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications. Ergodic Theory Dynam. Systems, 19(5):1233–1245, 1999.MathSciNetMATHCrossRef
35.
I.P. Cornfeld, S.V. Fomin, and Ya.G. Sinai. Ergodic Theory, volume 245 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New York, 1982.
36.
M. Demers, P. Wright, and L.-S. Young. Escape rates and physically relevant measures for billiards with small holes. Comm. Math. Phys., 294(2):353–388, 2010.MathSciNetMATHCrossRef
37.
M.F. Demers, P. Wright, and L.-S. Young. Entropy, Lyapunov exponents and escape rates in open systems. Ergodic Theory Dynam. Systems, 32(4):1270–1301, 2012.MathSciNetMATHCrossRef
38.
39.
40.
D. Dolgopyat and P. Nándori. Nonequilibrium density profiles in Lorentz tubes with thermostated boundaries. Comm. Pure Appl. Math., 69(4):649–692, 2016.MathSciNetMATHCrossRef
41.
D. Dolgopyat, D. Szász, and T. Varjú. Recurrence properties of planar Lorentz process. Duke Math. J., 142(2):241–281, 2008.MathSciNetMATHCrossRef
42.
D. Dolgopyat, D. Szász, and T. Varjú. Limit theorems for locally perturbed planar Lorentz processes. Duke Math. J., 148(3):459–499, 2009.MathSciNetMATHCrossRef
43.
P. Ehrenfest and T. Ehrenfest. Begriffliche Grundlagen der statistischen Auffassung in der Mechanik. Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912)., 1912.
44.
P. Gaspard and T. Gilbert. Heat conduction and fourier’s law by consecutive local mixing and thermalization. Phys. Rev. Lett., 101:020601, Jul 2008.
45.
B. Hasselblatt and A. Katok, editors. Handbook of Dynamical Systems. Volume 1A. Amsterdam: North-Holland, 2002.MATH
46.
A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems, volume 54 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1995.
47.
D. König and A. Szücs. Mouvement d’un point abandonné à l’intérieur d’un cube. Rend. Circ. Mat. Palermo, 36:79–90, 1913.MATHCrossRef
48.
N.S. Krylov. Works on the Foundations of Statistical Physics. Princeton University Press, Princeton, NJ, 1979.
49.
50.
H.A. Lorentz. Le mouvement des électrons dans les métaux. Arch. Néerl. 10 336–371 (1905). In Collected Paper. Vol. III, pages 180ff.
51.
E. Nelson. Dynamical Theories of Brownian Motion. Princeton University Press, Princeton, N.J., 1967.MATH
52.
53.
Y. Pesin. Sinai’s work on Markov partitions and SRB measures. In H. Holden, R. Piene, editors, The Abel Prize 2013–2017, pages 257–285. Springer, 2019.
54.
L. Rey-Bellet and L.-S. Young. Large deviations in non-uniformly hyperbolic dynamical systems. Ergodic Theory Dynam. Systems, 28(2):587–612, 2008.MathSciNetMATHCrossRef
55.
56.
N. Simányi. Further developments of Sinai’s ideas: the Boltzmann–Sinai hypothesis. In H. Holden, R. Piene, editors, The Abel Prize 2013–2017, pages 1–12. Springer, 2019.
57.
Ya.G. Sinai. Construction of Markov partitions. Funct. Anal. Appl., 2:245–253, 1968.MATHCrossRef
58.
59.
Ya.G. Sinai. Dynamical systems with elastic reflections. Russ. Math. Surv., 25(2):137–189, 1970.MATHCrossRef
60.
61.
62.
63.
D. Szász and T. Varjú. Local limit theorem for the Lorentz process and its recurrence in the plane. Ergodic Theory Dynam. Systems, 24(1):257–278, 2004.MathSciNetMATHCrossRef
64.
D. Szász and T. Varjú. Limit laws and recurrence for the planar Lorentz process with infinite horizon. J. Stat. Phys., 129(1):59–80, 2007.MathSciNetMATHCrossRef
65.
J. von Neumann. Proof of the quasi-ergodic hypothesis. Proc. Natl. Acad. Sci. USA, 18:70–82, 1932.MATHCrossRef
66.
67.
Metadaten
Titel
Markov Approximations and Statistical Properties of Billiards
verfasst von
Domokos Szász
Copyright-Jahr
2019
Buch
The Abel Prize 2013-2017

Print ISBN: 978-3-319-99027-9

Electronic ISBN: 978-3-319-99028-6

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-319-99028-6_13