Book Review
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2525742
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Book Information:
Authors:
Nikolai Chernov and
Roberto Markarian
Title:
Chaotic billiards
Additional book information:
Mathematical Surveys and Monographs, Vol. 127,
American Mathematical Society,
Providence, RI,
2006,
xii+316 pp.,
ISBN 0-8218-4096-7,
US $85.00$
V. M. Alekseev, Quasirandom dynamical systems, mat. Zametki 6 (1969), 489–498 (Russian). MR 0249754
Péter Bálint, Nikolai Chernov, Domokos Szász, and Imre Péter Tóth, Geometry of multi-dimensional dispersing billiards, Astérisque 286 (2003), xviii, 119–150 (English, with English and French summaries). Geometric methods in dynamics. I. MR 2052299
3. P. Balint and I. Melbourne, Decay of correlations and invariance principle for dispersing billiards with cusps, and related planar billiard flows, Preprint.
4. L. A. Bunimovich, On billiards close to dispersing, Math. USSR Sb. 23 (1974), 45-67.
L. A. Bunimovič, The ergodic properties of certain billiards, Funkcional. Anal. i Priložen. 8 (1974), no. 3, 73–74 (Russian). MR 0357736
L. A. Bunimovich, Many-dimensional nowhere dispersing billiards with chaotic behavior, Phys. D 33 (1988), no. 1-3, 58–64. Progress in chaotic dynamics. MR 984610, DOI 10.1016/S0167-2789(98)90009-4
L. A. Bunimovich, On absolutely focusing mirrors, Ergodic theory and related topics, III (Güstrow, 1990) Lecture Notes in Math., vol. 1514, Springer, Berlin, 1992, pp. 62–82. MR 1179172, DOI 10.1007/BFb0097528
Leonid A. Bunimovich, Mushrooms and other billiards with divided phase space, Chaos 11 (2001), no. 4, 802–808. MR 1875161, DOI 10.1063/1.1418763
Leonid A. Bunimovich and Jan Rehacek, How high-dimensional stadia look like, Comm. Math. Phys. 197 (1998), no. 2, 277–301. MR 1652730, DOI 10.1007/s002200050451
10. L. A. Bunimovich and A. Grigo, Focusing components in chaotic billiards should be absolutely focusing, Comm. Math. Phys., (to appear).
L. A. Bunimovich and Ya. G. Sinaĭ, Statistical properties of Lorentz gas with periodic configuration of scatterers, Comm. Math. Phys. 78 (1980/81), no. 4, 479–497. MR 606459
N. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys. 122 (2006), no. 6, 1061–1094. MR 2219528, DOI 10.1007/s10955-006-9036-8
N. Chernov, A stretched exponential bound on time correlations for billiard flows, J. Stat. Phys. 127 (2007), no. 1, 21–50. MR 2313061, DOI 10.1007/s10955-007-9293-1
Nikolai Chernov and Dmitry Dolgopyat, Hyperbolic billiards and statistical physics, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1679–1704. MR 2275665
N. I. Chernov and C. Haskell, Nonuniformly hyperbolic $K$-systems are Bernoulli, Ergodic Theory Dynam. Systems 16 (1996), no. 1, 19–44. MR 1375125, DOI 10.1017/S0143385700008695
16. G. Del Magno and R. Markarian, On the Bernoulli property of planar hyperbolic billiards, Preprint.
Victor J. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy, Ergodic Theory Dynam. Systems 8 (1988), no. 4, 531–553. MR 980796, DOI 10.1017/S0143385700004685
Victor J. Donnay, Using integrability to produce chaos: billiards with positive entropy, Comm. Math. Phys. 141 (1991), no. 2, 225–257. MR 1133266
E. Gutkin, Billiard dynamics: a survey with the emphasis on open problems, Regul. Chaotic Dyn. 8 (2003), no. 1, 1–13. MR 1963964, DOI 10.1070/RD2003v008n01ABEH000222
Anatole Katok, Jean-Marie Strelcyn, F. Ledrappier, and F. Przytycki, Invariant manifolds, entropy and billiards; smooth maps with singularities, Lecture Notes in Mathematics, vol. 1222, Springer-Verlag, Berlin, 1986. MR 872698, DOI 10.1007/BFb0099031
V. F. Lazutkin, Existence of caustics for the billiard problem in a convex domain, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 186–216 (Russian). MR 0328219
Howard Masur and Serge Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015–1089. MR 1928530, DOI 10.1016/S1874-575X(02)80015-7
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
Nándor Simányi, Proof of the Boltzmann-Sinai ergodic hypothesis for typical hard disk systems, Invent. Math. 154 (2003), no. 1, 123–178. MR 2004458, DOI 10.1007/s00222-003-0304-9
Nándor Simányi, Proof of the ergodic hypothesis for typical hard ball systems, Ann. Henri Poincaré 5 (2004), no. 2, 203–233. MR 2057672, DOI 10.1007/s00023-004-0166-8
Ja. G. Sinaĭ, Classical dynamic systems with countably-multiple Lebesgue spectrum. II, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 15–68 (Russian). MR 0197684
Ja. G. Sinaĭ, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Uspehi Mat. Nauk 25 (1970), no. 2 (152), 141–192 (Russian). MR 0274721
28. H. Spohn, Large scale dynamics of interacting particles, Springer, Berlin, 1991.
Serge Tabachnikov, Billiards, Panor. Synth. 1 (1995), vi+142 (English, with English and French summaries). MR 1328336
D. Szász (ed.), Hard ball systems and the Lorentz gas, Encyclopaedia of Mathematical Sciences, vol. 101, Springer-Verlag, Berlin, 2000. Mathematical Physics, II. MR 1805337
Maciej Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergodic Theory Dynam. Systems 5 (1985), no. 1, 145–161. MR 782793, DOI 10.1017/S0143385700002807
Maciej Wojtkowski, Principles for the design of billiards with nonvanishing Lyapunov exponents, Comm. Math. Phys. 105 (1986), no. 3, 391–414. MR 848647
Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585–650. MR 1637655, DOI 10.2307/120960
Lai-Sang Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153–188. MR 1750438, DOI 10.1007/BF02808180
- 1.
- V. M. Alekseev, Quasirandom dynamical systems, Mat. USSR Sbornik 7 (1969), 1-43. MR 0249754
- 2.
- P. Balint, N. Chernov, D. Szasz and I. P. Toth, Geometry of multidimensional dispersing billiards, Astérisque 286 (2003), 119-150. MR 2052299
- 3.
- P. Balint and I. Melbourne, Decay of correlations and invariance principle for dispersing billiards with cusps, and related planar billiard flows, Preprint.
- 4.
- L. A. Bunimovich, On billiards close to dispersing, Math. USSR Sb. 23 (1974), 45-67.
- 5.
- L. A. Bunimovich, The ergodic properties of certain billiards, Funk. Anal. Prilozh. 8 (1974), 73-74. MR 0357736
- 6.
- L. A. Bunimovich, Many-dimensional nowhere dispersing billiards with chaotic behavior, Physica D 33 (1988), 58-64. MR 0984610
- 7.
- L. A. Bunimovich, On absolutely focusing mirrors, In Ergodic Theory and Related Topics (Güstrow 1990). Edited by U. Krengel et al., Lect. Notes in Math. 1514, Springer, Berlin, 1992, pp. 62-82. MR 1179172
- 8.
- L. A. Bunimovich, Mushrooms and other billiards with divided phase space, Chaos 11 (2001), 802-808. MR 1875161
- 9.
- L. A. Bunimovich and J. Rehácek, How many-dimensional stadia look like, Comm. Math. Phys. 197 (1998), 277-301. MR 1652730
- 10.
- L. A. Bunimovich and A. Grigo, Focusing components in chaotic billiards should be absolutely focusing, Comm. Math. Phys., (to appear).
- 11.
- L. A. Bunimovich and Ya. G. Sinai, Statistical properties of Lorentz gas with periodic configuration of scatterers, Comm. Math. Phys. 78 (1981), 479-497. MR 0606459
- 12.
- N. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys. 122 (2006), 1061-1094. MR 2219528
- 13.
- N. Chernov, A stretched exponential bound on time correlations for billiard flows, J. Stat. Phys. 127 (2007), 21-50. MR 2313061
- 14.
- N. Chernov and D. Dolgopyat, Hyperbolic billiards and statistical physics, Int-l Congress of Math-ns, vol. II. Eur. Math. Soc., Zürich, 2006, pp. 1679-1704. MR 2275665
- 15.
- N. Chernov and C. Haskell, Non-uniformly hyperbolic
-systems are Bernoulli, Ergod. Th. and Dyn. Syst. 16 (1996), 19-44. MR 1375125
- 16.
- G. Del Magno and R. Markarian, On the Bernoulli property of planar hyperbolic billiards, Preprint.
- 17.
- V. J. Donnay, Geodesic flow on the two-sphere I: Positive measure entropy, Ergod. Th. and Dyn. Syst. 8 (1988), 531-553. MR 0980796
- 18.
- V. J. Donnay, Using integrability to produce chaos: Billiards with positive entropy, Comm. Math. Phys. 141 (1991), 225-257. MR 1133266
- 19.
- E. Gutkin, Billiard dynamics: a survey with the emphasis on open problems, Reg. Chaotic Dyn-s 8 (2003), 1-13. MR 1963964
- 20.
- A. Katok and J.-M. Strelcyn, with the collaboration of F. Ledrappier and F. Przytycki, Invariant manifolds, entropy and billiards; smooth maps with singularities, Lect. Notes Math. 1222, Springer, New York, 1986. MR 0872698
- 21.
- V. F. Lazutkin, Existence of caustics for the billiard ball problem in a convex domain, Math. USSR Izv. 37 (1973), 186-216. MR 0328219
- 22.
- H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook of Dynamical Systems, vol. 1A. Edited by B. Hasselblatt and A. Katok, Elsevier, Amsterdam, 2002. MR 1928530
- 23.
- V. I. Oseledec, A multiplicative ergodic theorem, Trans. Moscow Math. Soc. 19 (1968), 197-231. MR 0240280
- 24.
- N. Simanyi, Proof of the Boltzamnn-Sinai ergodic hypothesis for typical hard disk systems, Invent. Math. 154 (2003), 123-178. MR 2004458
- 25.
- N. Simanyi, Proof of the ergodic hypothesis for typical hard ball systems, Ann. Inst. Henri Poincaré 5 (2004), 203-233. MR 2057672
- 26.
- Ya. G. Sinai, Classical dynamical systems with countably-multiple Lebesgue spectrum. II, Izv. Akad. Nauk SSSR Ser. Math. 30 (1966), 15-68. MR 0197684
- 27.
- Ya. G. Sinai, Dynamical systems with elastic reflections, Ergodic properties of dispersing billiards, Russ. Math. Surv. 25 (1970), 137-189. MR 0274721
- 28.
- H. Spohn, Large scale dynamics of interacting particles, Springer, Berlin, 1991.
- 29.
- S. Tabachnikov, Billiards, Panor, Synth. No. 1, SMF, Paris, 1995. MR 1328336
- 30.
- D. Szasz (editor), Hard balls systems and Lorentz gas, Springer, Berlin, 2000. MR 1805337
- 31.
- M. Wojtkowski, Invariant families of cones and Lyapounov exponents, Ergod. Th. and Dyn. Syst. 5 (1985), 145-161. MR 0782793
- 32.
- M. Wojtkowski, Principles for the design of billiards with nonvanishing Lyapunov exponents, Comm. Math. Phys. 105 (1986), 391-414. MR 0848647
- 33.
- L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. 147 (1998), 585-650. MR 1637655
- 34.
- L.-S. Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153-188. MR 1750438
Review Information:
Reviewer:
Leonid Bunimovich
Affiliation:
Georgia Institute of Technology
Journal:
Bull. Amer. Math. Soc.
46 (2009), 683-690
DOI:
https://doi.org/10.1090/S0273-0979-09-01234-8
Published electronically:
March 23, 2009
Review copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
