Book Review
The AMS does not provide abstracts of book reviews.
You may download the entire review from the links below.
MathSciNet review:
2791778
Full text of review:
PDF
This review is available free of charge.
Book Information:
Author:
Richard Evan Schwartz
Title:
Outer billiards on kites
Additional book information:
Annals of Mathematics Studies, 171,
Princeton University Press,
Princeton, New Jersey,
2009,
xiv+306 pp.,
ISBN 978-0-691-14249-4
J. Cassaigne, P. Hubert, and S. Troubetzkoy, Complexity and growth for polygonal billiards, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 3, 835–847 (English, with English and French summaries). MR 1907389
Serge Tabachnikov and Filiz Dogru, Dual billiards, Math. Intelligencer 27 (2005), no. 4, 18–25. MR 2183864, DOI 10.1007/BF02985854
Dmitry Dolgopyat and Bassam Fayad, Unbounded orbits for semicircular outer billiard, Ann. Henri Poincaré 10 (2009), no. 2, 357–375. MR 2511890, DOI 10.1007/s00023-009-0409-9
Daniel Genin, Research announcement: boundedness of orbits for trapezoidal outer billiards, Electron. Res. Announc. Math. Sci. 15 (2008), 71–78. MR 2457051
Eugene Gutkin and Nándor Simányi, Dual polygonal billiards and necklace dynamics, Comm. Math. Phys. 143 (1992), no. 3, 431–449. MR 1145593
Eugene Gutkin and Serge Tabachnikov, Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards on surfaces of constant curvature, Mosc. Math. J. 6 (2006), no. 4, 673–701, 772. MR 2291158, DOI 10.17323/1609-4514-2006-6-4-673-701
RafałKołodziej, The antibilliard outside a polygon, Bull. Polish Acad. Sci. Math. 37 (1989), no. 1-6, 163–168 (1990) (English, with Russian summary). MR 1101465
Long Li, On Moser’s boundedness problem of dual billiards, Ergodic Theory Dynam. Systems 29 (2009), no. 2, 613–635. MR 2486786, DOI 10.1017/S0143385708000515
Jürgen Moser, Stable and random motions in dynamical systems, Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. MR 0442980
Jürgen Moser, Is the solar system stable?, Math. Intelligencer 1 (1978/79), no. 2, 65–71. MR 495314, DOI 10.1007/BF03023062
B. Neumann, Sharing ham and eggs. Iota, Manchester University, 1959.
Richard Evan Schwartz, Obtuse triangular billiards. I. Near the $(2,3,6)$ triangle, Experiment. Math. 15 (2006), no. 2, 161–182. MR 2253003
Richard Evan Schwartz, Unbounded orbits for outer billiards. I, J. Mod. Dyn. 1 (2007), no. 3, 371–424. MR 2318496, DOI 10.3934/jmd.2007.1.371
Richard Evan Schwartz, Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories, Experiment. Math. 18 (2009), no. 2, 137–171. MR 2549685
Richard Evan Schwartz, Outer billiards on kites, Annals of Mathematics Studies, vol. 171, Princeton University Press, Princeton, NJ, 2009. MR 2562898, DOI 10.1515/9781400831975
R. Schwartz, Outer Billiards and the Pinwheel Map. arXiv:1004.3025.
R. Schwartz, Outer Billiards, Arithmetic Graphs, and the Octagon. arXiv:1006.2782.
S. Tabachnikov, On the dual billiard problem, Adv. Math. 115 (1995), no. 2, 221–249. MR 1354670, DOI 10.1006/aima.1995.1055
S. Tabachnikov, Billiards, Soc. Math. France “Panoramas et Syntheses”, No 1, 1995.
Serge Tabachnikov, Geometry and billiards, Student Mathematical Library, vol. 30, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005. MR 2168892, DOI 10.1090/stml/030
Serge Tabachnikov, A proof of Culter’s theorem on the existence of periodic orbits in polygonal outer billiards, Geom. Dedicata 129 (2007), 83–87. MR 2353984, DOI 10.1007/s10711-007-9196-y
Franco Vivaldi and Anna V. Shaidenko, Global stability of a class of discontinuous dual billiards, Comm. Math. Phys. 110 (1987), no. 4, 625–640. MR 895220
References
- J. Cassaigne, P. Hubert, and S. Troubetzkoy, Complexity and growth for polygonal billiards. Ann. Inst. Fourier (Grenoble) 52 (2002), 835–847. MR 1907389 (2003d:37096)
- F. Dogru, S. Tabachnikov. Dual billiards. Math. Intelligencer 27 No 4 (2005), 18–25. MR 2183864 (2006i:37121)
- D. Dolgopyat, B. Fayad, Unbounded orbits for semicircular outer billiard. Ann. Henri Poincaré 10 (2009), 357–375. MR 2511890 (2010d:37076)
- D. Genin, Research announcement: boundedness of orbits for trapezoidal outer billiards. Electron. Res. Announc. Math. Sci. 15 (2008), 71–78. MR 2457051 (2009k:37036)
- E. Gutkin, N. Simanyi, Dual polygonal billiards and necklace dynamics. Comm. Math. Phys. 143 (1991), 431–450. MR 1145593 (92k:58139)
- E. Gutkin, S. Tabachnikov, Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards on surfaces of constant curvature. Mosc. Math. J. 6 (2006), 673–701. MR 2291158 (2008f:37080)
- R. Kolodziej, The antibilliard outside a polygon. Bull. Polish Acad. Sci. 37 (1989), 163–168. MR 1101465 (92g:52002)
- L. Li, On Moser’s boundedness problem of dual billiards. Ergodic Theory Dynam. Syst. 29 (2009), 613–635. MR 2486786 (2010b:37121)
- J. Moser, Stable and random motions in dynamical systems, Ann. of Math. Studies, 77, Princeton Univ. Press, Princeton, NJ, 1973. MR 0442980 (56:1355)
- J. Moser, Is the solar system stable? Math. Intelligencer 1 (1978), 65–71. MR 0495314 (58:14029)
- B. Neumann, Sharing ham and eggs. Iota, Manchester University, 1959.
- R. Schwartz, Obtuse triangular billiards. I. Near the $(2,3,6)$ triangle. Experiment. Math. 15 (2006), 161–182. MR 2253003 (2007c:37034)
- R. Schwartz, Unbounded orbits for outer billiards. I. J. Mod. Dyn. 1 (2007), 371–424. MR 2318496 (2008f:37082)
- R. Schwartz, Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories. Experiment. Math. 18 (2009), 137–171. MR 2549685 (2010g:37060)
- R. Schwartz, Outer billiards on kites, Ann. of Math. Studies, 171, Princeton Univ. Press, Princeton, NJ, 2009. MR 2562898
- R. Schwartz, Outer Billiards and the Pinwheel Map. arXiv:1004.3025.
- R. Schwartz, Outer Billiards, Arithmetic Graphs, and the Octagon. arXiv:1006.2782.
- S. Tabachnikov, On the dual billiard problem. Adv. Math. 115 (1995), 221–249. MR 1354670 (96g:58154)
- S. Tabachnikov, Billiards, Soc. Math. France “Panoramas et Syntheses”, No 1, 1995.
- S. Tabachnikov, Geometry and billiards, Student Math. Library, 30. Amer. Math. Soc., Providence, RI, 2005. MR 2168892 (2006h:51001)
- S. Tabachnikov, A proof of Culter’s theorem on the existence of periodic orbits in polygonal outer billiards. Geom. Dedicata 129 (2007), 83–87. MR 2353984 (2008m:37062)
- F. Vivaldi, A. Shaidenko, Global stability of a class of discontinuous dual billiards. Comm. Math. Phys. 110 (1987), 625–640. MR 895220 (89c:58067)
Review Information:
Reviewer:
Serge Tabachnikov
Affiliation:
Department of Mathematics, Penn State, University Park, Pennsylvania 16802
Email:
tabachni@math.psu.edu
Journal:
Bull. Amer. Math. Soc.
48 (2011), 285-291
DOI:
https://doi.org/10.1090/S0273-0979-2010-01313-6
Published electronically:
October 20, 2010
Review copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
