The conformal geometry of billiards

DeMarco, Laura

doi:10.1090/S0273-0979-2010-01322-7

The conformal geometry of billiards
HTML articles powered by AMS MathViewer

by Laura DeMarco PDF

Bull. Amer. Math. Soc. 48 (2011), 33-52 Request permission

Abstract:

This article provides an introduction to some recent results in billiard dynamics. We present results that follow from a study of compact Riemann surfaces (equipped with a holomorphic 1-form) and an $\mathrm {SL}_2\mathbb {R}$ action on the moduli spaces of these surfaces. We concentrate on the progress toward classification of “optimal” billiard tables, those with the simplest trajectory structure.

References

M. Bainbridge and M. Möller. Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus three. Preprint, 2009.
I. Bouw and M. Möller. Teichmüller curves, triangle groups, and Lyapunov exponents. To appear, Ann. of Math. (2).
Kariane Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc. 17 (2004), no. 4, 871–908. MR 2083470, DOI 10.1090/S0894-0347-04-00461-8
Yitwah Cheung, Hausdorff dimension of the set of nonergodic directions, Ann. of Math. (2) 158 (2003), no. 2, 661–678. With an appendix by M. Boshernitzan. MR 2018932, DOI 10.4007/annals.2003.158.661
Yitwah Cheung, Pascal Hubert, and Howard Masur, Topological dichotomy and strict ergodicity for translation surfaces, Ergodic Theory Dynam. Systems 28 (2008), no. 6, 1729–1748. MR 2465598, DOI 10.1017/S0143385708000126
Y. Cheung, P. Hubert, and H. Masur. Dichotomy for the Hausdorff dimension of the set of nonergodic directions. Preprint, 2009.
Yitwah Cheung and Howard Masur, Minimal non-ergodic directions on genus-2 translation surfaces, Ergodic Theory Dynam. Systems 26 (2006), no. 2, 341–351. MR 2218764, DOI 10.1017/S0143385705000465
J. Ellenberg and D. B. McReynolds. Every curve is a Teichmüller curve. Preprint, 2009.
E. Gutkin, Billiards on almost integrable polyhedral surfaces, Ergodic Theory Dynam. Systems 4 (1984), no. 4, 569–584. MR 779714, DOI 10.1017/S0143385700002650
Eugene Gutkin and Chris Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J. 103 (2000), no. 2, 191–213. MR 1760625, DOI 10.1215/S0012-7094-00-10321-3
W. Patrick Hooper, Periodic billiard paths in right triangles are unstable, Geom. Dedicata 125 (2007), 39–46. MR 2322537, DOI 10.1007/s10711-007-9129-9
W. P. Hooper. Grid graphs and lattice surfaces. Preprint, 2009.
John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, 2006. Teichmüller theory; With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra; With forewords by William Thurston and Clifford Earle. MR 2245223
P. Hubert and T. A. Schmidt, Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 2, 461–495. MR 1824961
Pascal Hubert and Thomas A. Schmidt, Infinitely generated Veech groups, Duke Math. J. 123 (2004), no. 1, 49–69. MR 2060022, DOI 10.1215/S0012-7094-04-12312-8
A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons, Mat. Zametki 18 (1975), no. 2, 291–300 (Russian). MR 399423
Richard Kenyon and John Smillie, Billiards on rational-angled triangles, Comment. Math. Helv. 75 (2000), no. 1, 65–108. MR 1760496, DOI 10.1007/s000140050113
Steven Kerckhoff, Howard Masur, and John Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2) 124 (1986), no. 2, 293–311. MR 855297, DOI 10.2307/1971280
Howard Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J. 66 (1992), no. 3, 387–442. MR 1167101, DOI 10.1215/S0012-7094-92-06613-0
Howard Masur, Ergodic theory of translation surfaces, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 527–547. MR 2186247, DOI 10.1016/S1874-575X(06)80032-9
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
Curtis T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), no. 4, 857–885. MR 1992827, DOI 10.1090/S0894-0347-03-00432-6
Curtis T. McMullen, Teichmüller curves in genus two: discriminant and spin, Math. Ann. 333 (2005), no. 1, 87–130. MR 2169830, DOI 10.1007/s00208-005-0666-y
Curtis T. McMullen, Teichmüller curves in genus two: the decagon and beyond, J. Reine Angew. Math. 582 (2005), 173–199. MR 2139715, DOI 10.1515/crll.2005.2005.582.173
Curtis T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J. 133 (2006), no. 3, 569–590. MR 2228463, DOI 10.1215/S0012-7094-06-13335-5
Curtis T. McMullen, Teichmüller curves in genus two: torsion divisors and ratios of sines, Invent. Math. 165 (2006), no. 3, 651–672. MR 2242630, DOI 10.1007/s00222-006-0511-2
Curtis T. McMullen, Dynamics of $\textrm {SL}_2(\Bbb R)$ over moduli space in genus two, Ann. of Math. (2) 165 (2007), no. 2, 397–456. MR 2299738, DOI 10.4007/annals.2007.165.397
Martin Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math. 165 (2006), no. 3, 633–649. MR 2242629, DOI 10.1007/s00222-006-0510-3
Martin Möller, Affine groups of flat surfaces, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, Eur. Math. Soc., Zürich, 2009, pp. 369–387. MR 2497782, DOI 10.4171/055-1/11
Jan-Christoph Puchta, On triangular billiards, Comment. Math. Helv. 76 (2001), no. 3, 501–505. MR 1854695, DOI 10.1007/PL00013215
Marina Ratner, Interactions between ergodic theory, Lie groups, and number theory, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 157–182. MR 1403920
Richard Evan Schwartz, Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories, Experiment. Math. 18 (2009), no. 2, 137–171. MR 2549685
John Smillie and Barak Weiss, Veech’s dichotomy and the lattice property, Ergodic Theory Dynam. Systems 28 (2008), no. 6, 1959–1972. MR 2465608, DOI 10.1017/S0143385708000114
Kisao Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977), no. 1, 91–106. MR 429744, DOI 10.2969/jmsj/02910091
W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), no. 3, 553–583. MR 1005006, DOI 10.1007/BF01388890
Clayton C. Ward, Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems 18 (1998), no. 4, 1019–1042. MR 1645350, DOI 10.1017/S0143385798117479