|
Periodic billiard orbits are dense in rational polygons
Author(s):
M.
Boshernitzan;
G.
Galperin;
T.
Krüger;
S.
Troubetzkoy
Journal:
Trans. Amer. Math. Soc.
350
(1998),
3523-3535.
MSC (1991):
Primary 58F05
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We show that periodic orbits are dense in the phase space for billiards in polygons for which the angle between each pair of sides is a rational multiple of 
References:
- [BKM]
- C. Boldrighini, M. Keane, and F. Marchetti, Billiards in polygons, Ann. Prob. 6 (1978), 532-540. MR 58:31007b
- [Bo]
- M. Boshernitzan, Billiards and rational periodic directions in polygons, Amer. Math. Monthly 99 (1992), 522-529. MR 93d:51043
- [Bo1]
- M. Boshernitza, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J. 52 (1985) 723-752.MR 87i:28012
- [CFS]
- I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai, Ergodic theory, Springer Verlag, Berlin, 1982. MR 87f:28019
- [GKT]
- G. Galperin, T. Krüger, and S. Troubetzkoy, Local instability of orbits in polygonal and polyhedral billiards, Comm. Math. Phys. 169 (1995), 463-473. MR 96d:58106
- [GSV]
- G.A. Galperin, A.M. Stepin, and Ya.B. Vorobetz, Periodic billiard trajectories in polygons: generating mechanisms, Russian Math. Surv. 47:3 (1992), 5-80. MR 93h:58088
- [Gu]
- E. Gutkin, Billiards in polygons, Physica D 19 (1986), 311-333. MR 87k:58085
- [Gu1]
- E. Gutkin, Billiards on almost integrable polyhedral surfaces, Ergod. Th. Dyn. Sys. 4 (1984), 569-584. MR 86m:58123
- [K]
- M. Keane, Interval exchange transformations, Math. Z. 141 (1975), 25-31. MR 50:10207
- [KMS]
- S. Kerckhoff, H. Masur, and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals Math. 124 (1986), 293-311. MR 88f:58122
- [M]
- H. Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J. 53 (1986), 307-313. MR 87j:30107
- [M1]
- H. Masur, The growth rate of trajectories of a quadratic differential, Ergod. Th. Dyn. Sys. 10 (1990), 151-176. MR 91d:30042
- [V]
- W. Veech, The billiard in a regular polygon Geom. Func. Anal. 2 (1992), 341-379. MR 94a:11074
- [V1]
- W. Veech, Teichmuller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), 553-583. MR 91h:58083a
- [W]
- P. Walters, An introduction to ergodic theory, Springer-Verlag, 1982. MR 84e:28017
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(1991):
58F05
Retrieve articles in all Journals with MSC
(1991):
58F05
Additional Information:
M.
Boshernitzan
Affiliation:
Department of Mathematics, Rice University, Houston, Texas 77005
Email:
michael@math.rice.edu
G.
Galperin
Affiliation:
Forschungszentrum BiBos, Universität Bielefeld, Bielefeld, Germany
Address at time of publication:
Department of Mathematics, Eastern Illinois University
Email:
cfgg@eiu.edu
T.
Krüger
Affiliation:
Forschungszentrum BiBos, Universität Bielefeld, Bielefeld, Germany
S.
Troubetzkoy
Affiliation:
Forschungszentrum BiBos, Universität Bielefeld, Bielefeld, Germany and Institute for Mathematical Science, SUNY at Stony Brook, Stony Brook, New York 11794
Address at time of publication:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294
Email:
troubetz@math.uab.edu
DOI:
10.1090/S0002-9947-98-02089-3
PII:
S 0002-9947(98)02089-3
Received by editor(s):
July 29, 1996
Additional Notes:
MB is partially supported by NSF-DMS-9224667.
GG thanks the Alexander von Humboldt Stiftung for their support.
ST thanks the Deutsche Forschungsgemeinschaft for their support.
Copyright of article:
Copyright
1998,
American Mathematical Society
|
Comments: webmaster@ams.org