×

Billiards in rectangles with barriers. (English) Zbl 1038.37031

The paper studies billiards on the unit square with an interior barrier consisting of a vertical line segment of irrational height extending up from a rational base point on the lower side of the square. Asymptotic formulas are developed for the number of families of parallel closed orbits of length at most \(T\). These estimates are of the form of a rational multiple of \(\pi T^2\). The rational factor depends only on the denominator \(q\) of the base point \((p/q,0)\).

MSC:

37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37A17 Homogeneous flows
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] S. Bloch and A. Okounkov, The character of the infinite wedge representation , · Zbl 0978.17016 · doi:10.1006/aima.1999.1845
[2] S. G. Dani and G. A. Margulis, “Limit distributions of orbits of unipotent flows and values of quadratic forms” in I. M. Gelfand Seminar , ed. S. Gelfand and S. Gindikin, Adv. Soviet Math. 16 , Part I, Amer. Math. Soc., Providence, 1993, 91–137. · Zbl 0814.22003
[3] R. Dijkgraaf, “Mirror symmetry and elliptic curves” in The Moduli Space of Curves (Texel Island, The Netherlands, 1994) , ed. R. Dijkgraaf, C. Faber, G. van der Geer, Progr. Math. 129 , Birkhäuser, Boston, 1995, 149–163. · Zbl 0913.14007
[4] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces , Ergodic Theory Dynam. Systems 21 (2001), 443–478. · Zbl 1096.37501 · doi:10.1017/S0143385701001225
[5] A. Eskin, H. Masur, and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems and the Siegel-Veech constants , to appear in Inst. Hautes Études Sci. Publ. Math., · Zbl 1037.32013 · doi:10.1007/s10240-003-0015-1
[6] A. Eskin and A. Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials , Invent. Math. 145 (2001), 59–103. · Zbl 1019.32014 · doi:10.1007/s002220100142
[7] A. Eskin, A. Okounkov, and A. Zorich, Volumes of spaces of Abelian and quadratic differentials , in preparation.
[8] A. B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes , Math. Res. Lett. 5 (1998), 497–516. · Zbl 0961.11040 · doi:10.4310/MRL.1998.v5.n4.a7
[9] E. Gutkin, Billiards in polygons , Phys. D 19 (1986), 311–333. · Zbl 0593.58016 · doi:10.1016/0167-2789(86)90062-X
[10] E. Gutkin and C. Judge, private communication.
[11] –. –. –. –., Affine mappings of translation surfaces: Geometry and arithmetic , Duke Math. J. 103 (2000), 191–213. · Zbl 0965.30019 · doi:10.1215/S0012-7094-00-10321-3
[12] N. Jacobson, The Theory of Rings , Math. Surveys 2 , Amer. Math. Soc., New York, 1943. · Zbl 0060.07302
[13] H. Masur and S. Tabachnikov, “Rational billiards and flat structures” in Handbook of Dynamical Systems, Vol. 1A , ed. B. Hasselblatt and A. Katok, North-Holland, Amsterdam, 2002, 1015–1089. \CMP1 928 530. · Zbl 1057.37034
[14] M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups , Acta. Math. 165 (1990), 229–309. · Zbl 0745.28010 · doi:10.1007/BF02391906
[15] –. –. –. –., Strict measure rigidity for unipotent subgroups of solvable groups , Invent. Math. 101 (1990), 449–482. · Zbl 0745.28009 · doi:10.1007/BF01231511
[16] –. –. –. –., On Raghunathan’s measure conjecture , Ann. of Math. (2) 134 (1991), 545–607. JSTOR: · Zbl 0763.28012 · doi:10.2307/2944357
[17] –. –. –. –., Raghunathan’s topological conjecture and distributions of unipotent flows , Duke Math. J. 63 (1991), 235–280. · Zbl 0733.22007 · doi:10.1215/S0012-7094-91-06311-8
[18] N. Shah, Limit distributions of expanding translates of certain orbits on homogeneous spaces , Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 105–125. · Zbl 0864.22004 · doi:10.1007/BF02837164
[19] W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards , Invent. Math. 97 (1989), 553–583. · Zbl 0676.32006 · doi:10.1007/BF01388890
[20] –. –. –. –., Siegel measures , Ann. of Math. (2) 148 (1998), 895–944. JSTOR: · Zbl 0922.22003 · doi:10.2307/121033
[21] A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons (in Russian), Mat. Zametki 18 (1975), 291–300.; English translation in Math. Notes 18 (1975), 760–764. · Zbl 0323.58012 · doi:10.1007/BF01818045
[22] A. Zorich,“Square tiled surfaces and Teichmüller volumes of the moduli spaces of abelian differentials” in Rigidity in Dynamics and Geometry (Cambridge, 2000) , ed. M. Burger and A. Iozzi, Springer, Berlin, 2002, 459–471. \CMP1 919 417 · Zbl 1038.37015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.