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Veech’s dichotomy and the lattice property

Published online by Cambridge University Press:  15 September 2008

JOHN SMILLIE  and
BARAK WEISS
JOHN SMILLIE
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY, USA (email: [email protected])
BARAK WEISS
Affiliation:
Department of Mathematics, Ben Gurion University, Be’er Sheva, 84105, Israel (email: [email protected])
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Abstract

Veech showed that if a translation surface has a stabilizer which is a lattice in SL(2,ℝ), then any direction for the corresponding constant slope flow is either completely periodic or uniquely ergodic. We show that the converse does not hold: there are translation surfaces that satisfy Veech’s dichotomy but for which the corresponding stabilizer subgroup is not a lattice. The construction relies on work of Hubert and Schmidt.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1959 - 1972
Copyright
Copyright © 2008 Cambridge University Press

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References

[1] Bers, L.. Fiber spaces over Teichmüller spaces. Acta Math. 130 (1973), 89–126.CrossRefGoogle Scholar
[2] Cheung, Y., Hubert, P. and Masur, H.. Topological dichotomy and strict ergodicity for translation surfaces. Ergod. Th. & Dynam. Sys. to appear. Preprint, 2006.Google Scholar
[3] Earle, C.. Teichmüller Theory (Discrete Groups and Automorphic Functions). Ed. W. J. Harvey. Academic Press, London, New York, 1977.Google Scholar
[4] Earle, C. and Eells, J.. A fibre bundle description of Teichmüller theory. J. Differential Geom. 3 (1969), 19–43.CrossRefGoogle Scholar
[5] Earle, C. and Fowler, R.. Holomorphic families of open Riemann surfaces. Math. Ann. 270(2) (1985), 249–273.CrossRefGoogle Scholar
[6] Fox, R. H. and Kershner, R. B.. Concerning the transitive properties of geodesics on a rational polyhedron. Duke Math. J. 2(1) (1936), 147–150.CrossRefGoogle Scholar
[7] Farkas, H. and Kra, I.. Riemann Surfaces, 2nd edn. Springer, Berlin, 1992.CrossRefGoogle Scholar
[8] Gutkin, E.. Billiards on almost integrable polyhedral surfaces. Ergod. Th. & Dynam. Sys. 4 (1984), 569–584.CrossRefGoogle Scholar
[9] Gutkin, E., Hubert, P. and Schmidt, T.. Affine diffeomorphisms of translation surfaces: periodic points, fuchsian groups, and arithmeticity. Ann. Sci. École Norm. Sup. 36 (2003), 847–866.CrossRefGoogle Scholar
[10] Gutkin, E. and Judge, C.. Affine maps of translation surfaces: geometry and arithmetic. Duke Math. J. 103 (2000), 191–213.CrossRefGoogle Scholar
[11] Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers. Clarendon Press, Oxford, 1938.Google Scholar
[12] Hatcher, A.. Algebraic Topology. Cambridge University Press, Cambridge, 2002.Google Scholar
[13] Hubert, P. and Schmidt, T.. Infinitely generated Veech groups. Duke Math. J. 123(1) (2004), 49–69.CrossRefGoogle Scholar
[14] Masur, H.. Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Math. J. 66 (1992), 387–442.CrossRefGoogle Scholar
[15] Masur, H. and Tabachnikov, S.. Rational billiards and flat structures. Handbook of Dynamical Systems, Volume 1. Eds. B. Hasselblatt and A. Katok. Elsevier, Amsterdam, 2001, pp. 1015–1089.Google Scholar
[16] Natanzon, S. M.. The topological structure of the space of holomorphic morphisms of Riemann surfaces. Russian Math. Surveys 53 (1998), 398–400 (Translation from Russian).CrossRefGoogle Scholar
[17] Thurston, W.. The geometry and topology of 3-manifolds, Chapter 13. Available at:http://www.msri.org/communications/books/gt3m/PS.Google Scholar
[18] Veech, W. A.. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97(3) (1989), 553–583.CrossRefGoogle Scholar
[19] Veech, W. A.. The billiard in a regular polygon. Geom. Funct. Anal. 2 (1992), 341–379.CrossRefGoogle Scholar
[20] Vorobets, Ya. B.. Planar structures and billiards in rational polygons: the Veech alternative. Russian Math. Surveys 51(5) (1996), 779–817 (Translation from Russian).CrossRefGoogle Scholar