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Topological dichotomy and strict ergodicity for translation surfaces

Published online by Cambridge University Press:  15 October 2008

YITWAH CHEUNG ,
PASCAL HUBERT  and
HOWARD MASUR
YITWAH CHEUNG
Affiliation:
San Francisco State University, San Francisco, CA 94132, USA (email: [email protected])
PASCAL HUBERT
Affiliation:
LATP, Case cour A, Faculté des Sciences de Saint Jérôme, Avenue Escadrille Normandie-Niemen, 13397, Marseille Cedex 20, France (email: [email protected])
HOWARD MASUR
Affiliation:
University of Illinois at Chicago, Chicago, IL 60607, USA (email: [email protected])
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Abstract

In this paper the authors find examples of translation surfaces that have infinitely generated Veech groups, satisfy the topological dichotomy property that for every direction the flow in that direction is either completely periodic or minimal, and yet have minimal but non-uniquely ergodic directions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1729 - 1748
Copyright
Copyright © 2008 Cambridge University Press

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References

[1] Bouw, I. and Möller, M.. Teichmüller curves, triangle groups, and Lyapunov exponents. Preprint, 2005. Ann. of Math. (2), to appear.Google Scholar
[2] Calta, K.. Veech surfaces and complete periodicity in genus two. J. Amer. Math. Soc. 17(4) (2004), 871–908.CrossRefGoogle Scholar
[3] Cheung, Y. and Masur, H.. Minimal nonergodic directions on genus two translation surfaces. Ergod. Th. & Dynam. Sys. 26(2) (2006), 341–351.CrossRefGoogle Scholar
[4] Eskin, A., Marklof, J. and Witte-Morris, D.. Unipotent flows on the space of branched covers of Veech surfaces. Ergod. Th. & Dynam. Sys. 26(1) (2006), 129–162.CrossRefGoogle Scholar
[5] Hubert, P. and Schmidt, T. A.. Infinitely generated Veech groups. Duke Math. J. 123 (2004), 49–69.CrossRefGoogle Scholar
[6] Masur, H. and Smillie, J.. Hausdorff dimension of sets of nonergodic measured foliations. Ann. Math. (2) 134(3) (1991), 455–543.CrossRefGoogle Scholar
[7] Masur, H. and Tabachnikov, S.. Rational Billiards and Flat Structures (Handbook of Dynamical Systems, 1A). North-Holland, Amsterdam, 2002, pp. 1015–1089.Google Scholar
[8] McMullen, C.. Billiards and Teichmüller curves on Hilbert modular surfaces. J. Amer. Math. Soc. 16(4) (2003), 857–885.CrossRefGoogle Scholar
[9] McMullen, C.. Teichmüller geodesics of infinite complexity. Acta Math. 191(2) (2003), 191–223.CrossRefGoogle Scholar
[10] McMullen, C.. Teichmüller curves in genus two: Discriminant and spin. Math. Ann. 333(1) (2005), 87–130.Google Scholar
[11] McMullen, C.. Teichmüller curves in genus two: The decagon and beyond. J. Reine Angew. Math. 582 (2005), 173–199.CrossRefGoogle Scholar
[12] McMullen, C.. Teichmüller curves in genus two: Torsion divisors and ratios of sines. Invent. Math. 165(3) (2006), 651–672.CrossRefGoogle Scholar
[13] McMullen, C.. Prym varieties and Teichmüller curves. Duke Math. J. 133(3) (2006), 569–590.Google Scholar
[14] Möller, M.. Periodic points on Veech surfaces and the Mordell–Weil group of a Teichmüller curve. Invent. Math. 165(3) (2006), 633–649.CrossRefGoogle Scholar
[15] Smillie, J. and Weiss, B.. Veech dichotomy and the lattice property. Preprint, 2006. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
[16] Veech, W.. Strict ergodicity in zero dimensional dynamical systems and the Kronecker–Weyl theorem mod 2. Trans. Amer. Math Soc. 140 (1969), 1–34.Google Scholar
[17] Veech, W.. Teichmüller curves in modular space, Eisenstein series, and an application to triangular billiards. Invent. Math. 97 (1989), 553–583.CrossRefGoogle Scholar
[18] Zorich, A.. Square Tiled Surfaces and Teichmüller Volumes of the Moduli Spaces of Abelian Differentials (Rigidity in Dynamics and Geometry (Cambridge, 2000)). Springer, Berlin, 2002, pp. 459–471.Google Scholar