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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: smillie@math.cornell.edu)
BARAK WEISS
Affiliation:
Department of Mathematics, Ben Gurion University, Be’er Sheva, 84105, Israel (email: barakw@math.bgu.ac.il)
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Abstract

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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.

Information

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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