Veech's dichotomy and the lattice property

John Smillie; Barak Weiss

doi:10.1017/S0143385708000114

Veech's dichotomy and the lattice property

John Smillie^*, Barak Weiss

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

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.

Original language	English
Pages (from-to)	1959-1972
Number of pages	14
Journal	Ergodic Theory and Dynamical Systems
Volume	28
Issue number	6
DOIs	https://doi.org/10.1017/S0143385708000114
State	Published - Dec 2008
Externally published	Yes

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title = "Veech's dichotomy and the lattice property",

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.",

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AB - 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.

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Veech's dichotomy and the lattice property

Abstract

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