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

Department of Mathematics

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

9 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 - 1 Dec 2008

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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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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N2 - 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.

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.

UR - https://www.scopus.com/pages/publications/56549124450

U2 - 10.1017/S0143385708000114

DO - 10.1017/S0143385708000114

M3 - Article

AN - SCOPUS:56549124450

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SP - 1959

EP - 1972

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 6

ER -

Veech's dichotomy and the lattice property

Abstract

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