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Veech’s dichotomy and the lattice property

  • JOHN SMILLIE (a1) and BARAK WEISS (a2)

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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[1] L. Bers . Fiber spaces over Teichmüller spaces. Acta Math. 130 (1973), 89126.

[5] C. Earle and R. Fowler . Holomorphic families of open Riemann surfaces. Math. Ann. 270(2) (1985), 249273.

[6] R. H. Fox and R. B. Kershner . Concerning the transitive properties of geodesics on a rational polyhedron. Duke Math. J. 2(1) (1936), 147150.

[14] H. Masur . Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Math. J. 66 (1992), 387442.

[18] W. A. Veech . Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97(3) (1989), 553583.

[20] Ya. B. Vorobets . Planar structures and billiards in rational polygons: the Veech alternative. Russian Math. Surveys 51(5) (1996), 779817 (Translation from Russian).

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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