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  • Cited by 3
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Dankwart, Klaus 2014. Rigidity of flat surfaces under the boundary measure. Israel Journal of Mathematics, Vol. 199, Issue. 2, p. 623.

    Davis, Diana 2013. Cutting sequences, regular polygons, and the Veech group. Geometriae Dedicata, Vol. 162, Issue. 1, p. 231.

    Cohen, Meital and Weiss, Barak 2012. Parking garages with optimal dynamics. Geometriae Dedicata, Vol. 161, Issue. 1, p. 157.

  • Ergodic Theory and Dynamical Systems, Volume 28, Issue 6
  • December 2008, pp. 1959-1972

Veech’s dichotomy and the lattice property

  • JOHN SMILLIE (a1) and BARAK WEISS (a2)
  • DOI:
  • Published online: 01 December 2008

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.

[2]Y. Cheung , P. Hubert and H. Masur . Topological dichotomy and strict ergodicity for translation surfaces. Ergod. Th. & Dynam. Sys. to appear. Preprint, 2006.

[3]C. Earle . Teichmüller Theory (Discrete Groups and Automorphic Functions). Ed. W. J. Harvey. Academic Press, London, New York, 1977.

[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.

[7]H. Farkas and I. Kra . Riemann Surfaces, 2nd edn. Springer, Berlin, 1992.

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

[16]S. M. Natanzon . The topological structure of the space of holomorphic morphisms of Riemann surfaces. Russian Math. Surveys 53 (1998), 398400 (Translation from Russian).

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

[19]W. A. Veech . The billiard in a regular polygon. Geom. Funct. Anal. 2 (1992), 341379.

[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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