Advertisement

Israel Journal of Mathematics

, Volume 142, Issue 1, pp 249–260 | Cite as

Minimal sets for flows on moduli space

  • John SmillieEmail author
  • Barak Weiss
Article

Abstract

LetS be a compact orientable surface, letQ be the moduli space of quadratic differentials onS and letM be a stratum inQ. We explicitly describe the minimal sets for the (Teichmüller) horocycle flow onM and onQ, and show that these correspond to horizontal cylindrical decompositions ofS.

Keywords

Modulus Space Quadratic Differential Saddle Connection Interval Exchange Transformation Horizontal Foliation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Cal] K. Calta,Veech surfaces and complete periodicity in genus 2, preprint, 2003.Google Scholar
  2. [CofoSi] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai,Ergodic Theory, Springer, Berlin, 1982.zbMATHGoogle Scholar
  3. [KoZo] M. Kontsevich and A. Zorich,Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Inventiones Mathematicae153 (2003), 631–678.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [Kul] M. Kulikov,Infinitely generated Fuchsian groups and minimal sets of the geodesic and horocycle flow, Comptes Rendus de l’Académie des Sciences, Paris, to appear.Google Scholar
  5. [La] E. Lanneau,Classification of connected components of the strata of the moduli space of quadratic differentials, Ph.D Thesis, Université de Rennes 1, 2003. Available at: http://iml.univ-mrs.fr/lanneau/Google Scholar
  6. [MaSm1] H. Masur and J. Smillie,Hausdorff dimension of sets of nonergodic measured foliations, Annals of Mathematics134 (1991), 455–543.CrossRefMathSciNetGoogle Scholar
  7. [MaSm2] H. Masur and J. Smillie,Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Commentarii Mathematici Helvetici68 (1993), 289–307.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [MaTa] H. Masur and S. Tabachnikov,Rational billiards and flat structures, inHandbook of Dynamical Systems, Encyclopedia of Mathematical Sciences, Springer, Berlin, 2001.Google Scholar
  9. [MiWe] Y. Minsky and B. Weiss,Nondivergence of horocyclic flows on moduli spaces, Journal für die reine und angewandte Mathematik552 (2002), 131–177.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [St] K. Strebel,Quadratic Differentials, Springer, Berlin, 1984.zbMATHGoogle Scholar
  11. [Ve1] W. A. Veech,Moduli spaces of quadratic differentials, Journal d’Analyse Mathématique55 (1990), 117–171.zbMATHMathSciNetCrossRefGoogle Scholar
  12. [Ve2] W. A. Veech,Geometric realizations of hyperelliptic curves, inAlgorithms, Fractals, and Dynamics (Okayama/Kyoto, 1992), Plenum Press, New York, 1995, pp. 217–226.Google Scholar
  13. [Ve3] W. A. Veech,Measures supported on the set of uniquely ergodic directions of an arbitrary holomorphic 1-form, Ergodic Theory and Dynamical Systems19 (1999), 1093–1109.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 2004

Authors and Affiliations

  1. 1.Department of MathematicsCornell UniversityIthacaUSA
  2. 2.Department of MathematicsBen Gurion University of the NegevBe’er ShevaIsrael

Personalised recommendations