Advertisement

Geometric and Functional Analysis

, Volume 24, Issue 1, pp 360–386 | Cite as

On the ergodicity of flat surfaces of finite area

  • Rodrigo Treviño
Article

Abstract

We prove some ergodic theorems for flat surfaces of finite area. The first result concerns such surfaces whose Teichmüller orbits are recurrent to a compact set of \({SL(2,\mathbb{R})/SL(S,\alpha)}\) , where SL(S,α) is the Veech group of the surface. In this setting, this means that the translation flow on a flat surface can be renormalized through its Veech group. This result applies in particular to flat surfaces of infinite genus and finite area. Our second result is an criterion for ergodicity based on the control of deforming metric of a flat surface. Applied to translation flows on compact surfaces, it improves and generalizes a theorem of Cheung and Eskin et al. (Partially hyperbolic dynamics, laminations, and Teichmüller flow, Fields Inst. Commun., Vol. 51. Amer. Math. Soc., Providence, pp. 213–221, 2007).

Keywords and phrases

Flat surfaces translation flows Masur’s criterion Teichmüller dynamics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. AF07.
    Avila A., Forni G.: Weak mixing for interval exchange transformations and translation flows. Annals of Mathematics (2) 165(2), 637–664 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  2. AS60.
    L.V. Ahlfors and L. Sario. Riemann surfaces. In: [Princeton Mathematical Series, Vol. 26. Princeton University Press, Princeton (1960).Google Scholar
  3. Bow12.
    J. Bowman. The complete family of Arnoux–Yoccoz surfaces.[Geometriae Dedicata (2012), 1–18. doi: 10.1007/s10711-012-9762-9
  4. CE07.
    Y. Cheung and A. Eskin. Unique ergodicity of translation flows. [Partially hyperbolic dynamics, laminations, and Teichmüller flow, Fields Inst. Commun., Vol. 51. Amer. Math. Soc., Providence (2007), pp. 213–221.Google Scholar
  5. Cha04.
    R. Chamanara. Affine automorphism groups of surfaces of infinite type. In: In the tradition of Ahlfors and Bers, III, Contemp. Math., Vol. 355. Amer. Math., Soc., Providence (2004), pp. 123–145.Google Scholar
  6. For97.
    G. Forni. Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Annals of Mathematics (2), (2) 146 (1997), 295–344.Google Scholar
  7. For02.
    G. Forni. Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Annals of Mathematics (2), (1) 155 (2002), 1–103.Google Scholar
  8. FU11.
    K. Fra̦czek and C. Ulcigrai. Non-ergodic Z-periodic billiards and infinite translation surfaces. Arxiv preprint arXiv: 1109.4584 (2011).Google Scholar
  9. HHW11.
    P. Hooper, P. Hubert, and B. Weiss Dynamics on the infinite staircase. Discrete and Continuous Dynamical Systems (2011, To appear).Google Scholar
  10. HM79.
    J. Hubbard and H. Masur. Quadratic differentials and foliations. Acta Mathematica. (3–4) 142 (1979), 221–274.Google Scholar
  11. Hoo10a.
    W.P. Hooper. An infinite surface with the lattice property I: Veech groups and coding geodesics. Arxiv preprint arXiv:1011.0700 (2010)Google Scholar
  12. Hoo10b.
    W.P. Hooper. The invariant measures of some infinite interval exchange maps. Arxiv preprint arXiv:1005.1902 (2010).Google Scholar
  13. Hör90.
    L. Hörmander. An introduction to complex analysis in several variables, 3rd edn, Vol. 7. North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam (1990).Google Scholar
  14. HS06.
    P. Hubert and T.A. Schmidt. An introduction to Veech surfaces. Handbook of dynamical systems, Vol. 1B. Elsevier B. V., Amsterdam (2006), pp. 501–526.Google Scholar
  15. HS10.
    P. Hubert and G. Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. J. Mod. Dyn. (4) 4 (2010), 715–732.Google Scholar
  16. KMS86.
    S. Kerckhoff, H. Masur, and J. Smillie. Ergodicity of billiard flows and quadratic differentials. Annals of Mathematics (2), (2) 124 (1986), 293–311.Google Scholar
  17. Mas82.
    H. Masur. Interval exchange transformations and measured foliations. Annals of Mathematics (1). (2) 115 (1982), 169–200.Google Scholar
  18. Mas92.
    H. Masur. Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Mathematical Journal (3) 66 (1992), 387–442.Google Scholar
  19. Mas93.
    H. Masur. Logarithmic law for geodesics in moduli space. Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991). In: Contemporary Mathematics, Vol. 150. Amer. Math. Soc., Providence (1993), pp. 229–245.Google Scholar
  20. MS91.
    H. Masur and J. Smillie. Hausdorff dimension of sets of nonergodic measured foliations. Annals of Mathematics (2) (3) 134 (1991), 455–543.Google Scholar
  21. MT02.
    H. Masur and S. Tabachnikov. Rational billiards and flat structures. Handbook of dynamical systems, Vol. 1A. North-Holland, Amsterdam (2002), pp. 1015–1089.Google Scholar
  22. RT12.
    D. Ralston and S. Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. ArXiv e-prints (2012)Google Scholar
  23. Str84.
    K. Strebel. Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Results in Mathematics and Related Areas (3), Vol. 5. Springer, Berlin (1984).Google Scholar
  24. Vee86.
    Veech W.A.: The Teichmüller geodesic flow. Annals of Mathematics (2) 124(3), 441–530 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  25. Vee87.
    W.A. Veech. Boshernitzan’s criterion for unique ergodicity of an interval exchange transformation. Ergodic Theory and Dynamical Systems. (1) 7 (1987), 149–153.Google Scholar
  26. Vee89.
    Veech W.A.: curves in moduli space, Eisenstein series and an application to triangular billiards. Inventiones mathematicae. (3) 97, 553–583 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  27. Zor06.
    A. Zorich. Flat surfaces. In: Frontiers in Number Theory, Physics, and Geometry. I. Springer, Berlin (2006), pp. 437–583.Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations