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).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avila A., Forni G.: Weak mixing for interval exchange transformations and translation flows. Annals of Mathematics (2) 165(2), 637–664 (2007)
L.V. Ahlfors and L. Sario. Riemann surfaces. In: [Princeton Mathematical Series, Vol. 26. Princeton University Press, Princeton (1960).
J. Bowman. The complete family of Arnoux–Yoccoz surfaces.[Geometriae Dedicata (2012), 1–18. doi:10.1007/s10711-012-9762-9
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.
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.
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.
G. Forni. Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Annals of Mathematics (2), (1) 155 (2002), 1–103.
K. Fra̦czek and C. Ulcigrai. Non-ergodic Z-periodic billiards and infinite translation surfaces. Arxiv preprint arXiv: 1109.4584 (2011).
P. Hooper, P. Hubert, and B. Weiss Dynamics on the infinite staircase. Discrete and Continuous Dynamical Systems (2011, To appear).
J. Hubbard and H. Masur. Quadratic differentials and foliations. Acta Mathematica. (3–4) 142 (1979), 221–274.
W.P. Hooper. An infinite surface with the lattice property I: Veech groups and coding geodesics. Arxiv preprint arXiv:1011.0700 (2010)
W.P. Hooper. The invariant measures of some infinite interval exchange maps. Arxiv preprint arXiv:1005.1902 (2010).
L. Hörmander. An introduction to complex analysis in several variables, 3rd edn, Vol. 7. North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam (1990).
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.
P. Hubert and G. Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. J. Mod. Dyn. (4) 4 (2010), 715–732.
S. Kerckhoff, H. Masur, and J. Smillie. Ergodicity of billiard flows and quadratic differentials. Annals of Mathematics (2), (2) 124 (1986), 293–311.
H. Masur. Interval exchange transformations and measured foliations. Annals of Mathematics (1). (2) 115 (1982), 169–200.
H. Masur. Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Mathematical Journal (3) 66 (1992), 387–442.
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.
H. Masur and J. Smillie. Hausdorff dimension of sets of nonergodic measured foliations. Annals of Mathematics (2) (3) 134 (1991), 455–543.
H. Masur and S. Tabachnikov. Rational billiards and flat structures. Handbook of dynamical systems, Vol. 1A. North-Holland, Amsterdam (2002), pp. 1015–1089.
D. Ralston and S. Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. ArXiv e-prints (2012)
K. Strebel. Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Results in Mathematics and Related Areas (3), Vol. 5. Springer, Berlin (1984).
Veech W.A.: The Teichmüller geodesic flow. Annals of Mathematics (2) 124(3), 441–530 (1986)
W.A. Veech. Boshernitzan’s criterion for unique ergodicity of an interval exchange transformation. Ergodic Theory and Dynamical Systems. (1) 7 (1987), 149–153.
Veech W.A.: curves in moduli space, Eisenstein series and an application to triangular billiards. Inventiones mathematicae. (3) 97, 553–583 (1989)
A. Zorich. Flat surfaces. In: Frontiers in Number Theory, Physics, and Geometry. I. Springer, Berlin (2006), pp. 437–583.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Brin and Flagship Fellowships at the University of Maryland.
Rights and permissions
About this article
Cite this article
Treviño, R. On the ergodicity of flat surfaces of finite area. Geom. Funct. Anal. 24, 360–386 (2014). https://doi.org/10.1007/s00039-014-0269-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-014-0269-4