Advertisement

Inventiones mathematicae

, Volume 197, Issue 2, pp 241–298 | Cite as

Non-ergodic \(\mathbb{Z}\)-periodic billiards and infinite translation surfaces

  • Krzysztof Frączek
  • Corinna UlcigraiEmail author
Article

Abstract

We give a criterion which proves non-ergodicity for certain infinite periodic billiards and directional flows on \(\mathbb{Z}\)-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a \(\mathbb{Z}^{2}\)-periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding directional billiard flow is not ergodic and has uncountably many ergodic components. As another application, we show that for any recurrent \(\mathbb{Z}\)-cover of a square tiled surface of genus two the directional flow is not ergodic and has no invariant sets of finite measure for a full measure set of directions. In the language of essential values, we prove that the skew-products which arise as Poincaré maps of the above systems are associated to non-regular \(\mathbb{Z}\)-valued cocycles for interval exchange transformations.

Mathematics Subject Classification (2000)

37A40 37C40 

Notes

Acknowledgements

We would like to thank Vincent Delecroix, Giovanni Forni and Pascal Hubert for useful discussions and suggestions that helped us improve the paper and Artur Avila for suggesting the argument used in Sect. 8. We also thank J.-P. Conze, A. Eskin, P. Hooper, M. Lemańczyk, C. Matheus, B. Weiss for useful discussions and the referee for suggestions to improve the presentation.

The first author is partially supported by the Narodowe Centrum Nauki Grant DEC-2011/03/B/ST1/00407. The second author is currently supported by an RCUK Academic Fellowship and the EPSRC Grant EP/I019030/1, whose support is fully acknowledged.

References

  1. 1.
    Aaronson, J.: An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs, vol. 50. AMS, Providence (1997) zbMATHGoogle Scholar
  2. 2.
    Avila, A., Hubert, P.: Recurrence for the wind-tree model. Ann. Inst. Henri Poincaré C (2011, to appear) Google Scholar
  3. 3.
    Aulicino, D.: Progress toward classifying Teichmüller disks with completely degenerate Kontsevich-Zorich spectrum. Ph.D. thesis, University of Maryland (2012) Google Scholar
  4. 4.
    Bachurin, P., Khanin, K., Marklof, J., Plakhov, A.: Perfect retroreflectors and billiard dynamics. J. Mod. Dyn. 5, 33–48 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bainbridge, M.: Euler characteristics of Teichmüller curves in genus two. Geom. Topol. 11, 1887–2073 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bergelson, V., del Junco, A., Lemańczyk, M., Rosenblatt, J.: Rigidity and non-recurrence along sequences. Ergod. Theory Dyn. Syst. (2013). doi: 10.1017/etds.2013.5. arXiv:1103.0905 Google Scholar
  7. 7.
    Bowman, J.P., Valdez, F.: Wild singularities of translations surfaces. Isr. J. Math. (2013). doi: 10.1007/s11856-013-0022-y. arXiv:1110.1350 MathSciNetGoogle Scholar
  8. 8.
    Cheung, Y.: Hausdorff dimension of the set of nonergodic directions. Ann. Math. (2) 158, 661–678 (2003). With an appendix by M. Boshernitzan CrossRefzbMATHGoogle Scholar
  9. 9.
    Cheung, Y., Hubert, P., Masur, H.: Dichotomy for the Hausdorff dimension of the set of nonergodic directions. Invent. Math. 183, 337–383 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Conze, J.-P., Frączek, K.: Cocycles over interval exchange transformations and multivalued Hamiltonian flows. Adv. Math. 226, 4373–4428 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Conze, J.-P., Gutkin, E.: On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces. Ergod. Theory Dyn. Syst. 32, 491–515 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Delecroix, V.: Divergent directions in some periodic wind-tree models. J. Mod. Dyn. 7, 1–29 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Delecroix, V., Hubert, P., Lelièvre, S.: Diffusion for the periodic wind-tree model. Ann. Sci. Éc. Norm. Supér. (to appear). arXiv:1107.1810
  14. 14.
    Eskin, A., Chaika, J.: Every flat surface is Birkhoff and Osceledets generic in almost every direction. arXiv:1305.1104
  15. 15.
    Eskin, A., Masur, H., Schmoll, M.: Billiards in rectangles with barriers. Duke Math. J. 118, 427–463 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Forni, G.: Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann. Math. (2) 146, 295–344 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Forni, G.: Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. Math. (2) 155, 1–103 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Forni, G.: Sobolev regularity of solutions of the cohomological equation. arXiv:0707.0940
  19. 19.
    Frączek, K., Ulcigrai, C.: Ergodic properties of extensions of locally Hamiltonian flows. Math. Ann. 354, 1289–1367 (2012) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Frączek, K., Ulcigrai, C.: Ergodic directions for billiards in a strip with periodically located obstacles. Commun. Math. Phys. (to appear). arXiv:1208.5212
  21. 21.
    Fulton, W.: Algebraic Topology. A First Course. Graduate Texts in Mathematics, vol. 153. Springer, New York (1995) zbMATHGoogle Scholar
  22. 22.
    Gutkin, E.: Billiards on almost integrable polyhedral surfaces. Ergod. Theory Dyn. Syst. 4, 569–584 (1984) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Gutkin, E.: Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces. Regul. Chaotic Dyn. 15, 482–503 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Hardy, J., Weber, J.: Diffusion in a periodic wind-tree model. J. Math. Phys. (7) 21, 1802–1808 (1980) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Hooper, P.: The invariant measures of some infinite interval exchange maps. arXiv:1005.1902
  26. 26.
    Hooper, P.: An infinite surface with the lattice property, I: Veech groups and coding geodesics. arXiv:1011.0700
  27. 27.
    Hooper, P.: Dynamics on an infinite surface with the lattice property. arXiv:0802.0189
  28. 28.
    Hooper, P., Hubert, P., Weiss, B.: Dynamics on the infinite staircase. Discrete Contin. Dyn. Syst., Ser. A 33, 4341–4347 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Hooper, P., Weiss, B.: Generalized staircases: recurrence and symmetry. Ann. Inst. Fourier (Grenoble) 62, 1581–1600 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Hubert, P., Lelièvre, S., Troubetzkoy, S.: The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion. J. Reine Angew. Math. 656, 223–244 (2011) zbMATHMathSciNetGoogle Scholar
  31. 31.
    Hubert, P., Weiss, B.: Ergodicity for infinite periodic translation surfaces. Compos. Math. 149, 1364–1380 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Hubert, P., Schmithüsen, G.: Infinite translation surfaces with infinitely generated Veech groups. J. Mod. Dyn. 4, 715–732 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Katok, A., Zemljakov, A.: Topological transitivity of billiards in polygons. Mat. Zametki 18, 291–300 (1975) (in Russian) zbMATHMathSciNetGoogle Scholar
  34. 34.
    Kerckhoff, S., Masur, H., Smillie, J.: Ergodicity of billiard flows and quadratic differentials. Ann. Math. (2) 124, 293–311 (1986) CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Kontsevich, M., Zorich, A.: Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math. 153, 631–678 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Marmi, S., Moussa, P., Yoccoz, J.-C.: The cohomological equation for Roth-type interval exchange maps. J. Am. Math. Soc. 18, 823–872 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Masur, H.: Ergodic theory of translation surfaces. In: Handbook of Dynamical Systems, vol. 1B, pp. 527–547. Elsevier B.V., Amsterdam (2006) Google Scholar
  38. 38.
    Masur, H.: Interval exchange transformations and measured foliations. Ann. Math. (2) 115, 169–200 (1982) CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Masur, H., Tabachnikov, S.: Rational billiards and flat structures. In: Handbook of Dynamical Systems, vol. 1A, pp. 1015–1089. North-Holland, Amsterdam (2002) Google Scholar
  40. 40.
    Matheus, C., Yoccoz, J.-C.: The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. J. Mod. Dyn. 4, 453–486 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    McMullen, C.T.: Dynamics of \(\mathit{SL}_{2}(\mathbb{R})\) over moduli space in genus two. Ann. Math. (2) 165, 397–456 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Przytycki, P., Valdez, F., Weitze-Schmithuesen, G.: Veech groups of Loch Ness monsters. Ann. Inst. Fourier (Grenoble) 61, 673–687 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Ralston, D., Troubetzkoy, S.: Ergodic infinite group extensions of geodesic flows on translation surfaces. J. Mod. Dyn. 6, 477–497 (2012) zbMATHMathSciNetGoogle Scholar
  44. 44.
    Schmidt, K.: Cocycle of Ergodic Transformation Groups. Lecture Notes in Mathematics, vol. 1. Mac Milan, India (1977) Google Scholar
  45. 45.
    Schmoll, M.: Veech groups for holonomy-free torus covers. J. Topol. Anal. 3, 521–554 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Troubetzkoy, S.: Typical recurrence for the Ehrenfest wind-tree model. J. Stat. Phys. 141, 60–67 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Valdez, F.: Infinite genus surfaces and irrational polygonal billiards. Geom. Dedic. 143, 143–154 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Valdez, F.: Veech groups, irrational billiards and stable Abelian differentials. Discrete Contin. Dyn. Syst., Ser. A 32, 1055–1063 (2011) CrossRefMathSciNetGoogle Scholar
  49. 49.
    Valdez, F., Weitze-Schmithuesen, G.: On the geometry and arithmetic of infinite translation surfaces. arXiv:1102.0974
  50. 50.
    Veech, W.A.: Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2. Trans. Am. Math. Soc. 140, 1–34 (1969) zbMATHMathSciNetGoogle Scholar
  51. 51.
    Veech, W.A.: Gauss measures for transformations on the space of interval exchange maps. Ann. Math. (2) 115, 201–242 (1982) CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Viana, M.: Ergodic theory of interval exchange maps. Rev. Mat. Complut. 19, 7–100 (2006) zbMATHMathSciNetGoogle Scholar
  53. 53.
    Viana, M.: Dynamics of interval exchange transformations and Teichmüller flows. Lecture notes available from http://w3.impa.br/~viana/out/ietf.pdf
  54. 54.
    Yoccoz, J.-C.: Interval exchange maps and translation surfaces. In: Einsiedler, M., et al. (eds.) Homogeneous Ows, Moduli Spaces and Arithmetic. Proceedings of the Clay Mathematics Institute Summer School, Centro di Recerca Mathematica Ennio De Giorgi, Pisa, Italy, June 11–July 6, 2007. Clay Mathematics Proceedings, vol. 10, pp. 1–69. AMS/Clay Mathematics Institute, Providence/Cambridge (2010) Google Scholar
  55. 55.
    Zimmer, R.J.: Ergodic Theory and Semisimple Groups. Monographs in Mathematics, vol. 81. Birkhäuser, Basel (1984) CrossRefzbMATHGoogle Scholar
  56. 56.
    Zorich, A.: Deviation for interval exchange transformations. Ergod. Theory Dyn. Syst. 17, 1477–1499 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  57. 57.
    Zorich, A.: How do the leaves of a closed 1-form wind around a surface? Transl. Am. Math. Soc. 2(197), 135–178 (1999) MathSciNetGoogle Scholar
  58. 58.
    Zorich, A.: Flat surfaces. In: Frontiers in Number Theory, Physics, and Geometry, vol. I, pp. 437–583. Springer, Berlin (2006) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland
  2. 2.Department of MathematicsUniversity WalkBristolUK

Personalised recommendations