logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Previous article
W. Patrick Hooper; Barak Weiss
Generalized Staircases: Recurrence and Symmetry
(Espaliers généralisés, récurrence et symétrie)
Annales de l'institut Fourier, 62 no. 4 (2012), p. 1581-1600, doi: 10.5802/aif.2730
Article PDF | Reviews MR 3025751 | Zbl 1279.37035
Class. Math.: 11Y40, 12Y05, 37M99, 52C99
Keywords: Infinite translation surfaces, Veech groups, lattices, straightline flow

Résumé - Abstract

We study infinite translation surfaces which are $\mathbb{Z}$-covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, as well as surfaces with infinitely generated Veech groups.

Bibliography

[1] R. Chamanara, F. P. Gardiner & N. Lakic, “A hyperelliptic realization of the horseshoe and baker maps”, Ergodic Theory Dynam. Systems 26 (2006) no. 6, p. 1749-1768 Article |  MR 2279264 |  Zbl 1121.37036
[2] J.P. Conze & E. Gutkin 2010, in preparation
[3] David Fried, “Growth rate of surface homeomorphisms and flow equivalence”, Ergodic Theory Dynam. Systems 5 (1985) no. 4, p. 539-563 Article |  MR 829857 |  Zbl 0603.58020
[4] Eugene Gutkin & Chris Judge, “Affine mappings of translation surfaces: geometry and arithmetic”, Duke Math. J. 103 (2000) no. 2, p. 191-213 Article |  MR 1760625 |  Zbl 0965.30019
[5] Frank Herrlich & Gabriela Schmithüsen, “An extraordinary origami curve”, Math. Nachr. 281 (2008) no. 2, p. 219-237 Article |  MR 2387362 |  Zbl 1159.14012
[6] W. Patrick Hooper, “Dynamics on an infinite surface with the lattice property”, preprint, 2008
[7] Pascal Hubert & Thomas A. Schmidt, “Infinitely generated Veech groups”, Duke Math. J. 123 (2004) no. 1, p. 49-69 Article |  MR 2060022 |  Zbl 1056.30044
[8] Pascal Hubert & Gabriela Schmithüsen, “Infinite translation surfaces with infinitely generated Veech groups”, preprint, 2009  MR 2753950 |  Zbl 1219.30019
[9] Pascal Hubert & Barak Weiss, “DYNAMICS ON THE INFINITE STAIRCASE”, preprint, 2008
[10] Richard Kenyon & John Smillie, “Billiards on rational-angled triangles”, Comment. Math. Helv. 75 (2000) no. 1, p. 65-108 Article |  MR 1760496 |  Zbl 0967.37019
[11] Steven Kerckhoff, Howard Masur & John Smillie, “A rational billiard flow is uniquely ergodic in almost every direction”, Bull. Amer. Math. Soc. (N.S.) 13 (1985) no. 2, p. 141-142 Article |  MR 799797 |  Zbl 0574.58020
[12] Howard Masur & John Smillie, “Hausdorff dimension of sets of nonergodic measured foliations”, Ann. of Math. (2) 134 (1991) no. 3, p. 455-543 Article |  MR 1135877 |  Zbl 0774.58024
[13] Howard Masur & Serge Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, Vol. 1A, North-Holland, 2002, p. 1015–1089  MR 1928530 |  Zbl 1057.37034
[14] Katsuhiko Matsuzaki & Masahiko Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998, Oxford Science Publications  MR 1638795 |  Zbl 0892.30035
[15] Curtis T. McMullen, “Teichmüller geodesics of infinite complexity”, Acta Math. 191 (2003) no. 2, p. 191-223 Article |  MR 2051398 |  Zbl 1131.37052
[16] Piotr Przytycki, G. Schmithuesen & Ferran Valdez, “Veech groups of Loch Ness monsters” 2009, http://arxiv.org/abs/0906.5268, preprint.
[17] I. Richards, “On the classification of noncompact surfaces”, Trans. Amer. Math. Soc. 106 (1963), p. 259-269 Article |  MR 143186 |  Zbl 0156.22203
[18] Klaus Schmidt, Cocycles on ergodic transformation groups, Macmillan Company of India, Ltd., Delhi, 1977, Macmillan Lectures in Mathematics, Vol. 1  MR 578731 |  Zbl 0421.28017
[19] William P. Thurston, “On the geometry and dynamics of diffeomorphisms of surfaces”, Bull. Amer. Math. Soc. (N.S.) 19 (1988) no. 2, p. 417-431 Article |  MR 956596 |  Zbl 0674.57008
[20] William P. Thurston, “Minimal stretch maps between hyperbolic surfaces” 1998, eprint of 1986 preprint. See http://arxiv.org/abs/math/9801039.
[21] J.F. Valdez, “Infinite genus surfaces and irrational polygonal billiards”, Geom. Dedicata , To appear  MR 2576299 |  Zbl 1190.37040
[22] W. A. Veech, “Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards”, Invent. Math. 97 (1989) no. 3, p. 553-583 Article |  MR 1005006 |  Zbl 0676.32006
[23] A. N. Zemljakov & A. B. Katok, “Topological transitivity of billiards in polygons”, Mat. Zametki 18 (1975) no. 2, p. 291-300  MR 399423 |  Zbl 0315.58014
[24] Anton Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, 2006, p. 437–583  MR 2261104 |  Zbl 1129.32012
top