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Ergodicity for infinite periodic translation surfaces

  • Pascal Hubert (a1) and Barak Weiss (a2)

Abstract

For a $ \mathbb{Z} $ -cover $\widetilde {M} \rightarrow M$ of a translation surface, which is a lattice surface, and which admits infinite strips, we prove that almost every direction for the straightline flow is ergodic.

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References

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[ANSS02]Aaronson, J., Nakada, H., Sarig, O. and Solomyak, R., Invariant measures and asymptotics for some skew products, Israel J. Math. 128 (2002), 93134.
[Con76]Conze, J.-P., Equirépartition et ergodicité de transformations cylindriques, in Séminaire de Probabilité de Rennes (UER de Mathématiques et Informatique, 1976), 121.
[CF10]Conze, J.-P. and Fraczek, K., Cocycles over interval exchange transformations and multivalued Hamiltonian flows, Preprint (2010), http://arxiv.org/abs/1003.1808.
[EKP11]Einsiedler, M., Kadyrov, S. and Pohl, A., Escape of mass and entropy for diagonal flows in real rank one situations, Preprint (2011), http://arxiv.org/abs/1110.0910.
[FU11]Fraczek, K. and Ulcigrai, C., Non-ergodic $ \mathbb{Z} $-periodic billiards and infinite translation surfaces, Preprint (2011), http://arxiv.org/abs/1109.4584.
[Hoo08]Hooper, P., Dynamics on an infinite staircase with the lattice property, Preprint (2008), http://arxiv.org/abs/0802.0189.
[Hoo10]Hooper, P., The invariant measures of some infinite interval exchange maps, Preprint (2010), http://arxiv.org/abs/1005.1902.
[HHW08]Hooper, W. P., Hubert, P. and Weiss, B., Dynamics on the infinite staircase surface, Preprint (2008), http://www.math.bgu.ac.il/~barakw/staircase.pdf.
[HW12]Hooper, W. P. and Weiss, B., Generalized staircases: recurrence and symmetry, Ann. Inst. Fourier 62 (2012), 15811600.
[HS10]Hubert, P. and Schmithuesen, G., Infinite translation surfaces with infinitely generated Veech group, J. Mod. Dyn. 4 (2010), 715732.
[Kat92]Katok, S., Fuchsian groups, Chicago Lectures in Mathematics (University of Chicago Press, 1992).
[Kes66]Kesten, H., On a conjecture of Erdős and Szüsz related to uniform distribution mod $1$, Acta Arith. 12 (1966/1967), 193212.
[KW11]Kleinbock, D. and Weiss, B., Modified Schmidt games and a conjecture of Margulis, Preprint (2011).
[KS13]König, D. and Szücs, A., Mouvement d’un point abandonné à l’interieur d’un cube, Rend. Circ. Mat. Palermo 36 (1913), 7990.
[Mas92]Masur, H., Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J. 66 (1992), 387442.
[MT02]Masur, H. and Tabachnikov, S., Rational billiards and flat structures, in Handbook of dynamical systems, Vol. 1A (North-Holland, Amsterdam, 2002), 10151089.
[Mon06]Monteil, T., A homological condition for a dynamical and illuminatory classification of torus branched coverings, Preprint (2006), http://arxiv.org/abs/math/0603352.
[Pan09]Panov, D., Foliations with unbounded deviation on ${ \mathbb{T} }^{2} $, J. Mod. Dyn. 3 (2009), 589594.
[Pat76]Patterson, S. J., Diophantine approximation in Fuchsian groups, Phil. Trans. Roy. Soc. Lond. 282 (1976), 527563.
[Pet73]Petersen, K., On a series of cosecants related to a problem in ergodic theory, Compositio Math. 26 (1973), 313317.
[Ral11]Ralston, D., $1/ 2$-heavy sequences driven by rotations, Preprint (2011), http://arxiv.org/abs/1106.0577.
[Sch77]Schmidt, K., Cocycles of ergodic transformation groups, Chicago Lectures in Mathematics (MacMillan, India, 1977), available at http://www.mat.univie.ac.at/~kschmidt.
[Vee89]Veech, W. A., Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), 553583.
[Via]Viana, M., Dynamics of interval exchange maps and Teichmüller flows, Lecture notes of graduate courses taught at IMPA in 2005 and 2007. Working preliminary manuscript, http://w3.impa.br/~viana/out/ietf.pdf.
[Zor06]Zorich, A., Flat surfaces, Frontiers in number theory, physics, and geometry I (Springer, Berlin, 2006), 437583.
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