Dynamics on the infinite staircase

W. Patrick Hooper; Pascal Hubert; Barak Weiss

doi:10.3934/dcds.2013.33.4341

Article Contents

2013, Volume 33, Issue 9: 4341-4347. Doi: 10.3934/dcds.2013.33.4341

This issue Previous Article Some advances on generic properties of the Oseledets splitting Next Article

Dynamics on the infinite staircase

1.
Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY, USA 10031
2.
LATP, case cour A, Faculté des sciences Saint Jérôme, Avenue Escadrille Normandie Niemen, 13397 Marseille cedex 20
3.
Ben Gurion University, Be'er Sheva, Israel 84105

Received: July 2010

Revised: February 2011

Published: September 2013

Abstract / Introduction Related Papers Cited by

Abstract

For the 'infinite staircase' square tiled surface we classify the Radon invariant measures for the straight line flow, obtaining an analogue of the celebrated Veech dichotomy for an infinite genus lattice surface. The ergodic Radon measures arise from Lebesgue measure on a one parameter family of deformations of the surface. The staircase is a $\mathbb{Z}$-cover of the torus, reducing the question to the well-studied cylinder map.

Keywords:

Mathematics Subject Classification: Primary: 37D50; Secondary: 14H30, 30F30, 30F35, 37F30.

Citation:

Related Papers

Cited by

References

[1]	J. Aaronson, H. Nakada, O. Sarig and R. Solomyak, Invariant measures and asymptotics for some skew products, Isr. J. Math., 128 (2002), 93-134.doi: 10.1007/BF02785420.
[2]	J. P. Conze, Equirépartition et ergodicité de transformations cylindriques, in "Séminaire de Probabilité, I" (Univ. Rennes, Rennes, 1976),Exp. No. 2, Dépt. Math. Informat., Univ. Rennes, Rennes, (1976), 21 pp.
[3]	E. Gutkin, Billiards on almost integrable polyhedral surfaces, Erg. Th. Dyn. Sys., 4 (1984), 569-584.doi: 10.1017/S0143385700002650.
[4]	B. Hasselblatt and A. Katok, "Intoduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
[5]	M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle des rotations, Inst. Hautes Etudes Sci. Publ. Math., 49 (1979), 5-233.
[6]	W. P. Hooper, Dynamics on an infinite surface with the lattice property, preprint, (2007), arXiv:0802.0189.
[7]	W. P. Hooper, The invariant measures of some infinite interval exchange maps, preprint, (2010), arXiv:1005.1902.
[8]	D. Maharam, Incompressible transformations, Fund. Math., 56 (1964), 35-50.
[9]	H. Nakada, Piecewise linear homeomorphisms of type III and the ergodicity of cylinder flows, Keio Math. Sem. Rep. No., 7 (1982), 29-40.
[10]	F. Valdez, Billiards in polygons and homogeneous foliations on $\mathbbC^2$, Ergod. Th. & Dynam. Sys., 29 (2009), 255-271.doi: 10.1017/S0143385708000151.
[11]	W. A. Veech, Boshernitzan's criterion for unique ergodicity of an interval exchange transformation, Ergod. Th. & Dynam. Sys., 7 (1987), 149-153.doi: 10.1017/S0143385700003862.
[12]	W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.doi: 10.1007/BF01388890.