September  2013, 33(9): 4341-4347. doi: 10.3934/dcds.2013.33.4341

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, United States

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, Israel

Received  July 2010 Revised  February 2011 Published  March 2013

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.
Citation: W. Patrick Hooper, Pascal Hubert, Barak Weiss. Dynamics on the infinite staircase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4341-4347. doi: 10.3934/dcds.2013.33.4341
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.  doi: 10.1007/BF02785420.  Google Scholar

[2]

J. P. Conze, Equirépartition et ergodicité de transformations cylindriques,, in, (1976).   Google Scholar

[3]

E. Gutkin, Billiards on almost integrable polyhedral surfaces,, Erg. Th. Dyn. Sys., 4 (1984), 569.  doi: 10.1017/S0143385700002650.  Google Scholar

[4]

B. Hasselblatt and A. Katok, "Intoduction to the Modern Theory of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995).   Google Scholar

[5]

M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle des rotations,, Inst. Hautes Etudes Sci. Publ. Math., 49 (1979), 5.   Google Scholar

[6]

W. P. Hooper, Dynamics on an infinite surface with the lattice property,, preprint, (2007).   Google Scholar

[7]

W. P. Hooper, The invariant measures of some infinite interval exchange maps,, preprint, (2010).   Google Scholar

[8]

D. Maharam, Incompressible transformations,, Fund. Math., 56 (1964), 35.   Google Scholar

[9]

H. Nakada, Piecewise linear homeomorphisms of type III and the ergodicity of cylinder flows,, Keio Math. Sem. Rep. No., 7 (1982), 29.   Google Scholar

[10]

F. Valdez, Billiards in polygons and homogeneous foliations on $\mathbbC^2$,, Ergod. Th. & Dynam. Sys., 29 (2009), 255.  doi: 10.1017/S0143385708000151.  Google Scholar

[11]

W. A. Veech, Boshernitzan's criterion for unique ergodicity of an interval exchange transformation,, Ergod. Th. & Dynam. Sys., 7 (1987), 149.  doi: 10.1017/S0143385700003862.  Google Scholar

[12]

W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Invent. Math., 97 (1989), 553.  doi: 10.1007/BF01388890.  Google Scholar

show all references

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.  doi: 10.1007/BF02785420.  Google Scholar

[2]

J. P. Conze, Equirépartition et ergodicité de transformations cylindriques,, in, (1976).   Google Scholar

[3]

E. Gutkin, Billiards on almost integrable polyhedral surfaces,, Erg. Th. Dyn. Sys., 4 (1984), 569.  doi: 10.1017/S0143385700002650.  Google Scholar

[4]

B. Hasselblatt and A. Katok, "Intoduction to the Modern Theory of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995).   Google Scholar

[5]

M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle des rotations,, Inst. Hautes Etudes Sci. Publ. Math., 49 (1979), 5.   Google Scholar

[6]

W. P. Hooper, Dynamics on an infinite surface with the lattice property,, preprint, (2007).   Google Scholar

[7]

W. P. Hooper, The invariant measures of some infinite interval exchange maps,, preprint, (2010).   Google Scholar

[8]

D. Maharam, Incompressible transformations,, Fund. Math., 56 (1964), 35.   Google Scholar

[9]

H. Nakada, Piecewise linear homeomorphisms of type III and the ergodicity of cylinder flows,, Keio Math. Sem. Rep. No., 7 (1982), 29.   Google Scholar

[10]

F. Valdez, Billiards in polygons and homogeneous foliations on $\mathbbC^2$,, Ergod. Th. & Dynam. Sys., 29 (2009), 255.  doi: 10.1017/S0143385708000151.  Google Scholar

[11]

W. A. Veech, Boshernitzan's criterion for unique ergodicity of an interval exchange transformation,, Ergod. Th. & Dynam. Sys., 7 (1987), 149.  doi: 10.1017/S0143385700003862.  Google Scholar

[12]

W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Invent. Math., 97 (1989), 553.  doi: 10.1007/BF01388890.  Google Scholar

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