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 and Continuous Dynamical Systems, 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), 93-134. 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 $\mathbb{C}^2$, 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.

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), 93-134. 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 $\mathbb{C}^2$, 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.

[1]

W. Patrick Hooper. An infinite surface with the lattice property Ⅱ: Dynamics of pseudo-Anosovs. Journal of Modern Dynamics, 2019, 14: 243-276. doi: 10.3934/jmd.2019009

[2]

Renhai Wang, Bixiang Wang. Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2461-2493. doi: 10.3934/dcdsb.2020019

[3]

Kathryn Lindsey, Rodrigo Treviño. Infinite type flat surface models of ergodic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5509-5553. doi: 10.3934/dcds.2016043

[4]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281

[5]

Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353

[6]

S. Eigen, A. B. Hajian, V. S. Prasad. Universal skyscraper templates for infinite measure preserving transformations. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 343-360. doi: 10.3934/dcds.2006.16.343

[7]

Gang Bao, Junshan Lin. Near-field imaging of the surface displacement on an infinite ground plane. Inverse Problems and Imaging, 2013, 7 (2) : 377-396. doi: 10.3934/ipi.2013.7.377

[8]

Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017

[9]

Robert Carlson. Myopic models of population dynamics on infinite networks. Networks and Heterogeneous Media, 2014, 9 (3) : 477-499. doi: 10.3934/nhm.2014.9.477

[10]

Ian Melbourne, Dalia Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 2018, 12: 285-313. doi: 10.3934/jmd.2018011

[11]

Claudio Bonanno, Tanja I. Schindler. Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022113

[12]

Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051

[13]

John Erik Fornæss. Infinite dimensional complex dynamics: Quasiconjugacies, localization and quantum chaos. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 51-60. doi: 10.3934/dcds.2000.6.51

[14]

Brendan Weickert. Infinite-dimensional complex dynamics: A quantum random walk. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 517-524. doi: 10.3934/dcds.2001.7.517

[15]

Victor Zvyagin, Vladimir Orlov. On one problem of viscoelastic fluid dynamics with memory on an infinite time interval. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3855-3877. doi: 10.3934/dcdsb.2018114

[16]

Wenjing Liu, Rong Yang, Xin-Guang Yang. Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1907-1930. doi: 10.3934/cpaa.2021052

[17]

Leandro Arosio, Anna Miriam Benini, John Erik Fornæss, Han Peters. Dynamics of transcendental Hénon maps III: Infinite entropy. Journal of Modern Dynamics, 2021, 17: 465-479. doi: 10.3934/jmd.2021016

[18]

Jingyu Wang, Yejuan Wang, Tomás Caraballo. Multi-valued random dynamics of stochastic wave equations with infinite delays. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021310

[19]

Joachim von Below, José A. Lubary. Isospectral infinite graphs and networks and infinite eigenvalue multiplicities. Networks and Heterogeneous Media, 2009, 4 (3) : 453-468. doi: 10.3934/nhm.2009.4.453

[20]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (175)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]