American Institute of Mathematical Sciences

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  2014, 21: 89-108. doi: 10.3934/era.2014.21.89

Pseudo-Anosov eigenfoliations on Panov planes

Chris Johnson 1, and  Martin Schmoll 2,

1. 

Clemson University, E-1b Martin Hall, Clemson, SC 29634, United States
2. 

Clemson University, O-229 Martin Hall, Clemson, SC 29634, United States

Received  March 2012 Revised  March 2014 Published  May 2014

We study dynamical properties of direction foliations on the complex plane pulled back from direction foliations on a half-translation torus $T$, i.e., a torus equipped with a strict and integrable quadratic differential. If the torus $T$ admits a pseudo-Anosov map we give a homological criterion for the appearance of dense leaves and leaves with bounded deviation on the universal covering of $T$, called Panov plane. Our result generalizes Dmitri Panov's explicit construction of dense leaves for certain arithmetic half-translation tori [33]. Certain Panov planes are related to the polygonal table of the periodic wind-tree model. In fact, we show that the dynamics on periodic wind-tree billiards can be converted to the dynamics on a pair of singular planes.
    Possible strategies to generalize our main dynamical result to larger sets of directions are discussed. Particularly we include recent results of Frączek and Ulcigrai [17, 18] and Delecroix [6] for the wind-tree model. Implicitly Panov planes appear in Frączek and Schmoll [15], where the authors consider Eaton Lens distributions.
Mathematics Subject Classification: 14H15, 14H52, 30F30, 30F60, 37A60, 37C35, 37N20, 58D15, 58D2.
Citation: Chris Johnson, Martin Schmoll. Pseudo-Anosov eigenfoliations on Panov planes. Electronic Research Announcements, 2014, 21: 89-108. doi: 10.3934/era.2014.21.89
References:
[1]

A. Avila and P. Hubert, Recurrence for the wind-tree model,, preprint, (2011).   Google Scholar

[2]

P. Boyland, Transitivity of surface dynamics lifted to abelian covers,, \emph{Ergodic Theory and Dynamical Systems}, 29 (2009), 1417.  doi: 10.1017/S0143385708000783.  Google Scholar

[3]

K. Calta, Veech surfaces and complete periodicity in genus two,, \emph{J. Amer. Math. Soc.}, 17 (2004), 871.  doi: 10.1090/S0894-0347-04-00461-8.  Google Scholar

[4]

J. Chaika and A. Eskin, Every flat surface is Birkhoff and Osceledets generic in almost every direction,, \arXiv{1305.1104}, (2013).   Google Scholar

[5]

J.-P. Conze and E. Gutkin, On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces,, \emph{Ergodic Theory Dyn. Systems}, 32 (2012), 491.  doi: 10.1017/S0143385711001003.  Google Scholar

[6]

V. Delecroix, Divergent directions in some periodic wind-tree models,, \emph{Journal of Modern Dynamics}, 7 (2013), 1.  doi: 10.3934/jmd.2013.7.1.  Google Scholar

[7]

V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model,, preprint, 47 (2014).   Google Scholar

[8]

P. Ehrenfest and T. Ehrenfest, The Conceptual Foundations of the Statistical Approach in Mechanics,, Translated from the German by Michael J. Moravcsik, (1959).   Google Scholar

[9]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $\slr$ action on moduli space,, \arXiv{1302.3320}., ().   Google Scholar

[10]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, \emph{Publications Mathématiques de l'IHÉS}, (2013), 1.   Google Scholar

[11]

B. Farb and D. Margalit, A Primer on Mapping Class Groups,, Princeton Mathematical Series, (2012).   Google Scholar

[12]

A. Fathi, F. Laudenbach and V. Poénaru, Thurston's Work on Surfaces,, Translation from the 1979 French original by Djun M. Kim and Dan Margalit, (1979).   Google Scholar

[13]

S. Ferenczi and L. Q. Zamboni, Structure of K-interval-exchange transformations: Induction, trajectories, and distance theorems,, \emph{J. Anal. Math.}, 112 (2010), 289.  doi: 10.1007/s11854-010-0031-2.  Google Scholar

[14]

S. Ferenczi and L. Q. Zamboni, Eigenvalues and simplicity of interval-exchange transformations,, \emph{Ann. Sci. Éc. Norm. Sup. (4)}, 44 (2011), 361.   Google Scholar

[15]

K. Frączek and M. Schmoll, Directional localization of light rays in a periodic array of retro-reflector lenses,, to appear in \emph{Nonlinearity}., ().   Google Scholar

[16]

K. Frączek and M. Schmoll, Dynamics on quadratic differentials in the determinant locus,, in preparation., ().   Google Scholar

[17]

K. Frączek and C. Ulcigrai, Non-ergodic $\Z$-periodic billiards and infinite translation surfaces,, to appear in \emph{Inventiones Math.}, ().   Google Scholar

[18]

K. Frączek and C. Ulcigrai, Ergodic directions for billiards in a strip with periodically located obstacles,, to appear in \emph{Communications in Mathematical Physics}, ().   Google Scholar

[19]

J. Grivaux and P. Hubert, Loci in strata of meromorphic differentials with fully degenerate Lyapunov spectrum,, \arXiv{1307.3481v1}., ().   Google Scholar

[20]

E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, \emph{Duke Math. J.}, 103 (2000), 191.  doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar

[21]

J. Hardy and J. Weber, Diffusion in a periodic wind-tree model,, \emph{J. Math. Phys.}, 21 (1980), 1802.  doi: 10.1063/1.524633.  Google Scholar

[22]

W. P. Hooper, The invariant measures of some infinite interval exchange maps,, \arXiv{1005.1902}., ().   Google Scholar

[23]

W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, \arXiv{0905.3736v1}., ().   Google Scholar

[24]

, P. Hubert,, Oral communication., ().   Google Scholar

[25]

P. Hubert, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion,, \emph{J. Reine Angew. Math.}, 656 (2011), 223.  doi: 10.1515/CRELLE.2011.052.  Google Scholar

[26]

P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces,, \emph{Compos. Math.}, 149 (2013), 1364.  doi: 10.1112/S0010437X12000887.  Google Scholar

[27]

C. Johnson and M. Schmoll, Hyperelliptic translation surfaces and folded tori,, \emph{Topology and its Applications}, 161 (2014), 73.  doi: 10.1016/j.topol.2013.09.010.  Google Scholar

[28]

C. Johnson and M. Schmoll, Dynamics on Panov planes,, in final preparation., ().   Google Scholar

[29]

H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential,, \emph{Duke Math. J.}, 66 (1992), 387.  doi: 10.1215/S0012-7094-92-06613-0.  Google Scholar

[30]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in \emph{Handbook of Dynamical Systems, (2002), 1015.  doi: 10.1016/S1874-575X(02)80015-7.  Google Scholar

[31]

C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces,, \emph{J. Amer. Math. Soc.}, 16 (2003), 857.  doi: 10.1090/S0894-0347-03-00432-6.  Google Scholar

[32]

C. T. McMullen, Prym varieties and Teichmüller curves,, \emph{Duke Math. J.}, 133 (2006), 569.  doi: 10.1215/S0012-7094-06-13335-5.  Google Scholar

[33]

D. Panov, Foliations with unbounded deviation on $\mathbbT^2$,, \emph{J. Mod. Dyn.}, 3 (2009), 589.  doi: 10.3934/jmd.2009.3.589.  Google Scholar

[34]

M. Pollicott and R. Sharp, Pseudo-Anosov foliations on periodic surfaces,, \emph{Topology Appl.}, 154 (2007), 2365.  doi: 10.1016/j.topol.2007.01.021.  Google Scholar

[35]

W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 19 (1988), 417.  doi: 10.1090/S0273-0979-1988-15685-6.  Google Scholar

[36]

S. Vasilyev, Genus two Veech Surfaces Arising from General Quadratic Differentials,, Ph.D. Thesis, (2005).   Google Scholar

[37]

W. Veech, Teichmüller curves in the moduli space, Eisenstein series and an application to triangular billiards,, \emph{Invent. Math.}, 97 (1989), 533.  doi: 10.1007/BF01388890.  Google Scholar

[38]

A. Zorich, Flat surfaces,, in \emph{Frontiers in Number Theory, (2006), 437.  doi: 10.1007/978-3-540-31347-2_13.  Google Scholar

show all references

References:
[1]

A. Avila and P. Hubert, Recurrence for the wind-tree model,, preprint, (2011).   Google Scholar

[2]

P. Boyland, Transitivity of surface dynamics lifted to abelian covers,, \emph{Ergodic Theory and Dynamical Systems}, 29 (2009), 1417.  doi: 10.1017/S0143385708000783.  Google Scholar

[3]

K. Calta, Veech surfaces and complete periodicity in genus two,, \emph{J. Amer. Math. Soc.}, 17 (2004), 871.  doi: 10.1090/S0894-0347-04-00461-8.  Google Scholar

[4]

J. Chaika and A. Eskin, Every flat surface is Birkhoff and Osceledets generic in almost every direction,, \arXiv{1305.1104}, (2013).   Google Scholar

[5]

J.-P. Conze and E. Gutkin, On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces,, \emph{Ergodic Theory Dyn. Systems}, 32 (2012), 491.  doi: 10.1017/S0143385711001003.  Google Scholar

[6]

V. Delecroix, Divergent directions in some periodic wind-tree models,, \emph{Journal of Modern Dynamics}, 7 (2013), 1.  doi: 10.3934/jmd.2013.7.1.  Google Scholar

[7]

V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model,, preprint, 47 (2014).   Google Scholar

[8]

P. Ehrenfest and T. Ehrenfest, The Conceptual Foundations of the Statistical Approach in Mechanics,, Translated from the German by Michael J. Moravcsik, (1959).   Google Scholar

[9]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $\slr$ action on moduli space,, \arXiv{1302.3320}., ().   Google Scholar

[10]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, \emph{Publications Mathématiques de l'IHÉS}, (2013), 1.   Google Scholar

[11]

B. Farb and D. Margalit, A Primer on Mapping Class Groups,, Princeton Mathematical Series, (2012).   Google Scholar

[12]

A. Fathi, F. Laudenbach and V. Poénaru, Thurston's Work on Surfaces,, Translation from the 1979 French original by Djun M. Kim and Dan Margalit, (1979).   Google Scholar

[13]

S. Ferenczi and L. Q. Zamboni, Structure of K-interval-exchange transformations: Induction, trajectories, and distance theorems,, \emph{J. Anal. Math.}, 112 (2010), 289.  doi: 10.1007/s11854-010-0031-2.  Google Scholar

[14]

S. Ferenczi and L. Q. Zamboni, Eigenvalues and simplicity of interval-exchange transformations,, \emph{Ann. Sci. Éc. Norm. Sup. (4)}, 44 (2011), 361.   Google Scholar

[15]

K. Frączek and M. Schmoll, Directional localization of light rays in a periodic array of retro-reflector lenses,, to appear in \emph{Nonlinearity}., ().   Google Scholar

[16]

K. Frączek and M. Schmoll, Dynamics on quadratic differentials in the determinant locus,, in preparation., ().   Google Scholar

[17]

K. Frączek and C. Ulcigrai, Non-ergodic $\Z$-periodic billiards and infinite translation surfaces,, to appear in \emph{Inventiones Math.}, ().   Google Scholar

[18]

K. Frączek and C. Ulcigrai, Ergodic directions for billiards in a strip with periodically located obstacles,, to appear in \emph{Communications in Mathematical Physics}, ().   Google Scholar

[19]

J. Grivaux and P. Hubert, Loci in strata of meromorphic differentials with fully degenerate Lyapunov spectrum,, \arXiv{1307.3481v1}., ().   Google Scholar

[20]

E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, \emph{Duke Math. J.}, 103 (2000), 191.  doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar

[21]

J. Hardy and J. Weber, Diffusion in a periodic wind-tree model,, \emph{J. Math. Phys.}, 21 (1980), 1802.  doi: 10.1063/1.524633.  Google Scholar

[22]

W. P. Hooper, The invariant measures of some infinite interval exchange maps,, \arXiv{1005.1902}., ().   Google Scholar

[23]

W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, \arXiv{0905.3736v1}., ().   Google Scholar

[24]

, P. Hubert,, Oral communication., ().   Google Scholar

[25]

P. Hubert, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion,, \emph{J. Reine Angew. Math.}, 656 (2011), 223.  doi: 10.1515/CRELLE.2011.052.  Google Scholar

[26]

P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces,, \emph{Compos. Math.}, 149 (2013), 1364.  doi: 10.1112/S0010437X12000887.  Google Scholar

[27]

C. Johnson and M. Schmoll, Hyperelliptic translation surfaces and folded tori,, \emph{Topology and its Applications}, 161 (2014), 73.  doi: 10.1016/j.topol.2013.09.010.  Google Scholar

[28]

C. Johnson and M. Schmoll, Dynamics on Panov planes,, in final preparation., ().   Google Scholar

[29]

H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential,, \emph{Duke Math. J.}, 66 (1992), 387.  doi: 10.1215/S0012-7094-92-06613-0.  Google Scholar

[30]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in \emph{Handbook of Dynamical Systems, (2002), 1015.  doi: 10.1016/S1874-575X(02)80015-7.  Google Scholar

[31]

C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces,, \emph{J. Amer. Math. Soc.}, 16 (2003), 857.  doi: 10.1090/S0894-0347-03-00432-6.  Google Scholar

[32]

C. T. McMullen, Prym varieties and Teichmüller curves,, \emph{Duke Math. J.}, 133 (2006), 569.  doi: 10.1215/S0012-7094-06-13335-5.  Google Scholar

[33]

D. Panov, Foliations with unbounded deviation on $\mathbbT^2$,, \emph{J. Mod. Dyn.}, 3 (2009), 589.  doi: 10.3934/jmd.2009.3.589.  Google Scholar

[34]

M. Pollicott and R. Sharp, Pseudo-Anosov foliations on periodic surfaces,, \emph{Topology Appl.}, 154 (2007), 2365.  doi: 10.1016/j.topol.2007.01.021.  Google Scholar

[35]

W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 19 (1988), 417.  doi: 10.1090/S0273-0979-1988-15685-6.  Google Scholar

[36]

S. Vasilyev, Genus two Veech Surfaces Arising from General Quadratic Differentials,, Ph.D. Thesis, (2005).   Google Scholar

[37]

W. Veech, Teichmüller curves in the moduli space, Eisenstein series and an application to triangular billiards,, \emph{Invent. Math.}, 97 (1989), 533.  doi: 10.1007/BF01388890.  Google Scholar

[38]

A. Zorich, Flat surfaces,, in \emph{Frontiers in Number Theory, (2006), 437.  doi: 10.1007/978-3-540-31347-2_13.  Google Scholar

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