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

Pseudo-Anosov eigenfoliations on Panov planes

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.
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).

[2]

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

[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.

[4]

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

[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.

[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.

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[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).

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[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}, ().

[19]

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

[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.

[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.

[22]

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

[23]

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

[24]

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

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[36]

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

[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.

[38]

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

show all references

References:
[1]

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

[2]

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

[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.

[4]

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

[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.

[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.

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[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).

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[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}, ().

[19]

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

[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.

[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.

[22]

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

[23]

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

[24]

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

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[36]

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

[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.

[38]

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

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