Pseudo-Anosov eigenfoliations on Panov planes

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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

Abstract
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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).
[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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