# American Institute of Mathematical Sciences

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, 2011/2012. Google Scholar [2] P. Boyland, Transitivity of surface dynamics lifted to abelian covers, Ergodic Theory and Dynamical Systems, 29 (2009), 1417-1449. doi: 10.1017/S0143385708000783.  Google Scholar [3] K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908. 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, Ergodic Theory Dyn. Systems, 32 (2012), 491-515. doi: 10.1017/S0143385711001003.  Google Scholar [6] V. Delecroix, Divergent directions in some periodic wind-tree models, Journal of Modern Dynamics, 7 (2013), 1-29. 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, arXiv:1107.1810v3, Annales Scientifiques de l'Ecole Normale Supérieure, 47 (2014), 28 pp. 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, Reprint of the 1959 English edition, Dover Publications, Inc., New York, 1990.  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, Publications Mathématiques de l'IHÉS, (2013), 1-127, arXiv:1112.5872. Google Scholar [11] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 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, Mathematical Notes, 48, Princeton University Press, Princeton, NJ, 2012.  Google Scholar [13] S. Ferenczi and L. Q. Zamboni, Structure of K-interval-exchange transformations: Induction, trajectories, and distance theorems, J. Anal. Math., 112 (2010), 289-328. doi: 10.1007/s11854-010-0031-2.  Google Scholar [14] S. Ferenczi and L. Q. Zamboni, Eigenvalues and simplicity of interval-exchange transformations, Ann. Sci. Éc. Norm. Sup. (4), 44 (2011), 361-392.  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, Duke Math. J., 103 (2000), 191-213. doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar [21] J. Hardy and J. Weber, Diffusion in a periodic wind-tree model, J. Math. Phys., 21 (1980), 1802-1808. 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, J. Reine Angew. Math., 656 (2011), 223-244. doi: 10.1515/CRELLE.2011.052.  Google Scholar [26] P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces, Compos. Math., 149 (2013), 1364-1380. doi: 10.1112/S0010437X12000887.  Google Scholar [27] C. Johnson and M. Schmoll, Hyperelliptic translation surfaces and folded tori, Topology and its Applications, 161 (2014), 73-94. 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, Duke Math. J., 66 (1992), 387-442. doi: 10.1215/S0012-7094-92-06613-0.  Google Scholar [30] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.  Google Scholar [31] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 857-885 (electronic). doi: 10.1090/S0894-0347-03-00432-6.  Google Scholar [32] C. T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5.  Google Scholar [33] D. Panov, Foliations with unbounded deviation on $\mathbbT^2$, J. Mod. Dyn., 3 (2009), 589-594. doi: 10.3934/jmd.2009.3.589.  Google Scholar [34] M. Pollicott and R. Sharp, Pseudo-Anosov foliations on periodic surfaces, Topology Appl., 154 (2007), 2365-2375. doi: 10.1016/j.topol.2007.01.021.  Google Scholar [35] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431. 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, The University of Chicago,, 2005.  Google Scholar [37] W. Veech, Teichmüller curves in the moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 533-683. doi: 10.1007/BF01388890.  Google Scholar [38] A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, 437-583. 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, 2011/2012. Google Scholar [2] P. Boyland, Transitivity of surface dynamics lifted to abelian covers, Ergodic Theory and Dynamical Systems, 29 (2009), 1417-1449. doi: 10.1017/S0143385708000783.  Google Scholar [3] K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908. 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, Ergodic Theory Dyn. Systems, 32 (2012), 491-515. doi: 10.1017/S0143385711001003.  Google Scholar [6] V. Delecroix, Divergent directions in some periodic wind-tree models, Journal of Modern Dynamics, 7 (2013), 1-29. 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, arXiv:1107.1810v3, Annales Scientifiques de l'Ecole Normale Supérieure, 47 (2014), 28 pp. 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, Reprint of the 1959 English edition, Dover Publications, Inc., New York, 1990.  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, Publications Mathématiques de l'IHÉS, (2013), 1-127, arXiv:1112.5872. Google Scholar [11] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 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, Mathematical Notes, 48, Princeton University Press, Princeton, NJ, 2012.  Google Scholar [13] S. Ferenczi and L. Q. Zamboni, Structure of K-interval-exchange transformations: Induction, trajectories, and distance theorems, J. Anal. Math., 112 (2010), 289-328. doi: 10.1007/s11854-010-0031-2.  Google Scholar [14] S. Ferenczi and L. Q. Zamboni, Eigenvalues and simplicity of interval-exchange transformations, Ann. Sci. Éc. Norm. Sup. (4), 44 (2011), 361-392.  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, Duke Math. J., 103 (2000), 191-213. doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar [21] J. Hardy and J. Weber, Diffusion in a periodic wind-tree model, J. Math. Phys., 21 (1980), 1802-1808. 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, J. Reine Angew. Math., 656 (2011), 223-244. doi: 10.1515/CRELLE.2011.052.  Google Scholar [26] P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces, Compos. Math., 149 (2013), 1364-1380. doi: 10.1112/S0010437X12000887.  Google Scholar [27] C. Johnson and M. Schmoll, Hyperelliptic translation surfaces and folded tori, Topology and its Applications, 161 (2014), 73-94. 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, Duke Math. J., 66 (1992), 387-442. doi: 10.1215/S0012-7094-92-06613-0.  Google Scholar [30] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.  Google Scholar [31] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 857-885 (electronic). doi: 10.1090/S0894-0347-03-00432-6.  Google Scholar [32] C. T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5.  Google Scholar [33] D. Panov, Foliations with unbounded deviation on $\mathbbT^2$, J. Mod. Dyn., 3 (2009), 589-594. doi: 10.3934/jmd.2009.3.589.  Google Scholar [34] M. Pollicott and R. Sharp, Pseudo-Anosov foliations on periodic surfaces, Topology Appl., 154 (2007), 2365-2375. doi: 10.1016/j.topol.2007.01.021.  Google Scholar [35] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431. 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, The University of Chicago,, 2005.  Google Scholar [37] W. Veech, Teichmüller curves in the moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 533-683. doi: 10.1007/BF01388890.  Google Scholar [38] A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, 437-583. doi: 10.1007/978-3-540-31347-2_13.  Google Scholar
 [1] Kariane Calta, Thomas A. Schmidt. Infinitely many lattice surfaces with special pseudo-Anosov maps. Journal of Modern Dynamics, 2013, 7 (2) : 239-254. doi: 10.3934/jmd.2013.7.239 [2] S. Öykü Yurttaş. Dynnikov and train track transition matrices of pseudo-Anosov braids. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 541-570. doi: 10.3934/dcds.2016.36.541 [3] Hieu Trung Do, Thomas A. Schmidt. New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant. Journal of Modern Dynamics, 2016, 10: 541-561. doi: 10.3934/jmd.2016.10.541 [4] Juan Alonso, Nancy Guelman, Juliana Xavier. Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1817-1827. doi: 10.3934/dcds.2015.35.1817 [5] Todd A. Drumm and William M. Goldman. Crooked planes. Electronic Research Announcements, 1995, 1: 10-17. [6] João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837 [7] Osama Khalil. Geodesic planes in geometrically finite manifolds. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 881-903. doi: 10.3934/dcds.2019037 [8] Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287 [9] Ivan Landjev. On blocking sets in projective Hjelmslev planes. Advances in Mathematics of Communications, 2007, 1 (1) : 65-81. doi: 10.3934/amc.2007.1.65 [10] Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39 [11] J. D. Key, T. P. McDonough, V. C. Mavron. Codes from hall planes of odd order. Advances in Mathematics of Communications, 2017, 11 (1) : 179-185. doi: 10.3934/amc.2017011 [12] Dominic Veconi. Equilibrium states of almost Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 767-780. doi: 10.3934/dcds.2020061 [13] Tracy L. Payne. Anosov automorphisms of nilpotent Lie algebras. Journal of Modern Dynamics, 2009, 3 (1) : 121-158. doi: 10.3934/jmd.2009.3.121 [14] Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801 [15] Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185 [16] Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015 [17] Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201 [18] Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235 [19] Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082 [20] Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1871-1897. doi: 10.3934/dcdss.2020462

2019 Impact Factor: 0.5