# 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
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##### References:
 [1] preprint, 2011/2012. Google Scholar [2] Ergodic Theory and Dynamical Systems, 29 (2009), 1417-1449. doi: 10.1017/S0143385708000783.  Google Scholar [3] J. Amer. Math. Soc., 17 (2004), 871-908. doi: 10.1090/S0894-0347-04-00461-8.  Google Scholar [4] arXiv:1305.1104, (2013). Google Scholar [5] Ergodic Theory Dyn. Systems, 32 (2012), 491-515. doi: 10.1017/S0143385711001003.  Google Scholar [6] Journal of Modern Dynamics, 7 (2013), 1-29. doi: 10.3934/jmd.2013.7.1.  Google Scholar [7] preprint, arXiv:1107.1810v3, Annales Scientifiques de l'Ecole Normale Supérieure, 47 (2014), 28 pp. Google Scholar [8] 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] Publications Mathématiques de l'IHÉS, (2013), 1-127, arXiv:1112.5872. Google Scholar [11] Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012.  Google Scholar [12] 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] J. Anal. Math., 112 (2010), 289-328. doi: 10.1007/s11854-010-0031-2.  Google Scholar [14] 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] Duke Math. J., 103 (2000), 191-213. doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar [21] 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] J. Reine Angew. Math., 656 (2011), 223-244. doi: 10.1515/CRELLE.2011.052.  Google Scholar [26] Compos. Math., 149 (2013), 1364-1380. doi: 10.1112/S0010437X12000887.  Google Scholar [27] 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] Duke Math. J., 66 (1992), 387-442. doi: 10.1215/S0012-7094-92-06613-0.  Google Scholar [30] in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.  Google Scholar [31] J. Amer. Math. Soc., 16 (2003), 857-885 (electronic). doi: 10.1090/S0894-0347-03-00432-6.  Google Scholar [32] Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5.  Google Scholar [33] J. Mod. Dyn., 3 (2009), 589-594. doi: 10.3934/jmd.2009.3.589.  Google Scholar [34] Topology Appl., 154 (2007), 2365-2375. doi: 10.1016/j.topol.2007.01.021.  Google Scholar [35] Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431. doi: 10.1090/S0273-0979-1988-15685-6.  Google Scholar [36] Ph.D. Thesis, The University of Chicago,, 2005.  Google Scholar [37] Invent. Math., 97 (1989), 533-683. doi: 10.1007/BF01388890.  Google Scholar [38] 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
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