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On Helly's theorem in geodesic spaces
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 |
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.
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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