Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology

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September 2016, 36(9): 4619-4635. doi: 10.3934/dcds.2016001

Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology

Fernando Alcalde Cuesta ^1, , Françoise Dal'Bo ^2, , Matilde Martínez ^3, and Alberto Verjovsky ^4,

1.	GeoDynApp - ECSING Group, Spain
2.	Institut de Recherche Mathématiques de Rennes, Université de Rennes 1, F-35042 Rennes, France
3.	Instituto de Matemática y Estadística Rafael Laguardia, Facultad de Ingeniería, Universidad de la República, J. Herrera y Reissig 565, C.P. 11300 Montevideo
4.	Universidad Nacional Autónoma de México, Apartado Postal 273, Admon. de correos #3, C.P. 62251 Cuernavaca, Morelos

Received June 2015 Revised March 2016 Published May 2016

Abstract
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We consider a minimal compact lamination by hyperbolic surfaces. We prove that if no leaf is simply connected, then the horocycle flow on its unitary tangent bundle is minimal.

Keywords: geometrically infinite surfaces, minimality., hyperbolic laminations, horocycle flows, Hyperbolic surfaces.

Mathematics Subject Classification: 37C85, 37D40, 57R3.

Citation: Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4619-4635. doi: 10.3934/dcds.2016001

References:

[1]	F. Alcalde Cuesta and F. Dal'Bo, Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces, Expo. Math., 33 (2015), 431-451. doi: 10.1016/j.exmath.2015.07.006.
[2]	S. Alvarez and P. Lessa, The Teichmüller space of the Hirsch foliation, preprint, arXiv:1507.01531.
[3]	Ch. Bonatti, X. Gómez-Mont and R. Vila-Freyer, Statistical behaviour of the leaves of Riccati foliations, Ergodic Theory Dynam. Systems, 30 (2010), 67-96. doi: 10.1017/S0143385708001028.
[4]	A. Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup., 26 (1993), 489-516.
[5]	A. Candel and L. Conlon, Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000.
[6]	J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom., 17 (1982), 15-53.
[7]	F. Dal'bo, Topologie du feuilletage fortement stable, Ann. Inst. Fourier (Grenoble), 50 (2000), 981-993. doi: 10.5802/aif.1781.
[8]	F. Dal'Bo, Geodesic and Horocyclic Trajectories, Universitext. Translated from the 2007 French original, Springer-Verlag London, 2011. doi: 10.1007/978-0-85729-073-1.
[9]	D. B. A. Epstein, K. C. Millett and D. Tischler, Leaves without holonomy, J. London Math. Soc. (2), 16 (1977), 548-552.
[10]	G. Hector, Feuilletages en cylindres, in Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., Vol. 597, Springer, Berlin, (1977), 252-270.
[11]	G. Hector, S. Matsumoto and G. Meigniez, Ends of leaves of Lie foliations, J. Math. Soc. Japan, 57 (2005), 753-779. doi: 10.2969/jmsj/1158241934.
[12]	G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542. doi: 10.1215/S0012-7094-36-00246-6.
[13]	M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.
[14]	S. Hurder, Ergodic theory of foliations and a theorem of Sacksteder, in Dynamical systems (College Park, MD, 1986-87), Lecture Notes in Math., Springer, Berlin, 1342 (1988), 291-328. doi: 10.1007/BFb0082838.
[15]	V. A. Kaimanovich, Ergodic properties of the horocycle flow and classification of Fuchsian groups, J. Dynam. Control Systems, 6 (2000), 21-56. doi: 10.1023/A:1009517621605.
[16]	M. Kulikov, The horocycle flow without minimal sets, C. R. Math. Acad. Sci. Paris, 338 (2004), 477-480. doi: 10.1016/j.crma.2003.12.027.
[17]	M. Martínez, S. Matsumoto and A. Verjovsky, Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem, to appear in Journal of Modern Dynamics, 10 (2016).
[18]	S. Matsumoto, Dynamical systems without minimal sets, preprint, http://gf2003.ms.u-tokyo.ac.jp/abstract/data/GF2003/1058425808-matsumo.ps.
[19]	S. Matsumoto, Horocycle flows without minimal sets, preprint, arXiv:1412.0410v2.
[20]	T. Roblin, Ergodicitéet équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003), vi+96pp.
[21]	A. Sambusetti, Asymptotic properties of coverings in negative curvature, Geom. Topol., 12 (2008), 617-637. doi: 10.2140/gt.2008.12.617.
[22]	O. Sarig, The horocyclic flow and the Laplacian on hyperbolic surfaces of infinite genus, Geom. Funct. Anal., 19 (2010), 1757-1812. doi: 10.1007/s00039-010-0048-9.
[23]	A. N. Starkov, Fuchsian groups from the dynamical viewpoint, J. Dynam. Control Systems, 1 (1995), 427-445. doi: 10.1007/BF02269378.
[24]	A. Verjovsky, A uniformization theorem for holomorphic foliations, in The Lefschetz centennial conference, Part III (Mexico City, 1984), Contemp. Math. 58, Amer. Math. Soc., Providence, RI, 1987, 233-253.

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References:

[1]	F. Alcalde Cuesta and F. Dal'Bo, Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces, Expo. Math., 33 (2015), 431-451. doi: 10.1016/j.exmath.2015.07.006.
[2]	S. Alvarez and P. Lessa, The Teichmüller space of the Hirsch foliation, preprint, arXiv:1507.01531.
[3]	Ch. Bonatti, X. Gómez-Mont and R. Vila-Freyer, Statistical behaviour of the leaves of Riccati foliations, Ergodic Theory Dynam. Systems, 30 (2010), 67-96. doi: 10.1017/S0143385708001028.
[4]	A. Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup., 26 (1993), 489-516.
[5]	A. Candel and L. Conlon, Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000.
[6]	J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom., 17 (1982), 15-53.
[7]	F. Dal'bo, Topologie du feuilletage fortement stable, Ann. Inst. Fourier (Grenoble), 50 (2000), 981-993. doi: 10.5802/aif.1781.
[8]	F. Dal'Bo, Geodesic and Horocyclic Trajectories, Universitext. Translated from the 2007 French original, Springer-Verlag London, 2011. doi: 10.1007/978-0-85729-073-1.
[9]	D. B. A. Epstein, K. C. Millett and D. Tischler, Leaves without holonomy, J. London Math. Soc. (2), 16 (1977), 548-552.
[10]	G. Hector, Feuilletages en cylindres, in Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., Vol. 597, Springer, Berlin, (1977), 252-270.
[11]	G. Hector, S. Matsumoto and G. Meigniez, Ends of leaves of Lie foliations, J. Math. Soc. Japan, 57 (2005), 753-779. doi: 10.2969/jmsj/1158241934.
[12]	G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542. doi: 10.1215/S0012-7094-36-00246-6.
[13]	M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.
[14]	S. Hurder, Ergodic theory of foliations and a theorem of Sacksteder, in Dynamical systems (College Park, MD, 1986-87), Lecture Notes in Math., Springer, Berlin, 1342 (1988), 291-328. doi: 10.1007/BFb0082838.
[15]	V. A. Kaimanovich, Ergodic properties of the horocycle flow and classification of Fuchsian groups, J. Dynam. Control Systems, 6 (2000), 21-56. doi: 10.1023/A:1009517621605.
[16]	M. Kulikov, The horocycle flow without minimal sets, C. R. Math. Acad. Sci. Paris, 338 (2004), 477-480. doi: 10.1016/j.crma.2003.12.027.
[17]	M. Martínez, S. Matsumoto and A. Verjovsky, Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem, to appear in Journal of Modern Dynamics, 10 (2016).
[18]	S. Matsumoto, Dynamical systems without minimal sets, preprint, http://gf2003.ms.u-tokyo.ac.jp/abstract/data/GF2003/1058425808-matsumo.ps.
[19]	S. Matsumoto, Horocycle flows without minimal sets, preprint, arXiv:1412.0410v2.
[20]	T. Roblin, Ergodicitéet équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003), vi+96pp.
[21]	A. Sambusetti, Asymptotic properties of coverings in negative curvature, Geom. Topol., 12 (2008), 617-637. doi: 10.2140/gt.2008.12.617.
[22]	O. Sarig, The horocyclic flow and the Laplacian on hyperbolic surfaces of infinite genus, Geom. Funct. Anal., 19 (2010), 1757-1812. doi: 10.1007/s00039-010-0048-9.
[23]	A. N. Starkov, Fuchsian groups from the dynamical viewpoint, J. Dynam. Control Systems, 1 (1995), 427-445. doi: 10.1007/BF02269378.
[24]	A. Verjovsky, A uniformization theorem for holomorphic foliations, in The Lefschetz centennial conference, Part III (Mexico City, 1984), Contemp. Math. 58, Amer. Math. Soc., Providence, RI, 1987, 233-253.

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