# American Institute of Mathematical Sciences

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

 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

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.
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 & Continuous Dynamical Systems - A, 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. doi: 10.1016/j.exmath.2015.07.006. Google Scholar [2] S. Alvarez and P. Lessa, The Teichmüller space of the Hirsch foliation,, preprint, (). Google Scholar [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. doi: 10.1017/S0143385708001028. Google Scholar [4] A. Candel, Uniformization of surface laminations,, Ann. Sci. École Norm. Sup., 26 (1993), 489. Google Scholar [5] A. Candel and L. Conlon, Foliations. I,, Graduate Studies in Mathematics, (2000). Google Scholar [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. Google Scholar [7] F. Dal'bo, Topologie du feuilletage fortement stable,, Ann. Inst. Fourier (Grenoble), 50 (2000), 981. doi: 10.5802/aif.1781. Google Scholar [8] F. Dal'Bo, Geodesic and Horocyclic Trajectories,, Universitext. Translated from the 2007 French original, (2007). doi: 10.1007/978-0-85729-073-1. Google Scholar [9] D. B. A. Epstein, K. C. Millett and D. Tischler, Leaves without holonomy,, J. London Math. Soc. (2), 16 (1977), 548. Google Scholar [10] G. Hector, Feuilletages en cylindres,, in Geometry and topology (Proc. III Latin Amer. School of Math., (1977), 252. Google Scholar [11] G. Hector, S. Matsumoto and G. Meigniez, Ends of leaves of Lie foliations,, J. Math. Soc. Japan, 57 (2005), 753. doi: 10.2969/jmsj/1158241934. Google Scholar [12] G. A. Hedlund, Fuchsian groups and transitive horocycles,, Duke Math. J., 2 (1936), 530. doi: 10.1215/S0012-7094-36-00246-6. Google Scholar [13] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, (1977). Google Scholar [14] S. Hurder, Ergodic theory of foliations and a theorem of Sacksteder,, in Dynamical systems (College Park, 1342 (1988), 1986. doi: 10.1007/BFb0082838. Google Scholar [15] V. A. Kaimanovich, Ergodic properties of the horocycle flow and classification of Fuchsian groups,, J. Dynam. Control Systems, 6 (2000), 21. doi: 10.1023/A:1009517621605. Google Scholar [16] M. Kulikov, The horocycle flow without minimal sets,, C. R. Math. Acad. Sci. Paris, 338 (2004), 477. doi: 10.1016/j.crma.2003.12.027. Google Scholar [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). Google Scholar [18] S. Matsumoto, Dynamical systems without minimal sets,, preprint, (). Google Scholar [19] S. Matsumoto, Horocycle flows without minimal sets,, preprint, (). Google Scholar [20] T. Roblin, Ergodicitéet équidistribution en courbure négative,, Mém. Soc. Math. Fr. (N.S.), 95 (2003). Google Scholar [21] A. Sambusetti, Asymptotic properties of coverings in negative curvature,, Geom. Topol., 12 (2008), 617. doi: 10.2140/gt.2008.12.617. Google Scholar [22] O. Sarig, The horocyclic flow and the Laplacian on hyperbolic surfaces of infinite genus,, Geom. Funct. Anal., 19 (2010), 1757. doi: 10.1007/s00039-010-0048-9. Google Scholar [23] A. N. Starkov, Fuchsian groups from the dynamical viewpoint,, J. Dynam. Control Systems, 1 (1995), 427. doi: 10.1007/BF02269378. Google Scholar [24] A. Verjovsky, A uniformization theorem for holomorphic foliations,, in The Lefschetz centennial conference, (1984), 233. Google Scholar

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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. doi: 10.1016/j.exmath.2015.07.006. Google Scholar [2] S. Alvarez and P. Lessa, The Teichmüller space of the Hirsch foliation,, preprint, (). Google Scholar [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. doi: 10.1017/S0143385708001028. Google Scholar [4] A. Candel, Uniformization of surface laminations,, Ann. Sci. École Norm. Sup., 26 (1993), 489. Google Scholar [5] A. Candel and L. Conlon, Foliations. I,, Graduate Studies in Mathematics, (2000). Google Scholar [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. Google Scholar [7] F. Dal'bo, Topologie du feuilletage fortement stable,, Ann. Inst. Fourier (Grenoble), 50 (2000), 981. doi: 10.5802/aif.1781. Google Scholar [8] F. Dal'Bo, Geodesic and Horocyclic Trajectories,, Universitext. Translated from the 2007 French original, (2007). doi: 10.1007/978-0-85729-073-1. Google Scholar [9] D. B. A. Epstein, K. C. Millett and D. Tischler, Leaves without holonomy,, J. London Math. Soc. (2), 16 (1977), 548. Google Scholar [10] G. Hector, Feuilletages en cylindres,, in Geometry and topology (Proc. III Latin Amer. School of Math., (1977), 252. Google Scholar [11] G. Hector, S. Matsumoto and G. Meigniez, Ends of leaves of Lie foliations,, J. Math. Soc. Japan, 57 (2005), 753. doi: 10.2969/jmsj/1158241934. Google Scholar [12] G. A. Hedlund, Fuchsian groups and transitive horocycles,, Duke Math. J., 2 (1936), 530. doi: 10.1215/S0012-7094-36-00246-6. Google Scholar [13] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, (1977). Google Scholar [14] S. Hurder, Ergodic theory of foliations and a theorem of Sacksteder,, in Dynamical systems (College Park, 1342 (1988), 1986. doi: 10.1007/BFb0082838. Google Scholar [15] V. A. Kaimanovich, Ergodic properties of the horocycle flow and classification of Fuchsian groups,, J. Dynam. Control Systems, 6 (2000), 21. doi: 10.1023/A:1009517621605. Google Scholar [16] M. Kulikov, The horocycle flow without minimal sets,, C. R. Math. Acad. Sci. Paris, 338 (2004), 477. doi: 10.1016/j.crma.2003.12.027. Google Scholar [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). Google Scholar [18] S. Matsumoto, Dynamical systems without minimal sets,, preprint, (). Google Scholar [19] S. Matsumoto, Horocycle flows without minimal sets,, preprint, (). Google Scholar [20] T. Roblin, Ergodicitéet équidistribution en courbure négative,, Mém. Soc. Math. Fr. (N.S.), 95 (2003). Google Scholar [21] A. Sambusetti, Asymptotic properties of coverings in negative curvature,, Geom. Topol., 12 (2008), 617. doi: 10.2140/gt.2008.12.617. Google Scholar [22] O. Sarig, The horocyclic flow and the Laplacian on hyperbolic surfaces of infinite genus,, Geom. Funct. Anal., 19 (2010), 1757. doi: 10.1007/s00039-010-0048-9. Google Scholar [23] A. N. Starkov, Fuchsian groups from the dynamical viewpoint,, J. Dynam. Control Systems, 1 (1995), 427. doi: 10.1007/BF02269378. Google Scholar [24] A. Verjovsky, A uniformization theorem for holomorphic foliations,, in The Lefschetz centennial conference, (1984), 233. Google Scholar
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