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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

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 & 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

show all references

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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