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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, 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.  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), 67-96. doi: 10.1017/S0143385708001028.  Google Scholar

[4]

A. Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup., 26 (1993), 489-516.  Google Scholar

[5]

A. Candel and L. Conlon, Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 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), 15-53.  Google Scholar

[7]

F. Dal'bo, Topologie du feuilletage fortement stable, Ann. Inst. Fourier (Grenoble), 50 (2000), 981-993. doi: 10.5802/aif.1781.  Google Scholar

[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.  Google Scholar

[9]

D. B. A. Epstein, K. C. Millett and D. Tischler, Leaves without holonomy, J. London Math. Soc. (2), 16 (1977), 548-552.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542. 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, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), vi+96pp.  Google Scholar

[21]

A. Sambusetti, Asymptotic properties of coverings in negative curvature, Geom. Topol., 12 (2008), 617-637. 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), 1757-1812. 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), 427-445. doi: 10.1007/BF02269378.  Google Scholar

[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.  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), 431-451. 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), 67-96. doi: 10.1017/S0143385708001028.  Google Scholar

[4]

A. Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup., 26 (1993), 489-516.  Google Scholar

[5]

A. Candel and L. Conlon, Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 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), 15-53.  Google Scholar

[7]

F. Dal'bo, Topologie du feuilletage fortement stable, Ann. Inst. Fourier (Grenoble), 50 (2000), 981-993. doi: 10.5802/aif.1781.  Google Scholar

[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.  Google Scholar

[9]

D. B. A. Epstein, K. C. Millett and D. Tischler, Leaves without holonomy, J. London Math. Soc. (2), 16 (1977), 548-552.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542. 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, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), vi+96pp.  Google Scholar

[21]

A. Sambusetti, Asymptotic properties of coverings in negative curvature, Geom. Topol., 12 (2008), 617-637. 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), 1757-1812. 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), 427-445. doi: 10.1007/BF02269378.  Google Scholar

[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.  Google Scholar

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