2016, 10: 113-134. doi: 10.3934/jmd.2016.10.113

Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem

1. 

Instituto de Matemática y Estadística Rafael Laguardia, Facultad de Ingeniería, Universidad de la República, J. Herrera y Reissig 565, C.P. 11300 Montevideo, Uruguay

2. 

Department of Mathematics, College of Science and Technology, Nihon University, 1-8-14 Kanda, Surugadai, Chiyoda-ku, Tokyo, 101-8308

3. 

Universidad Nacional Autónoma de México, Apartado Postal 273, Admon. de correos #3, C.P. 62251 Cuernavaca, Morelos, Mexico

Received  April 2015 Published  May 2016

We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathfrak{F})$ of a compact minimal lamination $(M,\mathfrak{F})$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal and examples where this action is not minimal. In the first case, we prove that if $\mathfrak{F}$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.
Citation: Matilde Martínez, Shigenori Matsumoto, Alberto Verjovsky. Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem. Journal of Modern Dynamics, 2016, 10: 113-134. doi: 10.3934/jmd.2016.10.113
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.

[2]

F. Alcalde Cuesta, F. Dal'Bo, M. Martínez and A. Verjovsky, Minimality of the horocycle flow on foliations by hyperbolic surfaces with non-trivial topology,, Discrete Contin. Dyn. Syst., 36 (2016), 4619.

[3]

M. Asaoka, Nonhomogeneous locally free actions of the affine group,, Ann. of Math. (2), 175 (2012), 1. doi: 10.4007/annals.2012.175.1.1.

[4]

T. Barbot, Plane affine geometry and Anosov flows,, Ann. Scient. Éc. Norm. Sup., 34 (2001), 871. doi: 10.1016/S0012-9593(01)01079-5.

[5]

J. Bellissard, R. Benedetti and J.-M. Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling,, Comm. Math. Phys., 261 (2006), 1. doi: 10.1007/s00220-005-1445-z.

[6]

T. Büber and W. A. Kirk, Convexity structures and the existence of minimal sets,, Comment. Math. Prace Mat., 35 (1995), 71.

[7]

A. Candel, Uniformization of surface laminations,, Ann. Sci. École Norm. Sup. (4), 26 (1993), 489.

[8]

A. Candel and L. Conlon, Foliations. I,, Graduate Studies in Mathematics, (2000).

[9]

J. W. Cannon and W. P. Thurston, Group invariant Peano curves,, Geom. Topol., 11 (2007), 1315. doi: 10.2140/gt.2007.11.1315.

[10]

S. G. Dani and G. A. Margulis, Values of quadratic forms at primitive integral points,, Invent. Math., 98 (1989), 405. doi: 10.1007/BF01388860.

[11]

S. R. Fenley, The structure of branching in Anosov flows of 3-manifolds,, Comment. Math. Helv., 73 (1998), 259. doi: 10.1007/s000140050055.

[12]

P. Foulon and B. Hasselblatt, Contact Anosov flows on hyperbolic 3-manifolds,, Geom. Topol., 17 (2013), 1225. doi: 10.2140/gt.2013.17.1225.

[13]

L. Garnett, Foliations, the ergodic theorem and Brownian motion,, J. Funct. Anal., 51 (1983), 285. doi: 10.1016/0022-1236(83)90015-0.

[14]

É. Ghys, Laminations par surfaces de Riemann,, in Dynamique et Géométrie Complexes (Lyon, (1997), 49.

[15]

M. Gromov, Hyperbolic manifolds (according to Thurston and Jørgensen),, in Bourbaki Seminar, (1979), 40.

[16]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, With a supplementary chapter by Katok and L. Mendoza, (1995). doi: 10.1017/CBO9780511809187.

[17]

M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics,, J. Differential Geom., 47 (1997), 17.

[18]

G. A. Margulis, Discrete subgroups and ergodic theory,, in Number Theory, (1987), 377.

[19]

S. Matsumoto, Remarks on the horocycle flows for the foliations by hyperbolic surfaces,, Proc. Amer. Math. Soc., (). doi: 10.1090/proc/13184.

[20]

C. T. McMullen, Renormalization and 3-Manifolds Which Fiber Over the Circle,, Annals of Mathematics Studies, (1996). doi: 10.1515/9781400865178.

[21]

C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces,, J. Amer. Math. Soc., 16 (2003), 857. doi: 10.1090/S0894-0347-03-00432-6.

[22]

D. W. Morris, Ratner's Theorems on Unipotent Flows,, Chicago Lectures in Mathematics, (2005).

[23]

S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle,, Osaka J. Math., 32 (1995), 1.

[24]

J.-P. Otal, The Hyperbolization Theorem for Fibered 3-Manifolds,, Translated from the 1996 French original by L. D. Kay, (1996).

[25]

S. Petite, On invariant measures of finite affine type tilings,, Ergodic Theory Dynam. Systems, 26 (2006), 1159. doi: 10.1017/S0143385706000137.

[26]

J. F. Plante, Locally free affine group actions,, Trans. Amer. Math. Soc., 259 (1980), 449. doi: 10.2307/1998240.

[27]

J. F. Plante, Anosov flows,, Amer. J. Math., 94 (1972), 729. doi: 10.2307/2373755.

[28]

M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows,, Duke Math. J., 63 (1991), 235. doi: 10.1215/S0012-7094-91-06311-8.

[29]

R. M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable,, Ann. of Math. (2), 92 (1970), 1. doi: 10.2307/1970696.

[30]

D. Sullivan, Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers,, in Topological Methods in Modern Mathematics (Stony Brook, (1991), 543.

[31]

W. Thurston, Hyperbolic geometry and 3-manifolds,, in Low-Dimensional Topology (Bangor, (1979), 9.

[32]

W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry,, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 357. doi: 10.1090/S0273-0979-1982-15003-0.

[33]

W. P. Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds,, Ann. of Math. (2), 124 (1986), 203. doi: 10.2307/1971277.

[34]

A. Verjovsky, A uniformization theorem for holomorphic foliations,, in The Lefschetz Centennial Conference, (1984), 233.

[35]

D. van Dantzig, Über topologisch homogene Kontinua,, Fund. Math., 15 (1930), 102.

[36]

L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen,, Math. Ann., 97 (1927), 454. doi: 10.1007/BF01447877.

[37]

A. Zorich, Geodesics on flat surfaces,, in International Congress of Mathematicians. Vol. III, (2006), 121.

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.

[2]

F. Alcalde Cuesta, F. Dal'Bo, M. Martínez and A. Verjovsky, Minimality of the horocycle flow on foliations by hyperbolic surfaces with non-trivial topology,, Discrete Contin. Dyn. Syst., 36 (2016), 4619.

[3]

M. Asaoka, Nonhomogeneous locally free actions of the affine group,, Ann. of Math. (2), 175 (2012), 1. doi: 10.4007/annals.2012.175.1.1.

[4]

T. Barbot, Plane affine geometry and Anosov flows,, Ann. Scient. Éc. Norm. Sup., 34 (2001), 871. doi: 10.1016/S0012-9593(01)01079-5.

[5]

J. Bellissard, R. Benedetti and J.-M. Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling,, Comm. Math. Phys., 261 (2006), 1. doi: 10.1007/s00220-005-1445-z.

[6]

T. Büber and W. A. Kirk, Convexity structures and the existence of minimal sets,, Comment. Math. Prace Mat., 35 (1995), 71.

[7]

A. Candel, Uniformization of surface laminations,, Ann. Sci. École Norm. Sup. (4), 26 (1993), 489.

[8]

A. Candel and L. Conlon, Foliations. I,, Graduate Studies in Mathematics, (2000).

[9]

J. W. Cannon and W. P. Thurston, Group invariant Peano curves,, Geom. Topol., 11 (2007), 1315. doi: 10.2140/gt.2007.11.1315.

[10]

S. G. Dani and G. A. Margulis, Values of quadratic forms at primitive integral points,, Invent. Math., 98 (1989), 405. doi: 10.1007/BF01388860.

[11]

S. R. Fenley, The structure of branching in Anosov flows of 3-manifolds,, Comment. Math. Helv., 73 (1998), 259. doi: 10.1007/s000140050055.

[12]

P. Foulon and B. Hasselblatt, Contact Anosov flows on hyperbolic 3-manifolds,, Geom. Topol., 17 (2013), 1225. doi: 10.2140/gt.2013.17.1225.

[13]

L. Garnett, Foliations, the ergodic theorem and Brownian motion,, J. Funct. Anal., 51 (1983), 285. doi: 10.1016/0022-1236(83)90015-0.

[14]

É. Ghys, Laminations par surfaces de Riemann,, in Dynamique et Géométrie Complexes (Lyon, (1997), 49.

[15]

M. Gromov, Hyperbolic manifolds (according to Thurston and Jørgensen),, in Bourbaki Seminar, (1979), 40.

[16]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, With a supplementary chapter by Katok and L. Mendoza, (1995). doi: 10.1017/CBO9780511809187.

[17]

M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics,, J. Differential Geom., 47 (1997), 17.

[18]

G. A. Margulis, Discrete subgroups and ergodic theory,, in Number Theory, (1987), 377.

[19]

S. Matsumoto, Remarks on the horocycle flows for the foliations by hyperbolic surfaces,, Proc. Amer. Math. Soc., (). doi: 10.1090/proc/13184.

[20]

C. T. McMullen, Renormalization and 3-Manifolds Which Fiber Over the Circle,, Annals of Mathematics Studies, (1996). doi: 10.1515/9781400865178.

[21]

C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces,, J. Amer. Math. Soc., 16 (2003), 857. doi: 10.1090/S0894-0347-03-00432-6.

[22]

D. W. Morris, Ratner's Theorems on Unipotent Flows,, Chicago Lectures in Mathematics, (2005).

[23]

S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle,, Osaka J. Math., 32 (1995), 1.

[24]

J.-P. Otal, The Hyperbolization Theorem for Fibered 3-Manifolds,, Translated from the 1996 French original by L. D. Kay, (1996).

[25]

S. Petite, On invariant measures of finite affine type tilings,, Ergodic Theory Dynam. Systems, 26 (2006), 1159. doi: 10.1017/S0143385706000137.

[26]

J. F. Plante, Locally free affine group actions,, Trans. Amer. Math. Soc., 259 (1980), 449. doi: 10.2307/1998240.

[27]

J. F. Plante, Anosov flows,, Amer. J. Math., 94 (1972), 729. doi: 10.2307/2373755.

[28]

M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows,, Duke Math. J., 63 (1991), 235. doi: 10.1215/S0012-7094-91-06311-8.

[29]

R. M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable,, Ann. of Math. (2), 92 (1970), 1. doi: 10.2307/1970696.

[30]

D. Sullivan, Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers,, in Topological Methods in Modern Mathematics (Stony Brook, (1991), 543.

[31]

W. Thurston, Hyperbolic geometry and 3-manifolds,, in Low-Dimensional Topology (Bangor, (1979), 9.

[32]

W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry,, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 357. doi: 10.1090/S0273-0979-1982-15003-0.

[33]

W. P. Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds,, Ann. of Math. (2), 124 (1986), 203. doi: 10.2307/1971277.

[34]

A. Verjovsky, A uniformization theorem for holomorphic foliations,, in The Lefschetz Centennial Conference, (1984), 233.

[35]

D. van Dantzig, Über topologisch homogene Kontinua,, Fund. Math., 15 (1930), 102.

[36]

L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen,, Math. Ann., 97 (1927), 454. doi: 10.1007/BF01447877.

[37]

A. Zorich, Geodesics on flat surfaces,, in International Congress of Mathematicians. Vol. III, (2006), 121.

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