# American Institute of Mathematical Sciences

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), 431-451. doi: 10.1016/j.exmath.2015.07.006.  Google Scholar [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), no. 9, 4619-4635. Google Scholar [3] M. Asaoka, Nonhomogeneous locally free actions of the affine group, Ann. of Math. (2), 175 (2012), 1-21. doi: 10.4007/annals.2012.175.1.1.  Google Scholar [4] T. Barbot, Plane affine geometry and Anosov flows, Ann. Scient. Éc. Norm. Sup., 34 (2001), 871-889. doi: 10.1016/S0012-9593(01)01079-5.  Google Scholar [5] J. Bellissard, R. Benedetti and J.-M. Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling, Comm. Math. Phys., 261 (2006), 1-41. doi: 10.1007/s00220-005-1445-z.  Google Scholar [6] T. Büber and W. A. Kirk, Convexity structures and the existence of minimal sets, Comment. Math. Prace Mat., 35 (1995), 71-81.  Google Scholar [7] A. Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4), 26 (1993), 489-516.  Google Scholar [8] A. Candel and L. Conlon, Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000.  Google Scholar [9] J. W. Cannon and W. P. Thurston, Group invariant Peano curves, Geom. Topol., 11 (2007), 1315-1355. doi: 10.2140/gt.2007.11.1315.  Google Scholar [10] S. G. Dani and G. A. Margulis, Values of quadratic forms at primitive integral points, Invent. Math., 98 (1989), 405-424. doi: 10.1007/BF01388860.  Google Scholar [11] S. R. Fenley, The structure of branching in Anosov flows of 3-manifolds, Comment. Math. Helv., 73 (1998), 259-297. doi: 10.1007/s000140050055.  Google Scholar [12] P. Foulon and B. Hasselblatt, Contact Anosov flows on hyperbolic 3-manifolds, Geom. Topol., 17 (2013), 1225-1252. doi: 10.2140/gt.2013.17.1225.  Google Scholar [13] L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., 51 (1983), 285-311. doi: 10.1016/0022-1236(83)90015-0.  Google Scholar [14] É. Ghys, Laminations par surfaces de Riemann, in Dynamique et Géométrie Complexes (Lyon, 1997), Panor. Synthèses, 8, Soc. Math. France, Paris, 1999, ix, xi, 49-95.  Google Scholar [15] M. Gromov, Hyperbolic manifolds (according to Thurston and Jørgensen), in Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., 842, Springer, Berlin-New York, 1981, 40-53.  Google Scholar [16] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, With a supplementary chapter by Katok and L. Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar [17] M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Differential Geom., 47 (1997), 17-94.  Google Scholar [18] G. A. Margulis, Discrete subgroups and ergodic theory, in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, MA, 1989, 377-398.  Google Scholar [19] S. Matsumoto, Remarks on the horocycle flows for the foliations by hyperbolic surfaces,, Proc. Amer. Math. Soc., ().  doi: 10.1090/proc/13184.  Google Scholar [20] C. T. McMullen, Renormalization and 3-Manifolds Which Fiber Over the Circle, Annals of Mathematics Studies, 142, Princeton University Press, Princeton, NJ, 1996. doi: 10.1515/9781400865178.  Google Scholar [21] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 857-885 (electronic). doi: 10.1090/S0894-0347-03-00432-6.  Google Scholar [22] D. W. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005.  Google Scholar [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), 1-34.  Google Scholar [24] J.-P. Otal, The Hyperbolization Theorem for Fibered 3-Manifolds, Translated from the 1996 French original by L. D. Kay, SMF/AMS Texts and Monographs, 7, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001.  Google Scholar [25] S. Petite, On invariant measures of finite affine type tilings, Ergodic Theory Dynam. Systems, 26 (2006), 1159-1176. doi: 10.1017/S0143385706000137.  Google Scholar [26] J. F. Plante, Locally free affine group actions, Trans. Amer. Math. Soc., 259 (1980), 449-456. doi: 10.2307/1998240.  Google Scholar [27] J. F. Plante, Anosov flows, Amer. J. Math., 94 (1972), 729-754. doi: 10.2307/2373755.  Google Scholar [28] M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280. doi: 10.1215/S0012-7094-91-06311-8.  Google Scholar [29] R. M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2), 92 (1970), 1-56. doi: 10.2307/1970696.  Google Scholar [30] D. Sullivan, Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers, in Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, 543-564.  Google Scholar [31] W. Thurston, Hyperbolic geometry and 3-manifolds, in Low-Dimensional Topology (Bangor, 1979), London Math. Soc. Lecture Note Ser., 48, Cambridge Univ. Press, Cambridge-New York, 1982, 9-25.  Google Scholar [32] W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 357-381. doi: 10.1090/S0273-0979-1982-15003-0.  Google Scholar [33] W. P. Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2), 124 (1986), 203-246. doi: 10.2307/1971277.  Google Scholar [34] 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 [35] D. van Dantzig, Über topologisch homogene Kontinua, Fund. Math., 15 (1930), 102-125. Google Scholar [36] L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann., 97 (1927), 454-472. doi: 10.1007/BF01447877.  Google Scholar [37] A. Zorich, Geodesics on flat surfaces, in International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, 121-146.  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), 431-451. doi: 10.1016/j.exmath.2015.07.006.  Google Scholar [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), no. 9, 4619-4635. Google Scholar [3] M. Asaoka, Nonhomogeneous locally free actions of the affine group, Ann. of Math. (2), 175 (2012), 1-21. doi: 10.4007/annals.2012.175.1.1.  Google Scholar [4] T. Barbot, Plane affine geometry and Anosov flows, Ann. Scient. Éc. Norm. Sup., 34 (2001), 871-889. doi: 10.1016/S0012-9593(01)01079-5.  Google Scholar [5] J. Bellissard, R. Benedetti and J.-M. Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling, Comm. Math. Phys., 261 (2006), 1-41. doi: 10.1007/s00220-005-1445-z.  Google Scholar [6] T. Büber and W. A. Kirk, Convexity structures and the existence of minimal sets, Comment. Math. Prace Mat., 35 (1995), 71-81.  Google Scholar [7] A. Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4), 26 (1993), 489-516.  Google Scholar [8] A. Candel and L. Conlon, Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000.  Google Scholar [9] J. W. Cannon and W. P. Thurston, Group invariant Peano curves, Geom. Topol., 11 (2007), 1315-1355. doi: 10.2140/gt.2007.11.1315.  Google Scholar [10] S. G. Dani and G. A. Margulis, Values of quadratic forms at primitive integral points, Invent. Math., 98 (1989), 405-424. doi: 10.1007/BF01388860.  Google Scholar [11] S. R. Fenley, The structure of branching in Anosov flows of 3-manifolds, Comment. Math. Helv., 73 (1998), 259-297. doi: 10.1007/s000140050055.  Google Scholar [12] P. Foulon and B. Hasselblatt, Contact Anosov flows on hyperbolic 3-manifolds, Geom. Topol., 17 (2013), 1225-1252. doi: 10.2140/gt.2013.17.1225.  Google Scholar [13] L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., 51 (1983), 285-311. doi: 10.1016/0022-1236(83)90015-0.  Google Scholar [14] É. Ghys, Laminations par surfaces de Riemann, in Dynamique et Géométrie Complexes (Lyon, 1997), Panor. Synthèses, 8, Soc. Math. France, Paris, 1999, ix, xi, 49-95.  Google Scholar [15] M. Gromov, Hyperbolic manifolds (according to Thurston and Jørgensen), in Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., 842, Springer, Berlin-New York, 1981, 40-53.  Google Scholar [16] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, With a supplementary chapter by Katok and L. Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar [17] M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Differential Geom., 47 (1997), 17-94.  Google Scholar [18] G. A. Margulis, Discrete subgroups and ergodic theory, in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, MA, 1989, 377-398.  Google Scholar [19] S. Matsumoto, Remarks on the horocycle flows for the foliations by hyperbolic surfaces,, Proc. Amer. Math. Soc., ().  doi: 10.1090/proc/13184.  Google Scholar [20] C. T. McMullen, Renormalization and 3-Manifolds Which Fiber Over the Circle, Annals of Mathematics Studies, 142, Princeton University Press, Princeton, NJ, 1996. doi: 10.1515/9781400865178.  Google Scholar [21] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 857-885 (electronic). doi: 10.1090/S0894-0347-03-00432-6.  Google Scholar [22] D. W. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005.  Google Scholar [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), 1-34.  Google Scholar [24] J.-P. Otal, The Hyperbolization Theorem for Fibered 3-Manifolds, Translated from the 1996 French original by L. D. Kay, SMF/AMS Texts and Monographs, 7, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001.  Google Scholar [25] S. Petite, On invariant measures of finite affine type tilings, Ergodic Theory Dynam. Systems, 26 (2006), 1159-1176. doi: 10.1017/S0143385706000137.  Google Scholar [26] J. F. Plante, Locally free affine group actions, Trans. Amer. Math. Soc., 259 (1980), 449-456. doi: 10.2307/1998240.  Google Scholar [27] J. F. Plante, Anosov flows, Amer. J. Math., 94 (1972), 729-754. doi: 10.2307/2373755.  Google Scholar [28] M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280. doi: 10.1215/S0012-7094-91-06311-8.  Google Scholar [29] R. M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2), 92 (1970), 1-56. doi: 10.2307/1970696.  Google Scholar [30] D. Sullivan, Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers, in Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, 543-564.  Google Scholar [31] W. Thurston, Hyperbolic geometry and 3-manifolds, in Low-Dimensional Topology (Bangor, 1979), London Math. Soc. Lecture Note Ser., 48, Cambridge Univ. Press, Cambridge-New York, 1982, 9-25.  Google Scholar [32] W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 357-381. doi: 10.1090/S0273-0979-1982-15003-0.  Google Scholar [33] W. P. Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2), 124 (1986), 203-246. doi: 10.2307/1971277.  Google Scholar [34] 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 [35] D. van Dantzig, Über topologisch homogene Kontinua, Fund. Math., 15 (1930), 102-125. Google Scholar [36] L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann., 97 (1927), 454-472. doi: 10.1007/BF01447877.  Google Scholar [37] A. Zorich, Geodesics on flat surfaces, in International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, 121-146.  Google Scholar
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