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

Previous Article
Arithmeticity and topology of smooth actions of higher rank abelian groups
JMD Home
This Volume
Next Article
Effective decay of multiple correlations in semidirect product actions

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

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

Matilde Martínez ^1, , Shigenori Matsumoto ^2, and Alberto Verjovsky ^3,

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

Abstract
Related Papers

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.

Keywords: Hyperbolic surfaces, horocycle and geodesic flows, hyperbolic laminations, minimality..

Mathematics Subject Classification: Primary: 37C85, 37D40, 57R3.

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.
[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.
[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.
[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.
[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.
[6]	T. Büber and W. A. Kirk, Convexity structures and the existence of minimal sets, Comment. Math. Prace Mat., 35 (1995), 71-81.
[7]	A. Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4), 26 (1993), 489-516.
[8]	A. Candel and L. Conlon, Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Differential Geom., 47 (1997), 17-94.
[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.
[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, 142, Princeton University Press, Princeton, NJ, 1996. doi: 10.1515/9781400865178.
[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.
[22]	D. W. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 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), 1-34.
[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.
[25]	S. Petite, On invariant measures of finite affine type tilings, Ergodic Theory Dynam. Systems, 26 (2006), 1159-1176. doi: 10.1017/S0143385706000137.
[26]	J. F. Plante, Locally free affine group actions, Trans. Amer. Math. Soc., 259 (1980), 449-456. doi: 10.2307/1998240.
[27]	J. F. Plante, Anosov flows, Amer. J. Math., 94 (1972), 729-754. doi: 10.2307/2373755.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	D. van Dantzig, Über topologisch homogene Kontinua, Fund. Math., 15 (1930), 102-125.
[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.
[37]	A. Zorich, Geodesics on flat surfaces, in International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, 121-146.

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.
[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.
[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.
[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.
[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.
[6]	T. Büber and W. A. Kirk, Convexity structures and the existence of minimal sets, Comment. Math. Prace Mat., 35 (1995), 71-81.
[7]	A. Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4), 26 (1993), 489-516.
[8]	A. Candel and L. Conlon, Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Differential Geom., 47 (1997), 17-94.
[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.
[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, 142, Princeton University Press, Princeton, NJ, 1996. doi: 10.1515/9781400865178.
[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.
[22]	D. W. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 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), 1-34.
[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.
[25]	S. Petite, On invariant measures of finite affine type tilings, Ergodic Theory Dynam. Systems, 26 (2006), 1159-1176. doi: 10.1017/S0143385706000137.
[26]	J. F. Plante, Locally free affine group actions, Trans. Amer. Math. Soc., 259 (1980), 449-456. doi: 10.2307/1998240.
[27]	J. F. Plante, Anosov flows, Amer. J. Math., 94 (1972), 729-754. doi: 10.2307/2373755.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	D. van Dantzig, Über topologisch homogene Kontinua, Fund. Math., 15 (1930), 102-125.
[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.
[37]	A. Zorich, Geodesics on flat surfaces, in International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, 121-146.

[1]	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 and Continuous Dynamical Systems, 2016, 36 (9) : 4619-4635. doi: 10.3934/dcds.2016001
[2]	Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Corrigendum to "Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology". Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4585-4586. doi: 10.3934/dcds.2017196
[3]	Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022
[4]	Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94.
[5]	Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[6]	Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841
[7]	David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477
[8]	Luis Barreira, Christian Wolf. Dimension and ergodic decompositions for hyperbolic flows. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 201-212. doi: 10.3934/dcds.2007.17.201
[9]	François Ledrappier, Omri Sarig. Fluctuations of ergodic sums for horocycle flows on $\Z^d$--covers of finite volume surfaces. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 247-325. doi: 10.3934/dcds.2008.22.247
[10]	Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173
[11]	Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365
[12]	Dubi Kelmer, Hee Oh. Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds. Journal of Modern Dynamics, 2021, 17: 401-434. doi: 10.3934/jmd.2021014
[13]	Bryce Weaver. Growth rate of periodic orbits for geodesic flows over surfaces with radially symmetric focusing caps. Journal of Modern Dynamics, 2014, 8 (2) : 139-176. doi: 10.3934/jmd.2014.8.139
[14]	Shucheng Yu. Logarithm laws for unipotent flows on hyperbolic manifolds. Journal of Modern Dynamics, 2017, 11: 447-476. doi: 10.3934/jmd.2017018
[15]	Zhiping Li, Yunhua Zhou. Quasi-shadowing for partially hyperbolic flows. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2089-2103. doi: 10.3934/dcds.2020107
[16]	Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615
[17]	Giovanni Forni, Corinna Ulcigrai. Time-changes of horocycle flows. Journal of Modern Dynamics, 2012, 6 (2) : 251-273. doi: 10.3934/jmd.2012.6.251
[18]	Andreas Strömbergsson. On the deviation of ergodic averages for horocycle flows. Journal of Modern Dynamics, 2013, 7 (2) : 291-328. doi: 10.3934/jmd.2013.7.291
[19]	Yong Fang, Patrick Foulon, Boris Hasselblatt. Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows. Electronic Research Announcements, 2010, 17: 80-89. doi: 10.3934/era.2010.17.80
[20]	C.P. Walkden. Stable ergodicity of skew products of one-dimensional hyperbolic flows. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 897-904. doi: 10.3934/dcds.1999.5.897

2020 Impact Factor: 0.848

Tools

Metrics

Other articles
by authors

on AIMS
Matilde Martínez Shigenori Matsumoto Alberto Verjovsky
on Google Scholar
Matilde Martínez Shigenori Matsumoto Alberto Verjovsky

[Back to Top]