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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 |
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. |
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