American Institute of Mathematical Sciences

Advanced
  2018, 12: 17-54. doi: 10.3934/jmd.2018003

Joining measures for horocycle flows on abelian covers

Wenyu Pan

Department of Mathematics, Yale University, New Haven, CT 06511, USA

Received  January 31, 2017 Revised  November 09, 2017 Published  March 2018

A celebrated result of Ratner from the eighties says that two horocycle flows on hyperbolic surfaces of finite area are either the same up to algebraic change of coordinates, or they have no non-trivial joinings. Recently, Mohammadi and Oh extended Ratner's theorem to horocycle flows on hyperbolic surfaces of infinite area but finite genus. In this paper, we present the first joining classification result of a horocycle flow on a hyperbolic surface of infinite genus: a $\mathbb{Z}$ or $\mathbb{Z}^2$-cover of a general compact hyperbolic surface.

Keywords: Joining measures, horocycle flows, abelian covers.
Mathematics Subject Classification: Primary: 37A17, 37D40; Secondary: 37D35.
Citation: Wenyu Pan. Joining measures for horocycle flows on abelian covers. Journal of Modern Dynamics, 2018, 12: 17-54. doi: 10.3934/jmd.2018003
References:
[1]

M. Babillot, On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds, in Random Walks and Geometry (ed. Kaimanovich), Walter de Gruyter, Berlin, 2004,319-335.  Google Scholar

[2]

M. Babillot and F. Ledrappier, Lalley's theorem on period orbits of hyperbolic flows, Ergod. Th. Dynam. Syst., 18 (1998), 17-39.  doi: 10.1017/S0143385798100330.  Google Scholar

[3]

M. Babillot and F. Ledrappier, Geodesic paths and horocycle flow on abelian covers, in Lie Groups and Ergodic Theory (Mumbai, 1996), Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998, 1-32.  Google Scholar

[4]

Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces, Ann. of Math., 174 (2011), 1111-1162.  doi: 10.4007/annals.2011.174.2.8.  Google Scholar

[5]

Y. Benoist and H. Oh, Fuchsian groups and compact hyperbolic surfaces, Enseign. Math., 62 (2016), 189-198.  doi: 10.4171/LEM/62-1/2-11.  Google Scholar

[6]

R. Bowen and B. Marcus, Unique ergodicity for horocycle foliations, Israel J. Math., 26 (1977), 43-67.  doi: 10.1007/BF03007655.  Google Scholar

[7]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848.  Google Scholar

[8]

R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 153-170.   Google Scholar

[9]

M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar

[10]

M. Denker and W. Philipp, Approximation by Brownian motion for Gibbs measures and flows under a function, Ergod. Th. Dynam. Syst., 4 (1984), 541-552.   Google Scholar

[11]

H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., 318, Springer, Berlin, 1972, 95-115.  Google Scholar

[12]

L. Flaminio and R. J. Spatzier, Ratner's rigidity theorem for geometrically finite Fuchsian groups, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988,180-195.  Google Scholar

[13]

L. Flaminio and R. Spatzier, Geometrically finite groups, Patterson-Sullivan measures and Ratner's rigidity theorem, Invent. Math., 99 (1990), 601-626.  doi: 10.1007/BF01234433.  Google Scholar

[14]

R. A. Johnson, Atomic and nonatomic measures, Proc. Amer. Math. Soc., 25 (1970), 650-655.  doi: 10.1090/S0002-9939-1970-0279266-8.  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]

A. Katsuda and T. Sunada, Closed orbits in homology classes, Inst. Hautes études Sci. Publ. Math., 71 (1990), 5-32.   Google Scholar

[17]

D. Kleinbock and G. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math., 148 (1998), 339-360.  doi: 10.2307/120997.  Google Scholar

[18]

S. P. Lalley, Renewal theorems in symbolic dynamics. with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math., 163 (1989), 1-55.  doi: 10.1007/BF02392732.  Google Scholar

[19]

F. Ledrappier, Horospheres on abelian covers, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 363-375.  doi: 10.1007/BF01233398.  Google Scholar

[20]

F. Ledrappier and O. Sarig, Unique ergodicity for non-uniquely ergodic horocycle flows, Discrete Contin. Dyn. Syst., 16 (2006), 411-433.  doi: 10.3934/dcds.2006.16.411.  Google Scholar

[21]

G. Margulis, Indefinite quadratic forms and unipotent flows on homogeneous spaces, in Dynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publ., 23, PWN, Warsaw, 1989,399-309.  Google Scholar

[22]

G. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991.  Google Scholar

[23]

G. Margulis and G. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., 116 (1994), 347-392.  doi: 10.1007/BF01231565.  Google Scholar

[24]

C. McMullen, A. Mohammadi and H. Oh, Geodesic planes in hyperbolic 3-manifolds, Invent. Math., 209 (2017), 425-461.  doi: 10.1007/s00222-016-0711-3.  Google Scholar

[25]

A. Mohammadi and H. Oh, Classification of joinings for Kleinian groups, Duke Math. J., 165 (2016), 2155-2223.  doi: 10.1215/00127094-3476807.  Google Scholar

[26]

A. Mohammadi and H. Oh, Invariant Radon measures for unipotent flows and products of Kleinian groups, Proc. Amer. Math. Soc., 146 (2018), 1469U-1479.   Google Scholar

[27]

M. Pollicott and R. Sharp, Orbit counting for some discrete subgroups acting on simply connected manifolds with negative curvature, Invent. Math., 117 (1994), 275-302.  doi: 10.1007/BF01232242.  Google Scholar

[28]

M. Ratner, The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature, Israel J. Math., 16 (1973), 181-197.  doi: 10.1007/BF02757869.  Google Scholar

[29]

M. Ratner, Rigidity of horocycle flows, Ann. of Math., 2 (1982), 597-614.  doi: 10.2307/2007014.  Google Scholar

[30]

M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math., 118 (1983), 277-313.  doi: 10.2307/2007030.  Google Scholar

[31]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar

[32]

M. Rees, Checking ergodicity of some geodesic flows with inifinte Gibbs measure, Ergod. Th. Dynam. Sys., 1 (1981), 107-133.   Google Scholar

[33]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003), ⅵ+96 pp.   Google Scholar

[34]

V. A. Rohlin, On basic concepts of measure theory, Mat. Sbornik, 67 (1949), 107-150.   Google Scholar

[35]

O. Sarig, Invariant measures for the horocycle flow on Abelian covers, Invent. Math., 157 (2004), 519-551.  doi: 10.1007/s00222-004-0357-4.  Google Scholar

[36]

O. Sarig and B. Schapira, The generic points for the horocycle flow on a class of hyperbolic surfaces with infinite genus, Int. Math. Res. Not. IMRN 2008, Art. ID rnn 086, 37 pp.  Google Scholar

[37]

C. Series, Geometric Markov coding of geodesics on surfaces of constant negative curvature, Ergodic Theory Dynam. Systems, 6 (1986), 601-625.   Google Scholar

[38]

C. Series, Geometrical methods of symbolic coding, in Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces (Trieste, 1989) (eds. T. Bedford, M. Keane and C. Series), Oxford Sci. Publ., Oxford Univ. Press, New York, 1991.  Google Scholar

[39]

D. Winter, Mixing of frame flow for rank one locally symmetric manifold and measure classification, Israel J. Math., 201 (2015), 467-507.  doi: 10.1007/s11856-015-1258-5.  Google Scholar

show all references

References:
[1]

M. Babillot, On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds, in Random Walks and Geometry (ed. Kaimanovich), Walter de Gruyter, Berlin, 2004,319-335.  Google Scholar

[2]

M. Babillot and F. Ledrappier, Lalley's theorem on period orbits of hyperbolic flows, Ergod. Th. Dynam. Syst., 18 (1998), 17-39.  doi: 10.1017/S0143385798100330.  Google Scholar

[3]

M. Babillot and F. Ledrappier, Geodesic paths and horocycle flow on abelian covers, in Lie Groups and Ergodic Theory (Mumbai, 1996), Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998, 1-32.  Google Scholar

[4]

Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces, Ann. of Math., 174 (2011), 1111-1162.  doi: 10.4007/annals.2011.174.2.8.  Google Scholar

[5]

Y. Benoist and H. Oh, Fuchsian groups and compact hyperbolic surfaces, Enseign. Math., 62 (2016), 189-198.  doi: 10.4171/LEM/62-1/2-11.  Google Scholar

[6]

R. Bowen and B. Marcus, Unique ergodicity for horocycle foliations, Israel J. Math., 26 (1977), 43-67.  doi: 10.1007/BF03007655.  Google Scholar

[7]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848.  Google Scholar

[8]

R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 153-170.   Google Scholar

[9]

M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar

[10]

M. Denker and W. Philipp, Approximation by Brownian motion for Gibbs measures and flows under a function, Ergod. Th. Dynam. Syst., 4 (1984), 541-552.   Google Scholar

[11]

H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., 318, Springer, Berlin, 1972, 95-115.  Google Scholar

[12]

L. Flaminio and R. J. Spatzier, Ratner's rigidity theorem for geometrically finite Fuchsian groups, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988,180-195.  Google Scholar

[13]

L. Flaminio and R. Spatzier, Geometrically finite groups, Patterson-Sullivan measures and Ratner's rigidity theorem, Invent. Math., 99 (1990), 601-626.  doi: 10.1007/BF01234433.  Google Scholar

[14]

R. A. Johnson, Atomic and nonatomic measures, Proc. Amer. Math. Soc., 25 (1970), 650-655.  doi: 10.1090/S0002-9939-1970-0279266-8.  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]

A. Katsuda and T. Sunada, Closed orbits in homology classes, Inst. Hautes études Sci. Publ. Math., 71 (1990), 5-32.   Google Scholar

[17]

D. Kleinbock and G. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math., 148 (1998), 339-360.  doi: 10.2307/120997.  Google Scholar

[18]

S. P. Lalley, Renewal theorems in symbolic dynamics. with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math., 163 (1989), 1-55.  doi: 10.1007/BF02392732.  Google Scholar

[19]

F. Ledrappier, Horospheres on abelian covers, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 363-375.  doi: 10.1007/BF01233398.  Google Scholar

[20]

F. Ledrappier and O. Sarig, Unique ergodicity for non-uniquely ergodic horocycle flows, Discrete Contin. Dyn. Syst., 16 (2006), 411-433.  doi: 10.3934/dcds.2006.16.411.  Google Scholar

[21]

G. Margulis, Indefinite quadratic forms and unipotent flows on homogeneous spaces, in Dynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publ., 23, PWN, Warsaw, 1989,399-309.  Google Scholar

[22]

G. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991.  Google Scholar

[23]

G. Margulis and G. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., 116 (1994), 347-392.  doi: 10.1007/BF01231565.  Google Scholar

[24]

C. McMullen, A. Mohammadi and H. Oh, Geodesic planes in hyperbolic 3-manifolds, Invent. Math., 209 (2017), 425-461.  doi: 10.1007/s00222-016-0711-3.  Google Scholar

[25]

A. Mohammadi and H. Oh, Classification of joinings for Kleinian groups, Duke Math. J., 165 (2016), 2155-2223.  doi: 10.1215/00127094-3476807.  Google Scholar

[26]

A. Mohammadi and H. Oh, Invariant Radon measures for unipotent flows and products of Kleinian groups, Proc. Amer. Math. Soc., 146 (2018), 1469U-1479.   Google Scholar

[27]

M. Pollicott and R. Sharp, Orbit counting for some discrete subgroups acting on simply connected manifolds with negative curvature, Invent. Math., 117 (1994), 275-302.  doi: 10.1007/BF01232242.  Google Scholar

[28]

M. Ratner, The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature, Israel J. Math., 16 (1973), 181-197.  doi: 10.1007/BF02757869.  Google Scholar

[29]

M. Ratner, Rigidity of horocycle flows, Ann. of Math., 2 (1982), 597-614.  doi: 10.2307/2007014.  Google Scholar

[30]

M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math., 118 (1983), 277-313.  doi: 10.2307/2007030.  Google Scholar

[31]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar

[32]

M. Rees, Checking ergodicity of some geodesic flows with inifinte Gibbs measure, Ergod. Th. Dynam. Sys., 1 (1981), 107-133.   Google Scholar

[33]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003), ⅵ+96 pp.   Google Scholar

[34]

V. A. Rohlin, On basic concepts of measure theory, Mat. Sbornik, 67 (1949), 107-150.   Google Scholar

[35]

O. Sarig, Invariant measures for the horocycle flow on Abelian covers, Invent. Math., 157 (2004), 519-551.  doi: 10.1007/s00222-004-0357-4.  Google Scholar

[36]

O. Sarig and B. Schapira, The generic points for the horocycle flow on a class of hyperbolic surfaces with infinite genus, Int. Math. Res. Not. IMRN 2008, Art. ID rnn 086, 37 pp.  Google Scholar

[37]

C. Series, Geometric Markov coding of geodesics on surfaces of constant negative curvature, Ergodic Theory Dynam. Systems, 6 (1986), 601-625.   Google Scholar

[38]

C. Series, Geometrical methods of symbolic coding, in Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces (Trieste, 1989) (eds. T. Bedford, M. Keane and C. Series), Oxford Sci. Publ., Oxford Univ. Press, New York, 1991.  Google Scholar

[39]

D. Winter, Mixing of frame flow for rank one locally symmetric manifold and measure classification, Israel J. Math., 201 (2015), 467-507.  doi: 10.1007/s11856-015-1258-5.  Google Scholar

[1]

François Ledrappier, Omri Sarig. Fluctuations of ergodic sums for horocycle flows on $\Z^d$--covers of finite volume surfaces. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 247-325. doi: 10.3934/dcds.2008.22.247
[2]

Chenxi Wu. The relative cohomology of abelian covers of the flat pillowcase. Journal of Modern Dynamics, 2015, 9: 123-140. doi: 10.3934/jmd.2015.9.123
[3]

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
[4]

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
[5]

Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94.

[6]

Rafael Tiedra De Aldecoa. Spectral analysis of time changes of horocycle flows. Journal of Modern Dynamics, 2012, 6 (2) : 275-285. doi: 10.3934/jmd.2012.6.275
[7]

François Ledrappier, Omri Sarig. Unique ergodicity for non-uniquely ergodic horocycle flows. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 411-433. doi: 10.3934/dcds.2006.16.411
[8]

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
[9]

Livio Flaminio, Giovanni Forni. Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows. Electronic Research Announcements, 2019, 26: 16-23. doi: 10.3934/era.2019.26.002
[10]

Radu Saghin. On the number of ergodic minimizing measures for Lagrangian flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 501-507. doi: 10.3934/dcds.2007.17.501
[11]

Giovanni Forni, Carlos Matheus, Anton Zorich. Square-tiled cyclic covers. Journal of Modern Dynamics, 2011, 5 (2) : 285-318. doi: 10.3934/jmd.2011.5.285
[12]

Mark Pollicott. $\mathbb Z^d$-covers of horosphere foliations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 147-154. doi: 10.3934/dcds.2000.6.147
[13]

Alex Eskin, Maxim Kontsevich, Anton Zorich. Lyapunov spectrum of square-tiled cyclic covers. Journal of Modern Dynamics, 2011, 5 (2) : 319-353. doi: 10.3934/jmd.2011.5.319
[14]

Feng Zhang, Jinting Wang, Bin Liu. Equilibrium joining probabilities in observable queues with general service and setup times. Journal of Industrial & Management Optimization, 2013, 9 (4) : 901-917. doi: 10.3934/jimo.2013.9.901
[15]

Francesca Alessio, Carlo Carminati, Piero Montecchiari. Heteroclinic motions joining almost periodic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 569-584. doi: 10.3934/dcds.1999.5.569
[16]

V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511
[17]

John Franks, Michael Handel, Kamlesh Parwani. Fixed points of Abelian actions. Journal of Modern Dynamics, 2007, 1 (3) : 443-464. doi: 10.3934/jmd.2007.1.443
[18]

Sandra Carillo, Mauro Lo Schiavo, Cornelia Schiebold. Abelian versus non-Abelian Bäcklund charts: Some remarks. Evolution Equations & Control Theory, 2019, 8 (1) : 43-55. doi: 10.3934/eect.2019003
[19]

Magdalena Czubak, Robert L. Jerrard. Topological defects in the abelian Higgs model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1933-1968. doi: 10.3934/dcds.2015.35.1933
[20]

Eldho K. Thomas, Nadya Markin, Frédérique Oggier. On Abelian group representability of finite groups. Advances in Mathematics of Communications, 2014, 8 (2) : 139-152. doi: 10.3934/amc.2014.8.139

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (104)
  • HTML views (1319)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences