American Institute of Mathematical Sciences

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

Joining measures for horocycle flows on abelian covers

 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.

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

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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
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