# American Institute of Mathematical Sciences

April  2012, 6(2): 251-273. doi: 10.3934/jmd.2012.6.251

## Time-changes of horocycle flows

 1 Department of Mathematics, University of Maryland, College Park, MD 20742-4015 2 School of Mathematics, University of Bristol, University Walk, Clifton, BS8 1TW,Bristol, United Kingdom

Received  February 2012 Published  August 2012

We consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and the rate of mixing. We then derive results on the spectrum of smooth time-changes and show that the spectrum is absolutely continuous with respect to the Lebesgue measure on the real line and that the maximal spectral type is equivalent to Lebesgue.
Citation: 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
##### References:
 [1] V. Bargmann, Irreducible unitary representations of the Lorentz group,, Ann. of Math. (2), 48 (1947), 568. doi: 10.2307/1969129. [2] A. Bufetov and G. Forni, Limit theorems for horocycle flows,, preprint, (2011), 1. [3] M. Burger, Horocycle flow on geometrically finite surfaces,, Duke Math. J., 61 (1990), 779. doi: 10.1215/S0012-7094-90-06129-0. [4] S. G. Dani, Invariant measures and minimal sets of horospherical flows,, Invent. Math., 64 (1981), 357. doi: 10.1007/BF01389173. [5] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465. doi: 10.1215/S0012-7094-03-11932-8. [6] H. Furstenberg, The unique ergodicity of the horocycle flow,, in, 318 (1973), 95. [7] I. M. Gel'fand and S. V. Fomin, Unitary representations of Lie groups and geodesic flows on surfaces of constant negative curvature, (Russian),, Dokl. Akad. Nauk SSSR (N.S.), 76 (1951), 771. [8] I. M. Gelfand and M. Neumark, Unitary representations of the Lorentz group,, Acad. Sci. USSR. J. Phys., 10 (1946), 93. [9] B. M. Gurevič, The entropy of horocycle flows, (Russian),, Dokl. Akad. Nauk SSSR, 136 (1961), 768. [10] G. A. Hedlund, Fuchsian groups and transitive horocycles,, Duke Math. J., 2 (1936), 530. doi: 10.1215/S0012-7094-36-00246-6. [11] D. A. Hejhal, On the uniform equidistribution of long closed horocycles. Loo-Keng Hua: A great mathematician of the twentieth century,, Asian J. Math., 4 (2000), 839. [12] A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory,, in, (2006), 649. [13] A. G. Kušnirenko, Spectral properties of certain dynamical systems with polynomial dispersal,, Vestnik Moskovskogo Universiteta. Ser. I Matematika Meh., 29 (1974), 101. [14] B. Marcus, Unique ergodicity of the horocycle flow: Variable negative curvature case,, Conference on Ergodic Theory and Topological Dynamics (Kibbutz Lavi, 21 (1975), 133. [15] _____, Ergodic properties of horocycle flows for surfaces of negative curvature,, Ann. of Math. (2), 105 (1977), 81. [16] C. C. Moore, Exponential decay of correlation coefficients for geodesic flows,, in, 6 (1987), 163. [17] E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators,, Comm. Math. Phys., 78 (): 391. doi: 10.1007/BF01942331. [18] O. S. Parasyuk, Flows of horocycles on surfaces of constant negative curvature, (Russian),, Uspekhi Mat. Nauk (N.S.), 8 (1953), 125. [19] M. Ratner, Factors of horocycle flows,, Ergodic Theory Dynam. Systems, 2 (1982), 465. doi: 10.1017/S0143385700001723. [20] _____, Rigidity of horocycle flows,, Ann. of Math. (2), 115 (1982), 597. [21] _____, Horocycle flows, joinings and rigidity of products,, Ann. of Math. (2), 118 (1983), 277. doi: 10.2307/2007030. [22] _____, The rate of mixing for geodesic and horocycle flows,, Ergodic Theory Dynam. Systems, 7 (1987), 267. [23] P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series,, Comm. Pure Appl. Math., 34 (1981), 719. doi: 10.1002/cpa.3160340602. [24] A. Strömbergsson, On the uniform equidistribution of long closed horocycles,, Duke Math. J., 123 (2004), 507. doi: 10.1215/S0012-7094-04-12334-6. [25] R. Tiedra de Aldecoa, Spectral analysis of time changes for horocycle flows,, , (). [26] A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity,, Ann. of Math. (2), 172 (2010), 989. doi: 10.4007/annals.2010.172.989. [27] D. Zagier, Eisenstein series and the Riemann zeta function,, in, 10 (1981), 275.

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##### References:
 [1] V. Bargmann, Irreducible unitary representations of the Lorentz group,, Ann. of Math. (2), 48 (1947), 568. doi: 10.2307/1969129. [2] A. Bufetov and G. Forni, Limit theorems for horocycle flows,, preprint, (2011), 1. [3] M. Burger, Horocycle flow on geometrically finite surfaces,, Duke Math. J., 61 (1990), 779. doi: 10.1215/S0012-7094-90-06129-0. [4] S. G. Dani, Invariant measures and minimal sets of horospherical flows,, Invent. Math., 64 (1981), 357. doi: 10.1007/BF01389173. [5] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465. doi: 10.1215/S0012-7094-03-11932-8. [6] H. Furstenberg, The unique ergodicity of the horocycle flow,, in, 318 (1973), 95. [7] I. M. Gel'fand and S. V. Fomin, Unitary representations of Lie groups and geodesic flows on surfaces of constant negative curvature, (Russian),, Dokl. Akad. Nauk SSSR (N.S.), 76 (1951), 771. [8] I. M. Gelfand and M. Neumark, Unitary representations of the Lorentz group,, Acad. Sci. USSR. J. Phys., 10 (1946), 93. [9] B. M. Gurevič, The entropy of horocycle flows, (Russian),, Dokl. Akad. Nauk SSSR, 136 (1961), 768. [10] G. A. Hedlund, Fuchsian groups and transitive horocycles,, Duke Math. J., 2 (1936), 530. doi: 10.1215/S0012-7094-36-00246-6. [11] D. A. Hejhal, On the uniform equidistribution of long closed horocycles. Loo-Keng Hua: A great mathematician of the twentieth century,, Asian J. Math., 4 (2000), 839. [12] A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory,, in, (2006), 649. [13] A. G. Kušnirenko, Spectral properties of certain dynamical systems with polynomial dispersal,, Vestnik Moskovskogo Universiteta. Ser. I Matematika Meh., 29 (1974), 101. [14] B. Marcus, Unique ergodicity of the horocycle flow: Variable negative curvature case,, Conference on Ergodic Theory and Topological Dynamics (Kibbutz Lavi, 21 (1975), 133. [15] _____, Ergodic properties of horocycle flows for surfaces of negative curvature,, Ann. of Math. (2), 105 (1977), 81. [16] C. C. Moore, Exponential decay of correlation coefficients for geodesic flows,, in, 6 (1987), 163. [17] E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators,, Comm. Math. Phys., 78 (): 391. doi: 10.1007/BF01942331. [18] O. S. Parasyuk, Flows of horocycles on surfaces of constant negative curvature, (Russian),, Uspekhi Mat. Nauk (N.S.), 8 (1953), 125. [19] M. Ratner, Factors of horocycle flows,, Ergodic Theory Dynam. Systems, 2 (1982), 465. doi: 10.1017/S0143385700001723. [20] _____, Rigidity of horocycle flows,, Ann. of Math. (2), 115 (1982), 597. [21] _____, Horocycle flows, joinings and rigidity of products,, Ann. of Math. (2), 118 (1983), 277. doi: 10.2307/2007030. [22] _____, The rate of mixing for geodesic and horocycle flows,, Ergodic Theory Dynam. Systems, 7 (1987), 267. [23] P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series,, Comm. Pure Appl. Math., 34 (1981), 719. doi: 10.1002/cpa.3160340602. [24] A. Strömbergsson, On the uniform equidistribution of long closed horocycles,, Duke Math. J., 123 (2004), 507. doi: 10.1215/S0012-7094-04-12334-6. [25] R. Tiedra de Aldecoa, Spectral analysis of time changes for horocycle flows,, , (). [26] A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity,, Ann. of Math. (2), 172 (2010), 989. doi: 10.4007/annals.2010.172.989. [27] D. Zagier, Eisenstein series and the Riemann zeta function,, in, 10 (1981), 275.
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