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), 568-640. doi: 10.2307/1969129.  Google Scholar

[2]

A. Bufetov and G. Forni, Limit theorems for horocycle flows, preprint, arXiv:1104.4502v1, (2011), 1-52. Google Scholar

[3]

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

[4]

S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385. doi: 10.1007/BF01389173.  Google Scholar

[5]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar

[6]

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, (1973), 95-115.  Google Scholar

[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), 771-774.  Google Scholar

[8]

I. M. Gelfand and M. Neumark, Unitary representations of the Lorentz group, Acad. Sci. USSR. J. Phys., 10 (1946), 93-94.  Google Scholar

[9]

B. M. Gurevič, The entropy of horocycle flows, (Russian), Dokl. Akad. Nauk SSSR, 136 (1961), 768-770.  Google Scholar

[10]

G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542. doi: 10.1215/S0012-7094-36-00246-6.  Google Scholar

[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), 839-853.  Google Scholar

[12]

A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, in "Handbook of Dynamical Systems," Vol. 1B (eds. B. Hasselblatt and A. Katok), Elsevier B. V., Amsterdam, (2006), 649-743.  Google Scholar

[13]

A. G. Kušnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Vestnik Moskovskogo Universiteta. Ser. I Matematika Meh., 29 (1974), 101-108.  Google Scholar

[14]

B. Marcus, Unique ergodicity of the horocycle flow: Variable negative curvature case, Conference on Ergodic Theory and Topological Dynamics (Kibbutz Lavi, 1974), Israel J. Math., 21 (1975), 133-144.  Google Scholar

[15]

_____, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math. (2), 105 (1977), 81-105.  Google Scholar

[16]

C. C. Moore, Exponential decay of correlation coefficients for geodesic flows, in "Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics" (Berkeley, Calif., 1984) (ed. C. C. Moore), Mathematical Science Research Institute Publications, 6, Springer, New York, (1987), 163-181.  Google Scholar

[17]

E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators,, Comm. Math. Phys., 78 (): 391.  doi: 10.1007/BF01942331.  Google Scholar

[18]

O. S. Parasyuk, Flows of horocycles on surfaces of constant negative curvature, (Russian), Uspekhi Mat. Nauk (N.S.), 8 (1953), 125-126.  Google Scholar

[19]

M. Ratner, Factors of horocycle flows, Ergodic Theory Dynam. Systems, 2 (1982), 465-489. doi: 10.1017/S0143385700001723.  Google Scholar

[20]

_____, Rigidity of horocycle flows, Ann. of Math. (2), 115 (1982), 597-614.  Google Scholar

[21]

_____, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.  Google Scholar

[22]

_____, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.  Google Scholar

[23]

P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739. doi: 10.1002/cpa.3160340602.  Google Scholar

[24]

A. Strömbergsson, On the uniform equidistribution of long closed horocycles, Duke Math. J., 123 (2004), 507-547. doi: 10.1215/S0012-7094-04-12334-6.  Google Scholar

[25]

R. Tiedra de Aldecoa, Spectral analysis of time changes for horocycle flows,, , ().   Google Scholar

[26]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989-1094. doi: 10.4007/annals.2010.172.989.  Google Scholar

[27]

D. Zagier, Eisenstein series and the Riemann zeta function, in "Automorphic Forms, Representation Theory and Arithmetic" (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., 10, Tata Inst. Fund. Res., Bombay, (1981), 275-301.  Google Scholar

show all references

References:
[1]

V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. (2), 48 (1947), 568-640. doi: 10.2307/1969129.  Google Scholar

[2]

A. Bufetov and G. Forni, Limit theorems for horocycle flows, preprint, arXiv:1104.4502v1, (2011), 1-52. Google Scholar

[3]

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

[4]

S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385. doi: 10.1007/BF01389173.  Google Scholar

[5]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar

[6]

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, (1973), 95-115.  Google Scholar

[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), 771-774.  Google Scholar

[8]

I. M. Gelfand and M. Neumark, Unitary representations of the Lorentz group, Acad. Sci. USSR. J. Phys., 10 (1946), 93-94.  Google Scholar

[9]

B. M. Gurevič, The entropy of horocycle flows, (Russian), Dokl. Akad. Nauk SSSR, 136 (1961), 768-770.  Google Scholar

[10]

G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542. doi: 10.1215/S0012-7094-36-00246-6.  Google Scholar

[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), 839-853.  Google Scholar

[12]

A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, in "Handbook of Dynamical Systems," Vol. 1B (eds. B. Hasselblatt and A. Katok), Elsevier B. V., Amsterdam, (2006), 649-743.  Google Scholar

[13]

A. G. Kušnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Vestnik Moskovskogo Universiteta. Ser. I Matematika Meh., 29 (1974), 101-108.  Google Scholar

[14]

B. Marcus, Unique ergodicity of the horocycle flow: Variable negative curvature case, Conference on Ergodic Theory and Topological Dynamics (Kibbutz Lavi, 1974), Israel J. Math., 21 (1975), 133-144.  Google Scholar

[15]

_____, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math. (2), 105 (1977), 81-105.  Google Scholar

[16]

C. C. Moore, Exponential decay of correlation coefficients for geodesic flows, in "Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics" (Berkeley, Calif., 1984) (ed. C. C. Moore), Mathematical Science Research Institute Publications, 6, Springer, New York, (1987), 163-181.  Google Scholar

[17]

E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators,, Comm. Math. Phys., 78 (): 391.  doi: 10.1007/BF01942331.  Google Scholar

[18]

O. S. Parasyuk, Flows of horocycles on surfaces of constant negative curvature, (Russian), Uspekhi Mat. Nauk (N.S.), 8 (1953), 125-126.  Google Scholar

[19]

M. Ratner, Factors of horocycle flows, Ergodic Theory Dynam. Systems, 2 (1982), 465-489. doi: 10.1017/S0143385700001723.  Google Scholar

[20]

_____, Rigidity of horocycle flows, Ann. of Math. (2), 115 (1982), 597-614.  Google Scholar

[21]

_____, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.  Google Scholar

[22]

_____, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.  Google Scholar

[23]

P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739. doi: 10.1002/cpa.3160340602.  Google Scholar

[24]

A. Strömbergsson, On the uniform equidistribution of long closed horocycles, Duke Math. J., 123 (2004), 507-547. doi: 10.1215/S0012-7094-04-12334-6.  Google Scholar

[25]

R. Tiedra de Aldecoa, Spectral analysis of time changes for horocycle flows,, , ().   Google Scholar

[26]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989-1094. doi: 10.4007/annals.2010.172.989.  Google Scholar

[27]

D. Zagier, Eisenstein series and the Riemann zeta function, in "Automorphic Forms, Representation Theory and Arithmetic" (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., 10, Tata Inst. Fund. Res., Bombay, (1981), 275-301.  Google Scholar

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