Time-changes of horocycle flows

Previous Article
Spectral analysis of time changes of horocycle flows
JMD Home
This Issue
Next Article
Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization

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

Time-changes of horocycle flows

Giovanni Forni ^1, and Corinna Ulcigrai ^2,

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

Abstract
Related Papers

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.

Keywords: Time-changes, horocycle flows, quantitative mixing, spectral theory., quantitative equidistribution.

Mathematics Subject Classification: Primary: 37A25, 37A30, 37C10, 37C40, 37D4.

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.
[2]	A. Bufetov and G. Forni, Limit theorems for horocycle flows, preprint, arXiv:1104.4502v1, (2011), 1-52.
[3]	M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803. doi: 10.1215/S0012-7094-90-06129-0.
[4]	S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385. doi: 10.1007/BF01389173.
[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.
[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.
[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.
[8]	I. M. Gelfand and M. Neumark, Unitary representations of the Lorentz group, Acad. Sci. USSR. J. Phys., 10 (1946), 93-94.
[9]	B. M. Gurevič, The entropy of horocycle flows, (Russian), Dokl. Akad. Nauk SSSR, 136 (1961), 768-770.
[10]	G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542. 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), 839-853.
[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.
[13]	A. G. Kušnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Vestnik Moskovskogo Universiteta. Ser. I Matematika Meh., 29 (1974), 101-108.
[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.
[15]	_____, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math. (2), 105 (1977), 81-105.
[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.
[17]	E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys., 78 (1980/81), 391-408. 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), 125-126.
[19]	M. Ratner, Factors of horocycle flows, Ergodic Theory Dynam. Systems, 2 (1982), 465-489. doi: 10.1017/S0143385700001723.
[20]	_____, Rigidity of horocycle flows, Ann. of Math. (2), 115 (1982), 597-614.
[21]	_____, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.
[22]	_____, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.
[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.
[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.
[25]	R. Tiedra de Aldecoa, Spectral analysis of time changes for horocycle flows, arXiv:1202.4640.
[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.
[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.

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.
[2]	A. Bufetov and G. Forni, Limit theorems for horocycle flows, preprint, arXiv:1104.4502v1, (2011), 1-52.
[3]	M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803. doi: 10.1215/S0012-7094-90-06129-0.
[4]	S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385. doi: 10.1007/BF01389173.
[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.
[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.
[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.
[8]	I. M. Gelfand and M. Neumark, Unitary representations of the Lorentz group, Acad. Sci. USSR. J. Phys., 10 (1946), 93-94.
[9]	B. M. Gurevič, The entropy of horocycle flows, (Russian), Dokl. Akad. Nauk SSSR, 136 (1961), 768-770.
[10]	G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542. 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), 839-853.
[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.
[13]	A. G. Kušnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Vestnik Moskovskogo Universiteta. Ser. I Matematika Meh., 29 (1974), 101-108.
[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.
[15]	_____, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math. (2), 105 (1977), 81-105.
[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.
[17]	E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys., 78 (1980/81), 391-408. 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), 125-126.
[19]	M. Ratner, Factors of horocycle flows, Ergodic Theory Dynam. Systems, 2 (1982), 465-489. doi: 10.1017/S0143385700001723.
[20]	_____, Rigidity of horocycle flows, Ann. of Math. (2), 115 (1982), 597-614.
[21]	_____, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313. doi: 10.2307/2007030.
[22]	_____, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.
[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.
[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.
[25]	R. Tiedra de Aldecoa, Spectral analysis of time changes for horocycle flows, arXiv:1202.4640.
[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.
[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.

[1]	Adam Kanigowski, Davide Ravotti. Polynomial 3-mixing for smooth time-changes of horocycle flows. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5347-5371. doi: 10.3934/dcds.2020230
[2]	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
[3]	Siyuan Tang. New time-changes of unipotent flows on quotients of Lorentz groups. Journal of Modern Dynamics, 2022, 18: 13-67. doi: 10.3934/jmd.2022002
[4]	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
[5]	Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33
[6]	Kazuhiko Yamamoto, Kiyoshi Hosono, Hiroko Nakayama, Akio Ito, Yuichi Yanagi. Experimental data for solid tumor cells: Proliferation curves and time-changes of heat shock proteins. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 235-244. doi: 10.3934/dcdss.2012.5.235
[7]	Saloni Rathee, Nilam. Quantitative analysis of time delays of glucose - insulin dynamics using artificial pancreas. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3115-3129. doi: 10.3934/dcdsb.2015.20.3115
[8]	Alex Eskin, Gregory Margulis and Shahar Mozes. On a quantitative version of the Oppenheim conjecture. Electronic Research Announcements, 1995, 1: 124-130.
[9]	Mauro Cesa. A brief history of quantitative finance. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 6-. doi: 10.1186/s41546-017-0018-3
[10]	Lin Wang. Quantitative destruction of invariant circles. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1569-1583. doi: 10.3934/dcds.2021164
[11]	Matthias Täufer, Martin Tautenhahn. Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1719-1730. doi: 10.3934/cpaa.2017083
[12]	Stefano Galatolo, Rafael Lucena. Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1309-1360. doi: 10.3934/dcds.2020079
[13]	Jeffrey Boland. On rigidity properties of contact time changes of locally symmetric geodesic flows. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 645-650. doi: 10.3934/dcds.2000.6.645
[14]	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
[15]	Aki Pulkkinen, Ville Kolehmainen, Jari P. Kaipio, Benjamin T. Cox, Simon R. Arridge, Tanja Tarvainen. Approximate marginalization of unknown scattering in quantitative photoacoustic tomography. Inverse Problems and Imaging, 2014, 8 (3) : 811-829. doi: 10.3934/ipi.2014.8.811
[16]	Jon Chaika, David Constantine. A quantitative shrinking target result on Sturmian sequences for rotations. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5189-5204. doi: 10.3934/dcds.2018229
[17]	Max Fathi, Emanuel Indrei, Michel Ledoux. Quantitative logarithmic Sobolev inequalities and stability estimates. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6835-6853. doi: 10.3934/dcds.2016097
[18]	Angkana Rüland, Mikko Salo. Quantitative approximation properties for the fractional heat equation. Mathematical Control and Related Fields, 2020, 10 (1) : 1-26. doi: 10.3934/mcrf.2019027
[19]	Florian Dumpert. Quantitative robustness of localized support vector machines. Communications on Pure and Applied Analysis, 2020, 19 (8) : 3947-3956. doi: 10.3934/cpaa.2020174
[20]	Zhongqi Yin. A quantitative internal unique continuation for stochastic parabolic equations. Mathematical Control and Related Fields, 2015, 5 (1) : 165-176. doi: 10.3934/mcrf.2015.5.165

2021 Impact Factor: 0.641

Tools

Metrics

Other articles
by authors

on AIMS
Giovanni Forni Corinna Ulcigrai
on Google Scholar
Giovanni Forni Corinna Ulcigrai

[Back to Top]