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April  2012, 6(2): 275-285. doi: 10.3934/jmd.2012.6.275

Spectral analysis of time changes of horocycle flows

Rafael Tiedra De Aldecoa 1,

1. 

Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile

Received  February 2012 Published  August 2012

We prove (under the condition of A. G. Kushnirenko) that all time changes of the horocycle flow have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This provides an answer to a question of A. Katok and J.-P. Thouvenot on the spectral nature of time changes of horocycle flows. Our proofs rely on positive commutator methods for self-adjoint operators.
Keywords: Horocycle flow, commutator methods., time change, spectral analysis.
Mathematics Subject Classification: 37C10, 37D40, 81Q10, 58J5.
Citation: 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
References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," Second edition, revised and enlarged, With the assistance of TudorRaţiu and Richard Cushman, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.  Google Scholar

[2]

W. O. Amrein, Hilbert space methods in quantum mechanics. Fundamental Sciences, EPFL Press, Lausanne, distributed by CRC Press, Boca Raton, FL, 2009.  Google Scholar

[3]

W. O. Amrein, A. Boutet de Monveland and V. Georgescu, "$ C_0 $-Groups, Commutator Methods and Spectral Theory of $N$-Body Hamiltonians," Progress in Math., 135, Birkhäuser Verlag, Basel, 1996.  Google Scholar

[4]

A. Avila, G. Forni and C. Ulcigrai, Mixing for time-changes of heisenberg nilflows, J. Differential Geom., 89 (2011), 369-410.  Google Scholar

[5]

H. Baumgärtel and M. Wollenberg, Mathematical scattering theory, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], 59, Akademie-Verlag, Berlin, 1983.  Google Scholar

[6]

M. B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Mathematical Society Lecture Note Series, 269, Cambridge University Press, Cambridge, 2000.  Google Scholar

[7]

A. Boutet de Monvel and V. Georgescu, The method of differential inequalities, in "Recent Developments in Quantum Mechanics" (Poiana Braşov, 1989), Math. Phys. Stud., 12, Kluwer Acad. Publ., Dordrecht, (1991), 279-298.  Google Scholar

[8]

I. P. Cornfeld, S. V. Fomin and Y. G. Sinaĭ, "Ergodic Theory," Translated from the Russian by A. B. Sosinskiĭ, Grundlehren derMathematischen Wissenschaften [Fundamental Principles of MathematicalSciences], 245, Springer-Verlag, New York, 1982.  Google Scholar

[9]

B. Fayad, Partially mixing and locally rank 1 smooth transformations and flows on the torus Td,$d $≥$ 3$, J. London Math. Soc. (2), 64 (2001), 637-654.  Google Scholar

[10]

B. Fayad, Smooth mixing flows with purely singular spectra, Duke Math. J., 132 (2006), 371-391. doi: 10.1215/S0012-7094-06-13225-8.  Google Scholar

[11]

B. Fayad, A. Katok and A. Windsor, Mixed spectrum reparameterizations of linear flows on $\mathbb T^2$, Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary, Mosc. Math. J., 1 (2001), 521-537, 644.  Google Scholar

[12]

C. Fernández, S. Richard and R. Tiedra de Aldecoa, Commutator methods for unitary operators,, J. Spectr. Theory, ().   Google Scholar

[13]

G. Forni and C. Ulcigrai, Time-changes of horocycle flows,, preprint, ().   Google Scholar

[14]

K. Gelfert and A. E. Motter, (Non)invariance of dynamical quantities for orbit equivalent flows, Comm. Math. Phys., 300 (2010), 411-433. doi: 10.1007/s00220-010-1120-x.  Google Scholar

[15]

G. A. Hedlund, Fuchsian groups and mixtures, Ann. of Math. (2), 40 (1939), 370-383.  Google Scholar

[16]

P. D. Humphries, Change of velocity in dynamical systems, J. London Math. Soc. (2), 7 (1974), 747-757. doi: 10.1112/jlms/s2-7.4.747.  Google Scholar

[17]

A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, in "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 649-743. doi: 10.1016/S1874-575X(06)80036-6.  Google Scholar

[18]

A. G. Kushnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Moscow Univ. Math. Bull., 29 (1974), 82-87. Google Scholar

[19]

B. Marcus, The horocycle flow is mixing of all degrees, Invent. Math., 46 (1978), 201-209. doi: 10.1007/BF01390274.  Google Scholar

[20]

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

[21]

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

[22]

W. Parry, "Topics in Ergodic Theory," Cambridge Tracts in Mathematics, 75, Cambridge University Press, Cambridge-New York, 1981.  Google Scholar

[23]

J. Sahbani, The conjugate operator method for locally regular Hamiltonians, J. Operator Theory, 38 (1997), 297-322.  Google Scholar

[24]

H. Totoki, Time changes of flows, Mem. Fac. Sci. Kyushu Univ. Ser. A, 20 (1966), 27-55. doi: 10.2206/kyushumfs.20.27.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," Second edition, revised and enlarged, With the assistance of TudorRaţiu and Richard Cushman, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.  Google Scholar

[2]

W. O. Amrein, Hilbert space methods in quantum mechanics. Fundamental Sciences, EPFL Press, Lausanne, distributed by CRC Press, Boca Raton, FL, 2009.  Google Scholar

[3]

W. O. Amrein, A. Boutet de Monveland and V. Georgescu, "$ C_0 $-Groups, Commutator Methods and Spectral Theory of $N$-Body Hamiltonians," Progress in Math., 135, Birkhäuser Verlag, Basel, 1996.  Google Scholar

[4]

A. Avila, G. Forni and C. Ulcigrai, Mixing for time-changes of heisenberg nilflows, J. Differential Geom., 89 (2011), 369-410.  Google Scholar

[5]

H. Baumgärtel and M. Wollenberg, Mathematical scattering theory, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], 59, Akademie-Verlag, Berlin, 1983.  Google Scholar

[6]

M. B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Mathematical Society Lecture Note Series, 269, Cambridge University Press, Cambridge, 2000.  Google Scholar

[7]

A. Boutet de Monvel and V. Georgescu, The method of differential inequalities, in "Recent Developments in Quantum Mechanics" (Poiana Braşov, 1989), Math. Phys. Stud., 12, Kluwer Acad. Publ., Dordrecht, (1991), 279-298.  Google Scholar

[8]

I. P. Cornfeld, S. V. Fomin and Y. G. Sinaĭ, "Ergodic Theory," Translated from the Russian by A. B. Sosinskiĭ, Grundlehren derMathematischen Wissenschaften [Fundamental Principles of MathematicalSciences], 245, Springer-Verlag, New York, 1982.  Google Scholar

[9]

B. Fayad, Partially mixing and locally rank 1 smooth transformations and flows on the torus Td,$d $≥$ 3$, J. London Math. Soc. (2), 64 (2001), 637-654.  Google Scholar

[10]

B. Fayad, Smooth mixing flows with purely singular spectra, Duke Math. J., 132 (2006), 371-391. doi: 10.1215/S0012-7094-06-13225-8.  Google Scholar

[11]

B. Fayad, A. Katok and A. Windsor, Mixed spectrum reparameterizations of linear flows on $\mathbb T^2$, Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary, Mosc. Math. J., 1 (2001), 521-537, 644.  Google Scholar

[12]

C. Fernández, S. Richard and R. Tiedra de Aldecoa, Commutator methods for unitary operators,, J. Spectr. Theory, ().   Google Scholar

[13]

G. Forni and C. Ulcigrai, Time-changes of horocycle flows,, preprint, ().   Google Scholar

[14]

K. Gelfert and A. E. Motter, (Non)invariance of dynamical quantities for orbit equivalent flows, Comm. Math. Phys., 300 (2010), 411-433. doi: 10.1007/s00220-010-1120-x.  Google Scholar

[15]

G. A. Hedlund, Fuchsian groups and mixtures, Ann. of Math. (2), 40 (1939), 370-383.  Google Scholar

[16]

P. D. Humphries, Change of velocity in dynamical systems, J. London Math. Soc. (2), 7 (1974), 747-757. doi: 10.1112/jlms/s2-7.4.747.  Google Scholar

[17]

A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, in "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 649-743. doi: 10.1016/S1874-575X(06)80036-6.  Google Scholar

[18]

A. G. Kushnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Moscow Univ. Math. Bull., 29 (1974), 82-87. Google Scholar

[19]

B. Marcus, The horocycle flow is mixing of all degrees, Invent. Math., 46 (1978), 201-209. doi: 10.1007/BF01390274.  Google Scholar

[20]

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

[21]

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

[22]

W. Parry, "Topics in Ergodic Theory," Cambridge Tracts in Mathematics, 75, Cambridge University Press, Cambridge-New York, 1981.  Google Scholar

[23]

J. Sahbani, The conjugate operator method for locally regular Hamiltonians, J. Operator Theory, 38 (1997), 297-322.  Google Scholar

[24]

H. Totoki, Time changes of flows, Mem. Fac. Sci. Kyushu Univ. Ser. A, 20 (1966), 27-55. doi: 10.2206/kyushumfs.20.27.  Google Scholar

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