Spectral analysis of time changes of horocycle flows

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Time-changes of horocycle flows

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

Abstract
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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.
[2]	W. O. Amrein, Hilbert space methods in quantum mechanics. Fundamental Sciences, EPFL Press, Lausanne, distributed by CRC Press, Boca Raton, FL, 2009.
[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.
[4]	A. Avila, G. Forni and C. Ulcigrai, Mixing for time-changes of heisenberg nilflows, J. Differential Geom., 89 (2011), 369-410.
[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.
[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.
[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.
[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.
[9]	B. Fayad, Partially mixing and locally rank 1 smooth transformations and flows on the torus T^d,$d $≥$ 3$, J. London Math. Soc. (2), 64 (2001), 637-654.
[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.
[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.
[12]	C. Fernández, S. Richard and R. Tiedra de Aldecoa, Commutator methods for unitary operators, J. Spectr. Theory, to appear, arXiv:1112.0167.
[13]	G. Forni and C. Ulcigrai, Time-changes of horocycle flows, preprint, arXiv:1202.4986.
[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.
[15]	G. A. Hedlund, Fuchsian groups and mixtures, Ann. of Math. (2), 40 (1939), 370-383.
[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.
[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.
[18]	A. G. Kushnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Moscow Univ. Math. Bull., 29 (1974), 82-87.
[19]	B. Marcus, The horocycle flow is mixing of all degrees, Invent. Math., 46 (1978), 201-209. doi: 10.1007/BF01390274.
[20]	É. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys., 78 (1980/81), 391-408. doi: 10.1007/BF01942331.
[21]	O. S. Parasyuk, Flows of horocycles on surfaces of constant negative curvature, Uspehi Matem. Nauk (N.S.), 8 (1953), 125-126.
[22]	W. Parry, "Topics in Ergodic Theory," Cambridge Tracts in Mathematics, 75, Cambridge University Press, Cambridge-New York, 1981.
[23]	J. Sahbani, The conjugate operator method for locally regular Hamiltonians, J. Operator Theory, 38 (1997), 297-322.
[24]	H. Totoki, Time changes of flows, Mem. Fac. Sci. Kyushu Univ. Ser. A, 20 (1966), 27-55. doi: 10.2206/kyushumfs.20.27.

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.
[2]	W. O. Amrein, Hilbert space methods in quantum mechanics. Fundamental Sciences, EPFL Press, Lausanne, distributed by CRC Press, Boca Raton, FL, 2009.
[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.
[4]	A. Avila, G. Forni and C. Ulcigrai, Mixing for time-changes of heisenberg nilflows, J. Differential Geom., 89 (2011), 369-410.
[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.
[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.
[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.
[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.
[9]	B. Fayad, Partially mixing and locally rank 1 smooth transformations and flows on the torus T^d,$d $≥$ 3$, J. London Math. Soc. (2), 64 (2001), 637-654.
[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.
[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.
[12]	C. Fernández, S. Richard and R. Tiedra de Aldecoa, Commutator methods for unitary operators, J. Spectr. Theory, to appear, arXiv:1112.0167.
[13]	G. Forni and C. Ulcigrai, Time-changes of horocycle flows, preprint, arXiv:1202.4986.
[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.
[15]	G. A. Hedlund, Fuchsian groups and mixtures, Ann. of Math. (2), 40 (1939), 370-383.
[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.
[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.
[18]	A. G. Kushnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Moscow Univ. Math. Bull., 29 (1974), 82-87.
[19]	B. Marcus, The horocycle flow is mixing of all degrees, Invent. Math., 46 (1978), 201-209. doi: 10.1007/BF01390274.
[20]	É. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys., 78 (1980/81), 391-408. doi: 10.1007/BF01942331.
[21]	O. S. Parasyuk, Flows of horocycles on surfaces of constant negative curvature, Uspehi Matem. Nauk (N.S.), 8 (1953), 125-126.
[22]	W. Parry, "Topics in Ergodic Theory," Cambridge Tracts in Mathematics, 75, Cambridge University Press, Cambridge-New York, 1981.
[23]	J. Sahbani, The conjugate operator method for locally regular Hamiltonians, J. Operator Theory, 38 (1997), 297-322.
[24]	H. Totoki, Time changes of flows, Mem. Fac. Sci. Kyushu Univ. Ser. A, 20 (1966), 27-55. doi: 10.2206/kyushumfs.20.27.

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