American Institute of Mathematical Sciences

Advanced
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, (1978).   Google Scholar

[2]

W. O. Amrein, Hilbert space methods in quantum mechanics. Fundamental Sciences,, EPFL Press, (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, (1996).   Google Scholar

[4]

A. Avila, G. Forni and C. Ulcigrai, Mixing for time-changes of heisenberg nilflows,, J. Differential Geom., 89 (2011), 369.   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, (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, (2000).   Google Scholar

[7]

A. Boutet de Monvel and V. Georgescu, The method of differential inequalities,, in, 12 (1991), 279.   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, (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.   Google Scholar

[10]

B. Fayad, Smooth mixing flows with purely singular spectra,, Duke Math. J., 132 (2006), 371.  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, 1 (2001), 521.   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.  doi: 10.1007/s00220-010-1120-x.  Google Scholar

[15]

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

[16]

P. D. Humphries, Change of velocity in dynamical systems,, J. London Math. Soc. (2), 7 (1974), 747.  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, (2006), 649.  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.   Google Scholar

[19]

B. Marcus, The horocycle flow is mixing of all degrees,, Invent. Math., 46 (1978), 201.  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.   Google Scholar

[22]

W. Parry, "Topics in Ergodic Theory," Cambridge Tracts in Mathematics, 75,, Cambridge University Press, (1981).   Google Scholar

[23]

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

[24]

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

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics,", Second edition, (1978).   Google Scholar

[2]

W. O. Amrein, Hilbert space methods in quantum mechanics. Fundamental Sciences,, EPFL Press, (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, (1996).   Google Scholar

[4]

A. Avila, G. Forni and C. Ulcigrai, Mixing for time-changes of heisenberg nilflows,, J. Differential Geom., 89 (2011), 369.   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, (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, (2000).   Google Scholar

[7]

A. Boutet de Monvel and V. Georgescu, The method of differential inequalities,, in, 12 (1991), 279.   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, (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.   Google Scholar

[10]

B. Fayad, Smooth mixing flows with purely singular spectra,, Duke Math. J., 132 (2006), 371.  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, 1 (2001), 521.   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.  doi: 10.1007/s00220-010-1120-x.  Google Scholar

[15]

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

[16]

P. D. Humphries, Change of velocity in dynamical systems,, J. London Math. Soc. (2), 7 (1974), 747.  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, (2006), 649.  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.   Google Scholar

[19]

B. Marcus, The horocycle flow is mixing of all degrees,, Invent. Math., 46 (1978), 201.  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.   Google Scholar

[22]

W. Parry, "Topics in Ergodic Theory," Cambridge Tracts in Mathematics, 75,, Cambridge University Press, (1981).   Google Scholar

[23]

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

[24]

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

[1]

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
[2]

Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94.

[3]

Hong Lu, Ji Li, Mingji Zhang. Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020065
[4]

Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4619-4635. doi: 10.3934/dcds.2016001
[5]

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
[6]

Yuhong Dai, Ya-xiang Yuan. Analysis of monotone gradient methods. Journal of Industrial & Management Optimization, 2005, 1 (2) : 181-192. doi: 10.3934/jimo.2005.1.181
[7]

Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Corrigendum to "Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology". Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4585-4586. doi: 10.3934/dcds.2017196
[8]

Reimund Rautmann. Lower and upper bounds to the change of vorticity by transition from slip- to no-slip fluid flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1101-1109. doi: 10.3934/dcdss.2014.7.1101
[9]

Pikkala Vijaya Laxmi, Seleshi Demie. Performance analysis of renewal input $(a,c,b)$ policy queue with multiple working vacations and change over times. Journal of Industrial & Management Optimization, 2014, 10 (3) : 839-857. doi: 10.3934/jimo.2014.10.839
[10]

Thomas Schuster, Joachim Weickert. On the application of projection methods for computing optical flow fields. Inverse Problems & Imaging, 2007, 1 (4) : 673-690. doi: 10.3934/ipi.2007.1.673
[11]

Andrei V. Dmitruk, Nikolai P. Osmolovskii. Proof of the maximum principle for a problem with state constraints by the v-change of time variable. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2189-2204. doi: 10.3934/dcdsb.2019090
[12]

Roman Shvydkoy, Eitan Tadmor. Eulerian dynamics with a commutator forcing Ⅱ: Flocking. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5503-5520. doi: 10.3934/dcds.2017239
[13]

Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure & Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1
[14]

Vu Hoang Linh, Volker Mehrmann. Spectral analysis for linear differential-algebraic equations. Conference Publications, 2011, 2011 (Special) : 991-1000. doi: 10.3934/proc.2011.2011.991
[15]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665
[16]

Yin Yang, Yunqing Huang. Spectral Jacobi-Galerkin methods and iterated methods for Fredholm integral equations of the second kind with weakly singular kernel. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 685-702. doi: 10.3934/dcdss.2019043
[17]

Xijun Hu, Li Wu. Decomposition of spectral flow and Bott-type iteration formula. Electronic Research Archive, 2020, 28 (1) : 127-148. doi: 10.3934/era.2020008
[18]

Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677
[19]

Dongping Zhuang. Irrational stable commutator length in finitely presented groups. Journal of Modern Dynamics, 2008, 2 (3) : 499-507. doi: 10.3934/jmd.2008.2.499
[20]

Kim Dang Phung. Carleman commutator approach in logarithmic convexity for parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 899-933. doi: 10.3934/mcrf.2018040

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences