Spectral properties of horocycle flows for compact surfaces of constant negative curvature

Rafael Tiedra de Aldecoa

Resumen


We consider flows, called Wu flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of Wuflows and we show that Wu flows have purely absolutely continuous spectrum in the orthocomplement of the constant functions. As an application, we obtain that time changes of the classical horocycle flows for compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions for time changes in a regularity class slightly less than C2. This generalises recent results on time changes ofhorocycle flows.

Palabras clave


Horocycle flow; Anosov flow; strong mixing; continuous spectrum; commutator methods.

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Referencias


W. O. Amrein, A. Boutet de Monvel, and V. Georgescu. C0-groups, commutator methods and spectral theory of N-body Hamiltonians, volume 135 of Progress in Mathematics. Birkhäuser Verlag, Basel, (1996).

D. V. Anosov. Geodesic flows on closed Riemann manifolds with negative curvature. Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder. American Mathematical Society, Providence, R.I., 1969.

M. Beboutoff and W. Stepanoff. Sur la mesure invariante dans les systèmes dynamiques qui ne diffèrent que par le temps. Rec. Math. [Mat. Sbornik] N.S., 7 (49), pp. 143—166, (1940).

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

R. Bowen and B. Marcus. Unique ergodicity for horocycle foliations. Israel J. Math., 26(1), pp. 43—67, (1977).

G. Forni and C. Ulcigrai. Time-changes of horocycle flows. J. Mod. Dyn., 6(2), pp. 251—273, (2012).

L. W. Green. Remarks on uniformly expanding horocycle parameterizations. J. Differential Geom., 13(2), pp. 263—271, (1978).

G. A. Hedlund. Fuchsian groups and transitive horocycles. Duke Math. J., 2(3), pp. 530—542, (1936).

G. A. Hedlund. Fuchsian groups and mixtures. Ann. of Math. (2), 40(2), pp. 370—383, (1939).

E. Hopf. Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91, pp. 261— 304, (1939).

P. D. Humphries. Change of velocity in dynamical systems. J. London Math. Soc. (2), 7, pp. 747—757, (1974).

M. C. Irwin. Smooth dynamical systems, volume 17 of Advanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River Edge, NJ, (2001). Reprint of the 1980 original, With a foreword by R. S. MacKay.

A. G. Kushnirenko. Spectral properties of certain dynamical systems with polynomial dispersal. Moscow Univ. Math. Bull., 29(1), pp. 82— 87, (1974).

B. Marcus. Unique ergodicity of the horocycle flow: variable negative curvature case. Israel J. Math., 21(2-3), pp. 133—144, (1975). Conference on Ergodic Theory and Topological Dynamics Kibbutz Lavi, (1974).

B. Marcus. Ergodic properties of horocycle flows for surfaces of negative curvature. Ann. of Math. (2), 105 (1), pp. 81—105, (1977).

G. A. Margulis. Certain measures that are connected with U-flows on compact manifolds. Funkcional. Anal. i Priloˇ zen., 4 (1), pp. 62—76, (1970).

S. Matsumoto. Codimension one Anosov flows, volume 27 of Lecture Notes Series. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, (1995).

É. Mourre. Absence of singular continuous spectrum for certain self-adjoint operators. Comm. Math. Phys., 78 (3), pp. 391—408, 1980/81.

O. S. Parasyuk. Flows of horocycles on surfaces of constant negative curvature. Uspehi Matem. Nauk (N.S.), 8 (3(55)):125—126, 1953.

Y. B. Pesin. Geodesic flows with hyperbolic behavior of trajectories and objects connected with them. Uspekhi Mat. Nauk, 36 (4(220)), pp. 3—51, 247, (1981).

M. Reed and B. Simon. Methods of modern mathematical physics. I. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, second edition, (1980). Functional analysis.

J. Sahbani. The conjugate operator method for locally regular Hamiltonians. J. Operator Theory, 38(2), pp. 297—322, (1997).

R. Tiedra de Aldecoa. Spectral analysis of time changes of horocycle flows. J. Mod. Dyn., 6(2), pp. 275—285, (2012).

R. Tiedra de Aldecoa. The absolute continuous spectrum of skew products of compact lie groups. Israel J. Math., 208(1), pp. 323—350, (2015).

R. Tiedra de Aldecoa. Commutator methods for the spectral analysis of uniquely ergodic dynamical systems. Ergodic Theory Dynam. Systems, 35(3), pp. 944—967, (2015).




DOI: http://dx.doi.org/10.4067/S0716-09172017000100006

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