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Zero-dimensional singular continuous spectrum for smooth differential equations on the torus

Published online by Cambridge University Press:  01 August 1998

A. HOF  and
O. KNILL
A. HOF
Affiliation:
Division of Physics, Mathematics and Astronomy, California Institute of Technology 253-37, Pasadena, CA 91125, USA Current address: NCM, PO Box 473, 1000 AL Amsterdam, The Netherlands.
O. KNILL
Affiliation:
Division of Physics, Mathematics and Astronomy, California Institute of Technology 253-37, Pasadena, CA 91125, USA Current address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA.
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Abstract

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We study spectral properties of the flow $\dot x =1/F(x,y)$, $\dot y = 1/\lambda F(x,y)$ on the 2-torus. We show that, in general, the speed of approximation in cyclic approximation gives an upper bound on the Hausdorff dimension of the supports of spectral measures. We use this to prove that for generic pairs $(F,\lambda)$ the spectrum of the flow on the torus is singular continuous with all spectral measures supported on sets of zero Hausdorff dimension.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 18 , Issue 4 , August 1998 , pp. 879 - 888
Copyright
© 1998 Cambridge University Press