Ergodic Theory and Dynamical Systems

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

A. HOF a1p1 and O. KNILL a1p2
a1 Division of Physics, Mathematics and Astronomy, California Institute of Technology 253-37, Pasadena, CA 91125, USA


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.

(Received September 14 1996)
(Revised July 24 1997)

p1 Current address: NCM, PO Box 473, 1000 AL Amsterdam, The Netherlands.
p2 Current address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA.