In a work together with Bert Hof, we proved that in general, a
realanalytic differential equation on the two dimensional torus
with smooth invariant measure is weakly mixing and
has zero-dimensional spectrum.
(The last statement means that
the spectral measures are continuous but supported on zero dimensional
On this page you see an animation of such a flow, computed numerically.
Using a result of Strichartz in Fourier theory, we give a
new proof for a general result of Iwanik which estimates the dimension
of spectral measures of an ergodic automorphism in terms of the speed
of approximation. We proved also the corresponding result for flows.
Our paper has appeared in
in Ergod. Th. and Dyn. Systems, 18, 879-888, 1998.
See Slides AMS Meeting talk
This research has applications in the theory of Hamiltonian systems.
As an offspin of this work with Hof and using a suggestion of Wojtkowski,
we showed that every Hamiltonian system with an invariant KAM torus of
dimension d>1 can be perturbed such that the torus becomes weakly mixing.
The proof of this result
needs a theorem of Arnold and Moser on differential equations on
the torus and a theorem of Simon in Spectral theory. This paper appeared
in Commun. Math. Phys, 204, 85-88, 1999
-> Springer Link
See the Slides of Western States Talk