Toral flows


Realanalytic 
Flow on the torus 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 Borel sets). 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


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