The Spectrum of the Kinematic Dynamo Operator for an Ideally Conducting Fluid

The spectrum of the kinematic dynamo operator for an ideally conducting fluid and the spectrum of the corresponding group acting in the space of continuous divergence free vector fields on a compact Riemannian manifold are described. We prove that the spectrum of the kinematic dynamo operator is exactly one vertical strip whose boundaries can be determined in terms of the Lyapunov-Oseledets exponents with respect to all ergodic measures for the Eulerian flow. Also, we prove that the spectrum of the corresponding group is obtained from the spectrum of its generator by exponentiation. In particular, the growth bound for the group coincides with the spectral bound for the generator. ∗[email protected], supported by the NSF grant DMS-9303767; †[email protected], supported by the NSF grant DMS-9400518 and by the Summer Research Fellowship of the University of Missouri; ‡[email protected], supported by the NSF grant DMS-9201357;