The Spectrum of the Kinematic DynamoOperator for an Ideally Conducting

The spectrum of the kinematic dynamo operator for an ideally conducting uid and the spectrum of the corresponding group acting in the space of continuous divergence free vector elds on a compact Rie-mannian 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 ow. 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.