THE ANNULAR HULL THEOREMS FOR THE KINEMATIC DYNAMO OPERATOR FOR AN IDEALLY CONDUCTING FLUID

The group generated by the kinematic dynamo operator in the space of c ontinuous divergence-free sections of the tangent bundle of a smooth m anifold is studied. As shown in previous work, if the underlying Euler ian flow is aperiodic, then the spectrum of this group is obtained fro m the spectrum of its generator by exponentiation, but this result doe s not hold for flows with the set of periodic trajectories having none mpty interior. In the present paper, we consider Eulerian vector field s with periodic trajectories and prove the following annular hull theo rems: The spectrum of the group belongs to the annular hull of the exp onent of the spectrum of the kinematic dynamo operator, that is to the union of all circles centered at the origin and intersecting this set . Also, the annular hull of the spectrum of the group on the space of divergence free vector fields coincides with the smallest annulus, con taining the spectrum of the group on the space of all continuous vecto r fields. As a corollary, the spectral abscissa of the generator coinc ides with the growth bound for the group.