The subject of the kinematic dynamo theory is the origin of magnetic fields in electroconductive media, playing an essential role in dynamics of astrophysical processes. One of the aspects of the rapid kinematic dynamo is the study of fluid motions which generate the exponential growth of the magnetic field at small magnetic diffusion [V. I. Arnold and B. A. Khesin, Topological methods in hydrodynamics. New York, NY: Springer (1998; Zbl 0902.76001)]. In the simplified form, the question is reduced to the existence of a conservative diffeomorphism \(f\colon M\to M\) of the Riemannian manifold \(M\) such that the energy of the magnetic field defined on \(M\) grows exponentially under the action of the iterations of the diffeomorphism \(f\).
In the reviewed article, a volume-preserving diffeomorphism with positive entropy is constructed for the 3-sphere. It means that in the space of conservative diffeomorphisms there exists a neighborhood in which diffeomorphisms have positive topological entropy.