Topology of dynamical systems on the Fermi surface and galvanomagnetic phenomena in normal metals

We discuss here the electron dynamics in crystals in rather strong magnetic fields. The most non-trivial part of this topic is the Novikov problem, namely, the problem of classifying non-closed trajectories for particles with an arbitrary dispersion relation in presence of a magnetic field. Here we will try to review both the classical results obtained in the study of this problem and the most recent results related to its most difficult part. At the same time, the study of electron dynamics on the Fermi surface can also be carried out in a more general setting, namely, as the study of the topology of the corresponding dynamical system and the physical phenomena associated with it. This approach is actually related to the Novikov problem, but includes the study of a wider range of issues, as well as a wider range of experimentally observed phenomena. Here we will try to describe a number of general principles relating changes in the topological pattern of trajectories on the Fermi surface to the observed phenomena in strong magnetic fields. We also note here that the study of the described effects can be quite informative for the experimental study of dispersion relations in conductors.