Dynamic systems on two-dimensional surfaces and non-trivial conductivity regimes in metals

We discuss 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 be interested in trajectories of the most complex shape, arising on rather complex Fermi surfaces. The geometric properties of such trajectories are quite non-trivial and are revealed in the study of galvanomagnetic phenomena in strong magnetic fields. Many important properties of such trajectories can be described using ideas and methods that arise in the theory of dynamical systems on complex two-dimensional surfaces. The connection of such ideas and methods with the theory of galvanomagnetic phenomena represents one of the most interesting directions in the study of Novikov’s problem.