On the Novikov Problem with a Large Number of Quasiperiods and Its Generalizations

The paper is devoted to the Novikov problem of describing the geometry of level lines of quasiperiodic functions on the plane. We consider here the most general case, when the number of quasiperiods of a function is not limited. The main subject of investigation is the occurrence of either open level lines or closed level lines of arbitrarily large size, which play an important role in many dynamical systems related to the general Novikov problem. As can be shown, the results obtained here for quasiperiodic functions on the plane can be generalized to the multidimensional case. In this case, we are dealing with a generalized Novikov problem, namely, the problem of describing level surfaces of quasiperiodic functions in a space of arbitrary dimension. Like the Novikov problem on the plane, the generalized Novikov problem plays an important role in many systems containing quasiperiodic modulations.