On the Novikov problem with a large number of quasiperiods and its generalizations

TL;DR

This paper investigates the geometry of level lines of quasiperiodic functions with many quasi-periods, generalizing the Novikov problem to higher dimensions and analyzing the emergence of large or open level lines relevant to dynamical systems.

Contribution

It extends the Novikov problem to functions with unlimited quasi-periods and generalizes results to multidimensional cases, broadening understanding of level surface structures.

Findings

Open and closed level lines of arbitrarily large sizes can occur.

Results applicable to multidimensional quasiperiodic functions.

Generalization of the Novikov problem to higher dimensions.

Abstract

The paper considers the Novikov problem of describing the geometry of level lines of quasi-periodic functions on the plane. We consider here the most general case, when the number of quasi-periods of a function is not limited. The main subject of investigation is the arising of open level lines or closed level lines of arbitrarily large sizes, which play an important role in many dynamical systems related to the general Novikov problem. As can be shown also, the results obtained 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.