On the types of quasiperiodic potentials and level surfaces of quasiperiodic functions

Abstract

We examine the similarities and differences between two types of quasiperiodic functions: those obtained by restricting smooth periodic functions in $\mathbb{R}^N$ to lower-dimensional subspaces $\mathbb{R}^n \subset \mathbb{R}^N$, and those that arise as physical fields or potentials in quasicrystalline structures. Our analysis reveals that functions in these two classes share a number of common features while also exhibiting significant differences. As a key tool for investigating both classes of functions, we employ the Novikov problem - specifically, the problem of describing the global geometry of the level sets of quasiperiodic functions.