A survey on quasiperiodic topology

TL;DR

This survey reviews the mathematical and physical aspects of quasiperiodic topology, focusing on the Novikov problem related to the structure of foliations and level sets of multivalued functions on tori.

Contribution

It compiles and discusses the main analytical and numerical results on quasiperiodic functions and their level sets, highlighting recent progress and open problems.

Findings

Main results on level sets of quasiperiodic functions

Applications to physical phenomena involving quasiperiodic structures

Comparison of different cases of multivalued and singlevalued functions

Abstract

This article is a survey of the Novikov problem of the structure of leaves of the foliations induced by a collection of closed 1-forms in a compact manifold $M$. Equivalently, this is to the study of the level sets of multivalued functions on $M$. To date, this problem was thoroughly investigated only for $M=\Bbb T^n$ and multivalued maps $F:\Bbb T^n\to\Bbb R^{n-1}$ in three different particular cases: when all components of $F$ but one are multivalued, started by Novikov in 1981, when all components of $F$ but one are singlevalued, started by Zorich in 1994, when none of the components is singlevalued, started by Arnold in 1991. The first two problems can be formulated as the study of the level sets of certain quasiperiodic functions, the last as level sets of pseudoperiodic functions. In this survey we present the main analytical and numerical results to date and some physical phenomena where they play a fundamental role.