A survey on quasiperiodic topology

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.