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A Survey on Quasiperiodic Topology

  • Roberto De LeoEmail author
Chapter
Part of the STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health book series

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 \(M={\mathbb {T}}^n\) and multivalued maps \(F:{\mathbb {T}}^n\to \mathbb {R}^{n-1}\) in three different particular cases: when all components of F but one are multivalued, started by Novikov in 1982; 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.

Keywords

Quasiperiodic functions Quasiperiodic topology Closed 1-forms Foliations Multivalued functions 

Notes

Acknowledgements

The author gladly thanks S.P. Novikov for introducing him to the subject and for his encouragement and support throughout the years and A. Zorich and I. Gelbukh for several discussions and suggestions on the present survey. All numerical calculations by the author presented here were made on the computational clusters of the National Institute for Nuclear Physics (INFN) in Cagliari (Italy) and, at Howard University in Washington, DC (USA), on those of the College of Arts and Sciences and of the Center for Computational Biology and Bioinformatics. Part of this work was supported by an Advanced Summer Faculty Research Fellowship of Howard University.

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Authors and Affiliations

  1. 1.Department of MathematicsHoward UniversityWashington, DCUSA

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