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Communications in Mathematical Physics

, Volume 324, Issue 2, pp 549–588 | Cite as

Uncovering Fractional Monodromy

  • K. EfstathiouEmail author
  • H. W. Broer
Article

Abstract

The uncovering of the role of monodromy in integrable Hamiltonian fibrations has been one of the major advances in the study of integrable Hamiltonian systems in the past few decades: on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action-angle coordinates while, on the other hand, it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra. Fractional monodromy is a generalization of the concept of monodromy: instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres. In this paper we analyze fractional monodromy for n 1:(−n 2) resonant Hamiltonian systems with n 1, n 2 coprime natural numbers. We consider, in particular, systems that for n 1, n 2 > 1 contain one-parameter families of singular fibres which are ‘curled tori’. We simplify the geometry of the fibration by passing to an appropriate branched covering. In the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration: we essentially obtain the same type of generalized monodromy independently of n 1, n 2. Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.

Keywords

Homology Group Parallel Transport Original Space Covering Space Closed Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands
  2. 2.Department of Mathematical SciencesXi’an Jiaotong–Liverpool UniversitySuzhouChina

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