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The presence of focus-focus singularities in semi-toric integrables Hamiltonian systems is one of the reasons why there cannot exist global Action-Angle coordinates on such systems. At focus-focus critical points, the Liouville-Arnold-Mineur theorem does not apply. In particular, the affine structure of the image of the moment map around has non-trivial monodromy. In this article, we establish that the singular behavior and the multivaluedness of the Action integrals is given by a complex logarithm. This extends a previous result by San VU Ngoc to any dimension. We also calculate the monodromy matrix for these systems. (C) 2015 Elsevier B.V. All rights reserved.
Annalisa Buffa, Pablo Antolin Sanchez, Margarita Chasapi
Karim Achouri, Jean-Yves Duboz