Asymptotics of action variables near semi-toric singularities

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.