Monodromy of Hamiltonian systems with complexity 1 torus actions

K. Efstathiou, N. Martynchuk

Journal of Geometry and Physics, 115, 104-115 (2017)
10.1016/j.geomphys.2016.05.014
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Abstract

We consider the monodromy of $n$-torus bundles in $n$ degree of freedom integrable Hamiltonian systems with a complexity 1 torus action, that is, a Hamiltonian $T^{n−1}$ action. We show that orbits with $T^1$ isotropy are associated to non-trivial monodromy and we give a simple formula for computing the monodromy matrix in this case. In the case of $2$ degree of freedom systems such orbits correspond to fixed points of the $T^1$ action. Thus we demonstrate that, given a $T^{n−1}$ invariant Hamiltonian $H$, it is the $T^{n−1}$ action, rather than $H$, that determines monodromy.