Monodromy of Hamiltonian systems with complexity 1 torus actions


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.