Konstantinos Efstathiou
Publications:
Efstathiou K., Lin B., Waalkens H.
Loops of Infinite Order and Toric Foliations
2022, vol. 27, no. 3, pp. 320332
Abstract
In 2005 Dullin et al. proved that the
nonzero vector of Maslov indices is an eigenvector with eigenvalue
$1$ of the monodromy matrices of an integrable Hamiltonian system.
We take a close look at the geometry behind this result and extend
it to the more general context of possibly nonHamiltonian systems.
We construct a bundle morphism defined
on the lattice bundle of an (general) integrable system, which can
be seen as a generalization of the vector of Maslov indices. The nontriviality of this bundle morphism implies the existence of common eigenvectors with eigenvalue $1$
of the monodromy matrices, and gives rise to a corank $1$ toric foliation
refining the original one induced by the integrable system. Furthermore,
we show that, in the case where the system has $2$ degrees of freedom,
this implies the existence of a compatible free $S^{1}$ action on the regular part of the system.
