Loops of Infinite Order and Toric Foliations

    2022, Volume 27, Number 3, pp.  320-332

    Author(s): Efstathiou K., Lin B., Waalkens H.

    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 non-Hamiltonian 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.
    Keywords: integrable system, toric foliation, $S^{1}$ action, Maslov index, monodromy matrix
    Citation: Efstathiou K., Lin B., Waalkens H., Loops of Infinite Order and Toric Foliations, Regular and Chaotic Dynamics, 2022, Volume 27, Number 3, pp. 320-332

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