Loops of Infinite Order and Toric Foliations

TL;DR

This paper generalizes the concept of Maslov indices in integrable Hamiltonian systems, establishing a bundle morphism that reveals eigenvectors with eigenvalue 1 and leads to a refined toric foliation, especially in 2-degree-of-freedom systems.

Contribution

It introduces a new bundle morphism on the lattice bundle of integrable systems, extending Maslov index concepts and uncovering geometric structures like toric foliations and S^1 actions.

Findings

Existence of common eigenvectors with eigenvalue 1 for monodromy matrices.

Construction of a bundle morphism generalizing Maslov indices.

Implication of a free S^1 action in 2-degree-of-freedom systems.

Abstract

In 2005 Dullin et al. proved that the non-zero 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 a more general context. 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 non-triviality 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 global existence of a free S^{1} action.