Fractional Hamiltonian Monodromy from a Gauss-Manin Monodromy

TL;DR

This paper explores fractional Hamiltonian Monodromy, a generalization of classical monodromy, using Gauss-Manin Monodromy of Riemann surfaces, and applies it to specific resonant systems with non-isolated singularities.

Contribution

It introduces a new perspective on fractional Hamiltonian Monodromy via Gauss-Manin Monodromy and proves propositions for 1:-n and m:-n resonant systems.

Findings

Fractional Hamiltonian Monodromy analyzed through Gauss-Manin Monodromy.

Proved propositions for 1:-n and m:-n resonant systems.

Provides a geometric framework for non-isolated singularities.

Abstract

Fractional Hamiltonian Monodromy is a generalization of the notion of Hamiltonian Monodromy, recently introduced by N. N. Nekhoroshev, D. A. Sadovskii and B. I. Zhilinskii for energy-momentum maps whose image has a particular type of non-isolated singularities. In this paper, we analyze the notion of Fractional Hamiltonian Monodromy in terms of the Gauss-Manin Monodromy of a Riemann surface constructed from the energy-momentum map and associated to a loop in complex space which bypasses the line of singularities. We also prove some propositions on Fractional Hamiltonian Monodromy for 1:-n and m:-n resonant systems.