Faithful Semitoric Systems

[摘要] This paper consists of two parts. The first provides a review of the basic properties of integrable and almost-toric systems, with a particular emphasis on the integral affine structure associated to an integrable system. The second part introduces faithful semitoric systems, a generalization of semitoric systems (introduced by Vũ Ngoc and classified by Pelayo and Vũ Ngoc) that provides the language to develop surgeries on almost-toric systems in dimension 4. We prove that faithful semitoric systems are natural building blocks of almost-toric systems. Moreover, we show that they enjoy many of the properties that their (proper) semitoric counterparts do.

[关键词] completely integrable Hamiltonian systems;almost toric systems;semitoric systems;integral affine geometry;focus-focus singularities