ABSTRACT
In this chapter, the authors intend to classify as fully as they can the main types of one-turn maps they encounter in rings. This classification is based on perturbation theory, that is, maps viewed as k-jets. The authors examine the motion around the fixed point and make some statements about the full map based on normalization methods. They want to make use of the relationship between the compositional map and Hamiltonian techniques. The basic idea behind Lie methods applied to one-turn maps emerges from the realization that the one-turn compositional map can be expressed in terms of Lie transforms. Lie methods allow them to use Hamiltonian perturbation theory on a representation of the nonlinear one-turn map. It gives them a way to extend to the nonlinear problem, at least formally, the normalization done on the linear part ℒ. The authors reflect on what perturbation theory achieves or fails to achieve.
