ABSTRACT
In this chapter the author intend to achieve four goals: to display the homomorphism which exists between various brackets; to use these homomorphisms to derive the effect of s-dependent canonical transformations on the map equation; thus make contact with Hamiltonian theory; and finally, to extend the concept of the resonance basis to an arbitrary vector field. The authors review the case of the mixed generating function and its first order connection with the Lie transform. It is well-known to students of classical mechanics that it is possible to transform a Hamiltonian by a canonical transformation expressed in terms of a mixed generating function. The authors present the ultimate "map-like" computation of the Green's function of perturbation theory starting from the Hamiltonian. They review the pure "map-style" derivation. The authors examine the propagation of the Green's function in the potential-free region.
