Action diffusion in sympletic and volume preserving maps

Abstract
Polynomial Hénon like symplectic maps are the basic models in nonlinear beam dynamics. The action-frequency map allows to analyze their network of resonances in action space and the Fokker-Planck equation is adequate to describe the diffusion in action space induced by a random perturbation. The changes introduced by noise correlations and the presence of resonant structures are outlined. The modulated standard map in a solid torus is a volume preserving map, which shares the basic features of volume preserving integrators of the magnetic field lines in a toroidal vessel. The scenario of diffusion for this map is briefly discussed.