| Volume-preserving mappings are appropriate models for many systems, such as, fluid, magnetic field line flows, and the motion of perturbed comet-s. One fundamental problem for application is to understand transport in Volume-preserving mappings, correspondingly, this is to analyse the existence of invariant tori. It is well known in KAM theory that these tori with strongly non-resonant frequencies are preserved, and resonant tori are destroyed un-der perturbations. However, it is possible to discuss the existence of lower dimensional tori and dynamics around these destroyed tori. Hence, this thesis mainly studies under different ranks of resonance module, the reducibility in Volume-preserving mappings by averaging theory so that it is convenient to analyse dynamics by a lower dimensional system. It consists of five chapters:In Chapter 1, We introduce the development history and research status in the relevant field, introduce main content of the thesis;Chapter 2 introduces some preliminary knowledge which is used in dis-cussion of resonances and reducibility in Volume-preserving mappings;In Chapter 3, We discuss the reducibility of Volume-preserving mappings with one-action and rank-two module, and introduce some dynamics in action-angle-angle Volume-preserving mappings at the end of the chapter;In Chapter 4, We discuss the reducibility of Volume-preserving mappings with two-action and rank-two module, and introduce some dynamics in Hamil-tonian systems with two degree of freedom at the end of the chapter;In Chapter 5, We discuss the reducibility of Volume-preserving mappings with t-action and rank-t module, and introduce some dynamics in Hamiltonian systems with n-degree of freedom at the end of the chapter. |
