Converse KAM Theory For Positive Definite Hamiltonian Systems

By the Kolmogorov, Arnold and Moser (KAM) theory, we know that under certain non-degeneracy conditions, most (full Lebesgue measure) invariant tori of an integrable Hamiltonian system are preserved under small perturbations. By the efforts of Moser, Russman, Herman and Poschel ([58,59],[93,94],[102] and [104,105]), for non-degenerate Hamiltonian systems with d-degrees of freedom, it is obtained that certain invariant tori are persisted under arbitrarily small perturbations in the C2d+δ topology, where8is a small positive constant. In particular, Herman proved in [59] that for twist maps on annulus, certain invariant circles can be persisted under arbitrarily small perturbations in the C3topology. As the sizes of the perturbations increasing, those preserved invariant tori are destructed progressively. The problems of determining the critical boundary between the persistence and destruction of invariant tori motivates so called converse KAM theory ([23,33,37,50,54,58,70,78] etc.).In this thesis, we considered converse KAM theory for positive definite Hamilto-nian systems. More precisely, we were concerned about the following problem. For an integrable positive definite Hamiltonian with d (d≥2) degrees of freedom, if the Lagrangian torus with a given rotation vector or all Lagrangian tori can be destructed by an arbitrarily small perturbation in the Cr topology, then what is the maximum of r?For exact area-preserving twist maps on annulus, it was proved by Herman in [58] that invariant circles with given rotation numbers can be destructed by C3-δ arbitrarily small C∞perturbations. For certain rotation numbers, it was obtained by Mather (resp. Forni) in [78](resp.[50]) that the invariant circles with those rotation numbers can be destroyed by small perturbations in finer topology respectively. More precisely, Math-er considered Liouville rotation numbers and the topology of the perturbation induced by C∞metric. Forni was concerned about more special rotation numbers which can be approximated by rational ones exponentially and the topology of the perturbation in-duced by the supremum norm of Cω (real-analytic) function. Bessi extended the result to the systems with multi-degrees of freedom. He found that the invariant Lagrangian torus with certain rotation vector can be destructed by an arbitrarily small Cω pertur-bation for certain positive definite systems with multi-degrees of freedom in [23]. In [33], it was proved that KAM torus with a given rotation vector does not exist if one carefully construct perturbations arbitrarily small in C2d-δ topology. But this does not imply non-existence of invariant Lagrangian torus. Indeed, it exists in the example in [33]. In a joint work with C.-Q. Cheng, we obtained that the Lagrangian torus with a given rotation vector of an integrable positive definite Hamiltonian system with d (d≥2) degrees of freedom can be destructed by an arbitrarily small C∞perturbation in the C2d-δ. In contrast with it, it has been shown that KAM torus with constant type fre-quency persists under C2d+δ perturbations. Based on both results above, it means that our result is almost optimal except for the case with r=2d. Indeed, more precise re-sults are proved based on the arithmetic properties of given rotation vectors. An vector ω∈Rd is called τ-approximated if there exists a positive constant C as well infinitely many integer vectors k∈Zd such that|<ω,k>|<C|k|-d+1-τ. By [33], all d dimension-al vector are O-approximated. For the Lagrangian torus with a given τ-approximated rotation vector of an integrable positive definite Hamiltonian system with d (d≥2) degrees of freedom, we proved:(1) it can be destructed by an arbitrarily small C∞perturbation in the Cr topology, where r<2d+2τ;(2) it can be destructed by an arbitrarily small Cω perturbation in the Cr topology, where r<d+τ;(3) it can be destructed by an arbitrarily small Gevrey-α (α＞1) perturbation in the Cr topology, where r<2d+2τ-2/α(d+τ).It was proved that all of the invariant circles can be destructed by C1arbitrarily small C∞perturbations of the integrable twist maps [112]. C1was improved to be C2-δ by Herman in [58]. Moreover, he extended the result to systems with multi-degrees of freedom and found that all of the Lagrangian tori of an integrable symplectic twist map can be destructed by Cd+2-δ arbitrarily small C∞perturbations of the generating function [62]. Based on the correspondence between symplectic twist maps and pos-itive definite Hamiltonian systems ([53,95]), it shows that all of the Lagrangian tori of an integrable Hamiltonian system with d≥2degrees of freedom can be destructed by C∞perturbations which are arbitrarily small in the Cd+1-δ topology. In this thesis, we showed that all of the Lagrangian tori of an integrable positive definite Hamiltonian system with d (d≥2) degrees of freedom can be destructed by an arbitrarily small Cω perturbation in the Cd-δ topology.