Dynamical Systems

点击翻译按钮获取中文摘要

This proposal has two parts. Mather's part concerns using variational methods to prove the existence of "wandering" orbits in Hamiltonian systems. The sort of thing he hopes to prove are illustrated by the conjectures which go under the name "Arnold diffusion". Forni's part concerns area preserving flows on surfaces. He intends to study two conjectures: an extension of the KAM theorem, where Kronecker flows are replaced by area preserving flows on surfaces of higher genus; and a conjecture that would imply that the convergence in Birkhoff's ergodic theorem takes place with polynomial-like decay. Both parts of this proposal concern generalized KAM theory, which has important applications to questions of stability and randomness in conservative physical systems. For example, the long-term dynamics of the solar system is far from being understood, and this proposal could contribute to mathematician's understanding.