Punts fixos de difeomorfismes hamiltonians

Abstract

[en] Arnold’s conjecture asserts that every Hamiltonian diffeomorfism of a compact symplectic manifold has at least as many fixed points as a function on the manifold must have critical points. What’s more, if the fixed points are all non degenerate, then the number of fixed points is at least the minimal number of critical points for a Morse function on the manifold. In this project we will give meaning to all the concepts mentioned in the conjecture’s statement and we will study a very specific known result: the case in which the manifolds are 2-dimensional tori and the diffeomorfisms are close enough to identity. We will also generalize some results to 2n-dimensional tori to study the general case for every Hamiltonian diffeomorfism.