Type of critical points of Hamiltonian functions and of fixed points of symplectic diffeomorphisms

Abstract

Let H be a Hamiltonian and a Morse function on a symplectic manifold M. We prove: at a critical point x0 of H (i.e. at a fixed point of the corresponding Hamiltonian flow), the signature of D2H(x0) allows us to bound the diagonal elliptic dimension from below and we obtain some Morse-like inequalities for diagonal elliptic dimensions on compact manifolds. We prove also: we can construct a Hamiltonian function whose critical points all have hyperbolic dimension less than 2. If f is a symplectic diffeomorphism of M, at a fixed point x0 of f, the signature of a certain quadratic form defined on T(x0)M allows us to bound the diagonal elliptic dimension from below and we obtain again some Morse-like inequalities for diagonal elliptic dimensions on compact manifolds when f has a global generating function.