TYPE OF CRITICAL-POINTS OF HAMILTONIAN FUNCTIONS AND OF FIXED-POINTS OF SYMPLECTIC DIFFEOMORPHISMS

Let H be a Hamiltonian and a Morse function on a symplectic manifold M. We prove : at a critical point x(0) of H (i.e. at a fixed point of the corresponding Hamiltonian flow), the signature of (DH)-H-2(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.