Let $(M, \omega)$ be a symplectic manifold. V.Arnold conjectured that the number of fixed points of a Hamiltonian symplectomorphism is bounded below by the number of critical points of a smooth function on $M$.
The conjecture was proved in a weaker (homological) version using Floer homology, with two additional conditions:
- Compactness of $M$;
- Non-degeneracy of the fixed points of the considered Hamiltonian symplectomorphism.
In this setting, it is now fully understood that the number of fixed points is at least the sum of Betti number of $M$.
I am wondering where the general statement stands today. More precisely:
- is compactness really necessary to Floer's framework ?
- what is known for degenerate Hamiltonians ?
- what is known for the critical points lower bound, rather than the Betti sum ?
Thanks a lot