use Floer homology to prove the fixed points

Question

I read paper, in page 21, there is a proposition:

Let $(M,\omega)$ be a closed symplectic manifold with $\pi_2(M)=0$. Let {$f_t$}, $f_0$ = id, $f_1= f$ be a Hamiltonian path on M generated by a Hamiltonian function F. Assume that $f \ne$id. Then $f$ has a pair of fixed points x and y so that their orbits {$f_tx$} and {$f_ty$} are contractible and A(F, y) − A(F, x) $\ne$ 0.

Above it, it says

The following deep fact is proved in Sch by using Floer homology

But I cannot find this result of the paper he mentioned. So how could we prove the proposition? Thanks in advance.

Answer 1

This is, indeed, proved in that paper. Indeed, it is proved that the Hofer-Zehnder capacity for $M$ is well-defined and non-zero when $f\neq id$. This implies the following:

There has to be two contractible periodic orbits with action distance this capacity. Indeed, were this not the case one would get a contradiction with the definition of the capacity, and the fact that something in the Floer homology (of a small pertubation of $A$) would have to have generators (of the classes $[1]$ and $[M]$) close to this distance appart.