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.


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.


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