Action and periodic orbits on annulus

We consider the classical problem of area-preserving maps on annulus $\mathbb{A} = S^1 \times [0, 1]$ . Let $\mathcal{M}_f$ be the set of all invariant probability measures of an area-preserving, orientation preserving diffeomorphism $f$ on $\mathbb{A}$. Given any $\mu_1$ and $\mu_2$ in $\mathcal{M}_f$, Franks \cite{Franks1988}\cite{Franks1992}, generalizing Poincar\'e's last geometric theorem (Birkhoff \cite{Birkhoff1913}), showed that if their rotation numbers are different, then $f$ has infinitely many periodic orbits. In this paper, we show that if $\mu_1$ and $\mu_2$ have different actions, even if they have the same rotation number, then $f$ has infinitely many periodic orbits. In particular, if the action difference is larger than one, then $f$ has at least two fixed points. The same result is also true for area-preserving diffeomorphisms on unit disk, where no rotation number is available.