Poincaré-Birkhoff theorem in sympl. geometry On p. 274 of McDuff and Salamon's Introduction to symplectic topology a corollary to the Poincaré Birkhoff theorem is presented.
So we are given an area preserving map on an annulus $\psi(x,y)=(f(x,y),g(x,y)).$
The equation $8.2$ is the property
$f(x+1,y)=f(x,y)+1$ and $g(x+1,y)=g(x,y)$
The equation (8.3) is
$g(x,a)=a, g(x,b)=b.$
To conclude that $\psi$ satisfies the Poincaré Birkhoff- theorem (has two fixed points) we need to know that the twist-condition holds
$f(x,a)<x$ and $f(x,b)>x.$
Then $\psi$ has two fixed points on an annlus with radius $r^2$ between $a$ and $b$.
I think the corollary wants to apply this result to a particular case:

But all that this shows it that $f^{q}(x,a)<x+p$ and $f^{q}(x,b)>x+p$.
Now, if $p>0$ we can conclude that $f^{q}(x,b)>x$ and if $p<0$ we can conclude $f^{q}(x,a)<x$. But I don't see that both conditions have to hold which we require to apply Poincaré-Birkhoff to this corollary. Does anybody know how to do this?
 A: I recall this argument confusing me in the past, but I think the point is as follows:
First, let's agree that by "area preserving map on the annulus" of the form described above, we really mean a corresponding periodic map on the universal cover, the infinite strip. All of the arguments in the proof of the two fixed point version of Poincare-Birkoff (as in, e.g, Brown and Neumann's paper) become much more tractable to work with upon lifting to the universal cover, and a few simple observations as in the aforementioned paper suffice to show the P.B. theorem can be recast into an equivalent statement about maps on the cover.
As you point out, McDuff and Salamon have shown:
$f^{q}(x,a) \leq x + p$ and $f^{q}(x,b) \geq x + p$, where $\psi(x,y) = (f(x,y),g(x,y))$ is an area preserving homeomorphism of the annulus satisfying the conditions of P.B. theorem.
Consider $\psi^{q}(x,y)=(f^{q}(x,y),g^{q}(x,y))$, and $\widetilde{\psi^{q}}(x,y)= (f^{q}(x,y) - p,g^{q}(x,y))$. Then $\widetilde{\psi^{q}}$ and ${\psi^{q}}$ have the same number of fixed points (after going down to the annulus), but the P.B. theorem can be applied to $\widetilde{\psi^{q}}$, proving the claim in the text.
