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?