A fixed point theorem for twist maps

Poincaré's last geometric theorem (Poincaré-Birkhoff Theorem [ 2 ]) states that any area-preserving twist map of annulus has at least two fixed points. We replace the area-preserving condition with a weaker intersection property, which states that any essential simple closed curve intersects its image under \begin{document}$ f $\end{document} at least at one point. The conclusion is that any such map has at least one fixed point. Besides providing a new proof to Poincaré's geometric theorem, our result also has some applications to reversible systems.