A fixed point theorem for twist maps

Abstract

<p style='text-indent:20px;'>Poincaré's last geometric theorem (Poincaré-Birkhoff Theorem [<xref ref-type="bibr" rid="b2">2</xref>]) 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 <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> 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.</p>