Open AccessA fixed point theorem for twist maps
Author(s) -
Peizheng Yu,
Zhihong Xia
Publication year - 2022
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2022045
Subject(s) - mathematics , twist , intersection (aeronautics) , fixed point theorem , fixed point , annulus (botany) , type (biology) , pure mathematics , combinatorics , discrete mathematics , geometry , mathematical analysis , cartography , ecology , botany , biology , geography
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.