An elementary proof of Poincaré’s last geometric theorem

Abstract

It is shown that the Poincaré-Birkhoff fixed point theorem may be proved by extending the geometric approach originally devised by Henri Poincar´e himself, along with several results from elementary differential topology. Beginning with a sample application of the theorem, we proceed by systematically constructing and classifying a certain set of invariant curves and their critical points. This classification is then used to prove the correctness of a procedure which guarantees the existence of at least two fixed points for any twist map of the annulus admitting a positive integral invariant.