An Elementary Proof of Poincar\'e's Last Geometric Theorem

TL;DR

This paper provides an elementary geometric proof of Poincaré's Last Geometric Theorem by extending Poincaré's original approach and utilizing elementary differential topology to establish the existence of fixed points in twist maps.

Contribution

It introduces a new elementary geometric proof of the Poincaré-Birkhoff fixed point theorem, simplifying the understanding of the theorem's core concepts.

Findings

Proves the existence of at least two fixed points for twist maps with positive invariants.

Classifies invariant curves and their critical points systematically.

Validates a procedure ensuring fixed points in annulus twist maps.

Abstract

It is shown that the Poincar\'e-Birkhoff fixed point theorem may be proven by extending the geometric approach originally devised by Henri Poincar\'e himself, along with several results from elementary differential topology. Beginning with an example 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.