The Poincaré–Birkhoff fixed point theorem (also called Poincaré’s last geometric theorem) asserts the existence of at least two fixed points for a so-called area-preserving twist homeomorphism of the annulus. It was formulated as a conjecture and proved in special cases by Poincaré [1912], shortly before his death. In [1913] George Birkhoff published a proof which, though correct for one fixed point, overlooked the passibility that this fixed point might have index 0 in deducing the existence of a second fixed point. This error was corrected in his paper [1926], in which a generalization of the theorem in question is proven, with “area-preserving” replaced by a purely topological condition and “homeomorphism” replaced by a more general situation. However, some mathematicians have claimed that this proof too is incorrect, and the last few years have seen some extensive efforts to try to find a correct proof for the second fixed point.
We present here an elementary proof for two fixed points which is a simple modification of Birkhoff’s well known original proof for one fixed point. Our modification to get the second fixed point is essentially the same modification that Birkhoff sketches in the 1926 proof of his topological version to get from one fixed point to two.
This paper is therefore in a sense an expository paper, and to make the proof as transparent as possible we shall restrict to the simplest situation — a twist homeomorphism of the annulus which is just a rotation by a fixed angle on each boundary circle. As we point out in a final section, the proof goes through almost word for word without this restriction. It also extends to more general measures than the standard Lebesgue measure on the annulus.
Since our proof is so chase to Birkhoff’s proof, which has met with some skepticism, we have felt it advisable to give somewhat more detail than would otherwise be necessary. This is also in keeping with the view of this paper as an expository one.
