Proof of the Fixed Point Theorems of Poincaré and Birkhoff

Abstract

In 1912, shortly before his death, Poincaré (8) conjectured the following theorem in his investigation of the restricted problem of three bodies.Poincaré's Last Geometric Theorem. Given a ring 0 < a ⩽ r ⩽ b in the r, θ plane and a homeomorphic, area-preserving mapping T of the ring onto itself under which points on r = a advance and those on r = b regress, there will exist at least two points of the ring invariant under T.Poincaré was able to prove this theorem in only a few special cases. Shortly thereafter, Birkhoff was able to give a complete proof in (2) and in, (3) he gave a generalization of the theorem, dropping the assumption that the transformation was area-preserving. Birkhoff's proofs were very ingenious; however, they did not use standard topological arguments.