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Proof of the Fixed Point Theorems of Poincaré and Birkhoff

Published online by Cambridge University Press:  20 November 2018

Richard B. Barrar
Richard B. Barrar*
Affiliation:
System Development Corporation, Santa Monica, California
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 333 - 343
Copyright
Copyright © Canadian Mathematical Society 1967

References

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