✦ LIBER ✦
Combinatorial Proofs of the Conley-Zehnder-Franks Theorem on a Fixed Point for Torus Homeomorphisms
✍ Scribed by S. Alpern; V.S. Prasad
- Publisher
- Elsevier Science
- Year
- 1993
- Tongue
- English
- Weight
- 428 KB
- Volume
- 99
- Category
- Article
- ISSN
- 0001-8708
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✦ Synopsis
We give two combinatorial proofs of Franks' Theorem, that an area preserving torus homeomorphism with mean rotation zero has a fixed point. The first proof uses Lax's version of the Marriage Theorem. The second proof uses Euler's Theorem on circuits in graphs and an explicit method of Alpern for decomposing a Markov chain into cycles. Our methods apply equally well to the annulus and, if combined with a result of Oxtoby and Ulam on homeomorphic measures, yield another proof of Poincaré's Last Geometric Theorem. 1993 Academic Press, Inc.