Combinatorial Proofs of the Conley-Zehnder-Franks Theorem on a Fixed Point for Torus Homeomorphisms

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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 Poincarés Last Geometric Theorem.