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Transactions of the American Mathematical Society

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Removing index 0 fixed points for area preserving maps of two-manifolds

Author: Edward E. Slaminka
Journal: Trans. Amer. Math. Soc. 340 (1993), 429-445
MSC: Primary 58F20; Secondary 54H20, 58F10
MathSciNet review: 1145963
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Abstract: Using the method of free modifications developed by M. Brown and extended to area preserving homeomorphisms, we prove the following fixed point removal theorem.

Theorem. Let $ h:M \to M$ be an orientation preserving, area preserving homeomorphism of an orientable two-manifold $ M$ having an isolated fixed point $ p$ of index 0. Given any open neighborhood $ N$ of $ p$ such that $ N \cap \operatorname{Fix}(h) = p$, there exists an area preserving homeomorphism $ \hat h$ such that


$\displaystyle \hat h = h\;on\;\overline {M - N} $


(ii) $ \hat h$ is fixed point free on $ N$.

Two applications of this theorem are the second fixed point for the topological version of the Conley-Zehnder theorem on the two-torus, and a new proof of the second fixed point for the Poincaré-Birkhoff Fixed Point Theorem.

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Keywords: Fixed point, area preserving, fixed point index
Article copyright: © Copyright 1993 American Mathematical Society

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