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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Removing index $0$ fixed points for area preserving maps of two-manifolds
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by Edward E. Slaminka PDF
Trans. Amer. Math. Soc. 340 (1993), 429-445 Request permission

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 (i) \[ \hat h = h\;on\;\overline {M - N} \] and (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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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 340 (1993), 429-445
  • MSC: Primary 58F20; Secondary 54H20, 58F10
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1145963-5
  • MathSciNet review: 1145963