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

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.