# 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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by Edward E. Slaminka
Trans. Amer. Math. Soc. 306 (1988), 277-291 Request permission

## Abstract:

We prove a generalization of the Brouwer Translation Theorem which applies to a class of homeomorphisms (free homeomorphisms) which admit fixed points, but retain a dynamical property of fixed point free orientation preserving homeomorphsims. That is, if $h:{M^2} \to {M^2}$ is a free homeomorphism where ${M^2}$ is a surface, then whenever $D$ is a disc and $h(D) \cap D = \emptyset$, we have that ${h^n}(D) \cap D = \emptyset$ for all $n \ne 0$. Theorem. Let $h$ be a free homeomorphism of ${S^2}$, the two-sphere, with finite fixed point set $F$. Then each $p \in {S^2} - F$ lies in the image of an embedding ${\phi _p}:({R^2}, 0) \to ({S^2} - F, p)$ such that: (i) $h{\phi _p} = {\phi _p}\tau$, where $\tau (z) = z + 1$ is the canonical translation of the plane, and (ii) the image of each vertical line under ${\phi _p}$ is closed in ${S^2} - F$.
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