A Brouwer translation theorem for free homeomorphisms

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 ${S^2} - F$.