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 → M 2 h:{M^2} \to {M^2} is a free homeomorphism where M 2 {M^2} is a surface, then whenever D D is a disc and h ( D ) ∩ D = ∅ h(D) \cap D = \emptyset , we have that h n ( D ) ∩ D = ∅ {h^n}(D) \cap D = \emptyset for all n ≠ 0 n \ne 0 . Theorem. Let h h be a free homeomorphism of S 2 {S^2} , the two-sphere, with finite fixed point set F F . Then each p ∈ S 2 − F p \in {S^2} - F lies in the image of an embedding ϕ p : ( R 2 , 0 ) → ( S 2 − F , p ) {\phi _p}:({R^2},\,0) \to ({S^2} - F,\,p) such that: (i) h ϕ p = ϕ p τ h{\phi _p} = {\phi _p}\tau , where τ ( z ) = z + 1 \tau (z) = z + 1 is the canonical translation of the plane, and (ii) the image of each vertical line under ϕ p {\phi _p} is closed in S 2 − F {S^2} - F .