Ergodic Theory and Dynamical Systems

Research Article

A new proof of the Brouwer plane translation theorem

John Franksa1

a1 Department of Mathematics, Northwestern University, Evanston, IL 60201, USA

Abstract

Let f be an orientation-preserving homeomorphism of 2 which is fixed point free. The Brouwer ‘plane translation theorem’ asserts that every x0 2 is contained in a domain of translation for f i.e. an open connected subset of 2 whose boundary is L f(L) where L is the image of a proper embedding of in 2, such that L separates f(L) and f−1(L). In addition to a short new proof of this result we show that there exists a smooth Morse function g: 2 such that g(f(x)) < g(x) for all x and the level set of g containing x0 is connected and non-compact (and hence the image of a properly embedded line).

(Received February 04 1991)