Area-preserving homeomorphisms of the open disk without fixed points

Abstract

D. G. Bourgin has proved that every measure-preserving orientation-preserving homeomorphism of the open two-dimensional disk D D has a fixed point. He suggested that the "result is perhaps valid even if the condition of orientability preservation be dropped." We show that on the contrary there exist fixed point free homeomorphisms of D D which preserve any given finite nonatomic locally positive Borel measure. Examples are also constructed in all higher dimensions.