Stability and almost periodicity of asymptotically dominated semigroups ofpositive operators

We discuss conditions such that strong stability and strong asymptotic comp actness of a (discrete or continuous) semi ow defined on a subset in the po sitive cone of an ordered Banach space is preserved under asymptotic domina tion. This is used to show that on a Banach lattice with order continuous n orm strong stability and almost periodicity of a (discrete or strongly cont inuous) semigroup of positive operators is preserved under asymptotic domin ation.