Mean equicontinuous factor maps

Mean equicontinity is a well studied notion for actions. We propose a definition of mean equicontinuous factor maps that generalizes mean equicontinuity to the relative context. For this we work in the context of countable amenable groups. We show that a factor map is equicontinuous, if and only if it is mean equicontinuous and distal. Furthermore, we show that a factor map is topo-isomorphic, if and only if it is mean equicontinuous and proximal. We present that the notions of topo-isomorphy and Banach proximality coincide for all factor maps. In the second part of the paper we turn our attention to decomposition and composition properties. It is well known that a mean equicontinuous action is a topo-isomorphic extension of an equicontinuous action. In the context of minimal and the context of weakly mean equicontinuous actions, respectively, we show that any mean equicontinuous factor map can be decomposed into an equicontinuous factor map after a topo-isomorphic factor map. Furthermore, for factor maps between weakly mean equicontinuous actions we show that a factor map is mean equicontinuous, if and only if it is the composition of an equicontinuous factor map after a topo-isomorphic factor map. We will see that this decomposition is always unique up to conjugacy.