Sofic conditional mean dimension, relative sofic mean dimension and their localizations

Abstract

Let $\ddot{I}€:(X,G)\to (Y,G)$ be a factor map between continuous actions of a sofic group $G$, we study sofic conditional mean dimension and relative sofic mean dimension introduced in \cite{LBB2} and \cite{LB}, respectively. We obtain that if $\ddot{I}€$ has non-negative sofic conditional mean dimension (resp. relative sofic mean dimension), then $\ddot{I}€$ has the maximal zero sofic conditional mean dimension factor (resp. maximal relative zero sofic mean dimension factor). Additionally, the local properties of sofic conditional mean dimension and relative sofic mean dimension are studied. We introduce the sofic conditional mean dimension tuples and relative sofic mean dimension tuples, and show that $\ddot{I}€$ has positive sofic conditional mean dimension (resp. relative sofic mean dimension) if and only if the set of sofic conditional mean dimension tuples (resp. relative sofic mean dimension tuples) is nonempty.