Conditional sofic mean dimension.

The conditional mean dimension is a useful tool for obtaining a general upper bound of mean dimension for an extension system, serving as an analogue of the conditional topological entropy. However, this estimation presents a striking contrast with Tsukamoto’s counterexample, which challenges the classical Hurewicz theorem for mean dimension.

In this talk, we will provide an overview of the concept of conditional mean dimension for actions by amenable groups, highlighting its favorable properties. Subsequently, we will introduce the notion of conditional mean dimension for actions by sofic groups and explore new phenomena related to it.