Scaling entropy and universal zero entropy system

In works by A.M.Vershik, it was proposed to study measure-theoretic properties of an automorphism considering its action on functions of several variables, namely, measurable metrics. A sequence $\{h_n\}$ is called a scaling entropy sequence if for some (and then any) admissible metric the sequence of epsilon-entropies of its averages over $n$ iterations is equivalent to $h_n$. We say that a system is stable if it has a scaling entropy sequence. We will discuss examples of unstable transformations and a generalized definition which extends the notion of a scaling sequence to the general case.
In the final part, we will discuss an application of the theory of scaling entropy. Let $G$ be a non-periodic amenable group. Then there does not exist a topological action of $G$ for which the set of invariant measures coincides with the set of all measure-theoretic systems of zero entropy.