Non-existence of a universal zero entropy system for non-periodic amenable group actions

Scaling Entropy of Unstable Systems

“…A question that naturally arises is what are the possible asymptotics that may appear as values of such invariants. In recent papers [1,14,16,18,13], several attempts to answer such questions led to non-trivial applications. For the case of a single transformation, the possible values of scaling entropy were completely described in [11,25].…”

“…For amenable groups, this question becomes much harder. For any amenable group, it is shown in [18] that the scaling entropy of a generic p.m.p. action of such a group can not be bounded from above by a given non-trivial sequence.…”

“…Examples of amenable groups with a scaling entropy growth gap provided in [18] are not finitely generated. Namely, they are the matrix groups SL(2, Fp ) over the algebraic closure of a finite field F p , p > 2.…”

“…For any action α of an amenable group G there are two simple natural bounds for the scaling entropy (see, e. g., [16]):…”

“…Moreover, the scaling entropy is bounded (meaning that H(α, λ) ≍ 1) if and only if the action is compact (see [10,23]). Equivalence H(α, λ) ≍ |F n | characterizes exactly the systems with positive measure-theoretic entropy (see [16]).…”