On The Possible Growth Rate of a Scaling Entropy Sequence

Abstract

We define a scaling entropy sequence of a group action and give a family of examples that exhaust all possible growth rates of entropy scaling sequences for actions of the groups Z and ⊕Z 2 .

Universal adic approximation, invariant measures and scaled entropy

“…Theorem 7 ((see [27])). If ρ 1 , ρ 2 ∈ Adm(X, µ) are metrics, then H(X, µ, G, ρ 1 ) = H(X, µ, G, ρ 2 ).…”

“…To replace the metrics in this theorem by generating semimetrics, it is necessary to impose some conditions on the sequence of sets G n (see [27]). As in the case of a single transformation, we introduce the notion of a scaling sequence of an action of an equipped group; this is a scaling sequence of an arbitrary admissible metric.…”

Scaling Entropy of Unstable Systems

“…One could choose φ m (n) = log m (n), for instance. By [6], there is a family of corresponding ergodic systems S m = (X m , µ m , T m ) such that for all m, any admissible generating semimetric ρ m on (X m , µ m ), and all positive ε small enough…”

“…By the previous formula, h n does not exceed φ R(ε0) (n) asymptotically. However, in [6] it is proved that the scaling entropy sequence of a system grows not slower than the entropy sequence of a factor system. This implies that for any m h n φ m (n).…”

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

“…The main tool that we implement in the proof is the notion of scaling entropy which was proposed in works by A. M. Vershik [10,11]. The theory of scaling entropy was developed in [6,12,16,17]. In this work, we prove the lower bound for the ε-entropy of an averaging of independent metrics (Lemma 5) which allows us to estimate a scaling entropy for a special series of examples of G-actions.…”