Scaling Entropy of Unstable Systems

Abstract

In this paper, we study the slow entropy-type invariant of a dynamic system proposed by A. M. Vershik. We provide an explicit construction of a system that has an empty class of scaling entropy sequences. For this unstable case, we introduce an upgraded notion of the invariant, generalize the subadditivity results, and provide an exhaustive series of examples. Bibliography: 8 titles.

Dynamics of metrics in measure spaces and scaling entropy

“…In [14] this approach was called the 'reversed definition of mm-spaces'. This point of view was consistently pursued in [48]- [51], [78], [80], and [82]- [87], and we present it in this survey. Namely, a theory of metric triples (X, µ, ρ) -a space, a measure, and a metric -was developed.…”

“…In [48] (also see § 3.3.1), an important step was taken in the study of scaling entropy: the realization that the answer to the question about the existence of a universal equivalence class for the growth of entropies for averaged metrics can be negative for specially constructed metric triples, that is, the asymptotic behaviour can significantly depend on ε. Therefore, the definition of an equivalence class must involve functions of two variables: n (the number of the particular average we take) and ε.…”

“…For some time, the question of the existence of a scaling sequence for any ergodic automorphism T of a standard probability space (X, µ) remained open. An example of such an ergodic automorphism T and an admissible metric ρ on (X, µ) for which the scaling sequence in the sense of Definition 3.10 does not exist was constructed in [48]. The reason for this phenomenon is that for different ε > 0, the growth rate of H ε (X, µ, T n av ρ) with respect to n can be significantly different: for each ε > 0 there exists δ > 0 such that…”

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

“…shown by the author [9] that a scaling entropy sequence may not exist even for one automorphism. We will call a system stable if its class of scaling entropy sequences is not empty.…”

“…We will call a system stable if its class of scaling entropy sequences is not empty. That is, in [9], the examples of unstable systems were constructed.…”