Slow entropy of some combinatorial constructions

TL;DR

This paper investigates the properties of measure-theoretic slow entropy in dynamical systems, demonstrating flexibility in its values for certain transformations and establishing limitations on bounds for systems of finite rank.

Contribution

It provides new results on the variability of polynomial slow entropy and shows the absence of a universal upper bound for lower slow entropy in finite rank systems.

Findings

Flexibility results for upper and lower polynomial slow entropy.

No general upper bound for lower slow entropy in finite rank systems.

Analysis of slow entropy in systems with cyclic approximation.

Abstract

Measure-theoretic slow entropy is a more refined invariant than the classical measure-theoretic entropy to characterize the complexity of dynamical systems with subexponential growth rates of distinguishable orbit types. In this paper we prove flexibility results for the values of upper and lower polynomial slow entropy of rigid transformations as well as maps admitting a good cyclic approximation. Moreover, we show that there cannot exist a general upper bound on the lower measure-theoretic slow entropy for systems of finite rank.