Scaling Entropy of Unstable Systems

TL;DR

This paper investigates the scaling entropy of unstable dynamical systems, introduces a new invariant for these cases, and provides explicit examples and generalizations of existing results.

Contribution

It introduces an upgraded invariant for unstable systems, generalizes subadditivity results, and constructs explicit examples with empty scaling entropy classes.

Findings

Constructed a system with an empty class of scaling entropy sequences.

Generalized subadditivity results for unstable systems.

Provided a comprehensive series of examples illustrating the concepts.

Abstract

In this work, we study the slow entropy type invariant of a dynamical system proposed by A. M. Vershik. We provide an explicit construction of a system whose class of scaling entropy sequences is empty. For this unstable case, we introduce an upgraded notion of the invariant, generalize subadditivity results, and provide a complete series of examples.