Some remarks on modified power entropy

TL;DR

This paper critically examines modified power entropy, demonstrating its limitations in detecting certain dynamical transitions and its sensitivity to transient behaviors, contrasting it with other invariants like power entropy.

Contribution

It provides examples showing modified power entropy's sensitivity to transients and its inadequacy in detecting certain dynamical transitions, highlighting its limitations.

Findings

Modified power entropy is sensitive to transient dynamics.

It does not satisfy a variational principle.

It cannot detect the break of equicontinuity during system transitions.

Abstract

The aim of this note is to point out some observations concerning modified power entropy of $\Z$- and $\N$-actions. First, we provide an elementary example showing that this quantity is sensitive to transient dynamics, and therefore does not satisfy a variational principle. Further, we show that modified power entropy is not suitable to detect the break of equicontinuity which takes place during the transition from almost periodic to almost automorphic minimal systems. In this respect, it differs from power entropy and amorphic complexity, which are two further topological invariants for zero entropy systems (`slow entropies'). Finally, we construct an example of an irregular Toeplitz flow with zero modified power entropy.