A Study of Directional Entropy Arising from \(\mathbb{Z} \times \mathbb{Z}_+\) Semigroup Actions

TL;DR

This paper explores directional entropy in semigroup actions generated by linear cellular automata and shift transformations, providing a systematic analysis of topological and measure-theoretic entropy within a unified framework.

Contribution

It offers a comprehensive study of both topological and measure-theoretic directional entropy for semigroup actions on compact metric spaces, linking cellular automata and shift dynamics.

Findings

Characterization of topological directional entropy for LCAs

Analysis of measure-theoretic directional entropy via Kolmogorov--Sinai formalism

Unified framework connecting cellular automata and shift transformations

Abstract

In this chapter, we investigate directional entropy for semigroup actions generated by one-dimensional linear cellular automata (LCAs) and the shift transformation on the compact metric space $\mathbb{Z}_m^{\mathbb{N}}$. This work provides a systematic study of both \emph{topological directional entropy} (TDE) within Milnor's geometric framework and \emph{measure-theoretic directional entropy} via the Kolmogorov--Sinai formalism.