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

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.