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

Hasan Akin

TL;DR

This work analyzes directional entropy for $\ Z \times \nZ_+$ semigroup actions generated by a one-dimensional linear cellular automaton and a shift map on $\nZ_m^{\nZ}$. It develops both measure-theoretic (MTDE) and topological (TDE) directional entropy frameworks, deriving rigorous bounds for Bernoulli and Markov measures and obtaining explicit angle-dependent formulas for TDE over rings $\nZ_m$. The results reveal directional anisotropy in entropy, with tractable expressions in terms of local-rule permutivity, ring structure, and direction, and they illustrate these phenomena via concrete examples over primes and prime-power rings. By decomposing general rings into prime-power components and applying weak addition, the paper provides a unified approach to computing MTDE and TDE for $\mathbb{Z}^2$-actions arising from 1D LCAs and shifts, offering concrete benchmarks and guidance for future explorations of directional entropy in more complex CA-driven systems.

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.

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

TL;DR

This work analyzes directional entropy for semigroup actions generated by a one-dimensional linear cellular automaton and a shift map on . It develops both measure-theoretic (MTDE) and topological (TDE) directional entropy frameworks, deriving rigorous bounds for Bernoulli and Markov measures and obtaining explicit angle-dependent formulas for TDE over rings . The results reveal directional anisotropy in entropy, with tractable expressions in terms of local-rule permutivity, ring structure, and direction, and they illustrate these phenomena via concrete examples over primes and prime-power rings. By decomposing general rings into prime-power components and applying weak addition, the paper provides a unified approach to computing MTDE and TDE for -actions arising from 1D LCAs and shifts, offering concrete benchmarks and guidance for future explorations of directional entropy in more complex CA-driven systems.

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 . 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.
Paper Structure (14 sections, 8 theorems, 91 equations, 12 figures)

This paper contains 14 sections, 8 theorems, 91 equations, 12 figures.

Table of Contents

  1. Short introduction
  2. The MTDE of $\mathbb{Z}^2$-actions over the ring $\mathbb{Z}_{m}$
  3. Examples
  4. MTDE for Bernoulli and Markov measures
  5. The MTDE with respect to Bernoulli measure
  6. The MTDE with respect to Markov measure
  7. Methodological Note
  8. Comments and References
  9. Topological Directional Entropy
  10. DTE for ring $m=p^k$
  11. DTE for ring $m=p_1^{k_1}.p_2^{k_2} \ldots p_h^{k_h}$
  12. Key Observations
  13. Computational Scope
  14. Comments and references

Key Result

theorem 1

Sinai1985 Let $p>0, q>0$ have no common factor. Then for any interval $I = I(a,-q/p)$.

Figures (12)

  • Figure 1: The segment $I(a,\omega)$ on the plane joining the points $(a, 0)$ and $(a +\omega^{-l}, 1)$.
  • Figure 2: The graph of the entropy function given in \ref{['exam-MTDE-eq1']}.
  • Figure 3: The graph of the entropy function given in \ref{['exam-MTDE-eq2']}.
  • Figure 4: The graphs of the functions $|z_l|$ and $|z_r|$ associated with the local rule \ref{['exam-MTDE-eq2']}.
  • Figure 5: The graph of function $h_{\vec{v}}(\Phi )$ given in \ref{['MTE-EXAM3a']} in the interval $[0,\pi]$.
  • ...and 7 more figures

Theorems & Definitions (17)

  • definition 1
  • definition 2
  • theorem 1
  • definition 3
  • proposition 1
  • theorem 2
  • theorem 3
  • proof
  • theorem 4
  • proof
  • ...and 7 more