Directional dynamics of $\mathbb{Z}_+\times \mathbb{Z}$-actions generated by 1D-CA and the shift map

TL;DR

This paper investigates the directional sequence entropy and dynamics of $ abla_+ imes abla$-actions generated by cellular automata and shift maps, revealing positive entropy in all directions and connecting to number theory through ergodic theory.

Contribution

It introduces the computation of directional sequence entropy for these actions and establishes the existence of systems with positive entropy in all directions, linking dynamics to number theory.

Findings

Existence of a sequence with positive directional sequence entropy in all directions.

Computation of directional sequence entropy for cellular automata and shift map actions.

Application of mean ergodic theory to derive number theoretic results.

Abstract

In this short paper, we compute the directional sequence entropy for of $\mathbb{Z}_+\times \mathbb{Z}$-actions generated by cellular automata and the shift map. Meanwhile, we study the directional dynamics of this system. As a corollary, we prove that there exists a sequence such that for any direction, some of the systems above have positive directional sequence entropy. Moreover, with help of mean ergodic theory for directional weak mixing systems, we obtain a result of number theory about combinatorial numbers.