Directional Dynamics along Arbitrary Curves in Cellular Automata

TL;DR

This paper explores how cellular automata behave along arbitrary space-time curves, revealing complex dynamics such as equicontinuity and sensitivity, and characterizing slopes with equicontinuous behavior as computably enumerable numbers.

Contribution

It introduces a framework for analyzing cellular automata along arbitrary curves and characterizes slopes with equicontinuous dynamics as computably enumerable numbers.

Findings

Existence of automaton with equicontinuous parabola dynamics but sensitivity in linear directions.

Slopes with equicontinuous dynamics are exactly the computably enumerable numbers.

Demonstrates complex directional behaviors in cellular automata.

Abstract

This paper studies directional dynamics in cellular automata, a formalism previously introduced by the third author. The central idea is to study the dynamical behaviour of a cellular automaton through the conjoint action of its global rule (temporal action) and the shift map (spacial action): qualitative behaviours inherited from topological dynamics (equicontinuity, sensitivity, expansivity) are thus considered along arbitrary curves in space-time. The main contributions of the paper concern equicontinuous dynamics which can be connected to the notion of consequences of a word. We show that there is a cellular automaton with an equicontinuous dynamics along a parabola, but which is sensitive along any linear direction. We also show that real numbers that occur as the slope of a limit linear direction with equicontinuous dynamics in some cellular automaton are exactly the computably enumerable numbers.