Directional Dynamics along Arbitrary Curves 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.
Keywords
Cite
@article{arxiv.1001.5470,
title = {Directional Dynamics along Arbitrary Curves in Cellular Automata},
author = {Martin Delacourt and Victor Poupet and Mathieu Sablik and Guillaume Theyssier},
journal= {arXiv preprint arXiv:1001.5470},
year = {2010}
}