Topological Dynamics of Cellular Automata: Dimension Matters

TL;DR

This paper explores the complex topological dynamics of higher-dimensional cellular automata, revealing new phenomena and undecidability results that differ significantly from the well-understood one-dimensional case.

Contribution

It introduces novel constructions demonstrating the richer behavior of higher-dimensional CA and extends undecidability results to these cases.

Findings

Existence of non-sensitive CA without equicontinuous points

Non-recursivity of sensitivity constants in higher dimensions

Higher-dimensional CA exhibit more complex topological classification

Abstract

Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants, the existence of CA having only non-recursive equicontinuous points and the existence of CA having only countably many equicontinuous points. They all show a difference between dimension 1 and higher dimensions. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case. Finally, we show that the set of sensitive CA is only Pi_2 in dimension 1, but becomes Sigma_3-hard for dimension 3.