Topological Dynamics of 2D Cellular Automata

TL;DR

This paper explores the complex topological dynamics of 2D cellular automata, revealing new phenomena and undecidability results that distinguish them from 1D cases, including non-sensitive CA without equicontinuous points.

Contribution

It demonstrates novel properties of 2D CA, such as non-sensitive automata lacking equicontinuous points and the non-recursivity of sensitivity constants, extending 1D undecidability results.

Findings

Existence of non-sensitive 2D CA without equicontinuous points

Non-recursivity of sensitivity constants in 2D CA

Extension of undecidability results to 2D 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 2D CA and aims at showing that the situation is different and more complex. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants and the existence of CA having only non-recursive equicontinuous points. They all show a difference between the 1D and the 2D case. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case.