Topological Dynamics of Cellular Automata: Dimension Matters

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.

💡 Research Summary

The paper investigates the topological dynamics of cellular automata (CA) beyond the well‑studied one‑dimensional case, focusing on dimensions two and three. While in dimension 1 every non‑sensitive CA necessarily possesses an equicontinuous point (the classic “Equ / Sens” dichotomy proved by Kůrka), the authors demonstrate that this partition collapses as soon as the dimension is at least two.

The core of the work is a constructive 2‑dimensional CA, denoted F, built from a 12‑state alphabet split into a “solid” component S (nine symbols) and a “liquid” component L (three symbols: U, D, 0). The solid part obeys a finite‑type subshift Σ_S that forces solid cells to appear only as well‑spaced rectangular obstacles. The liquid part consists of particles formed by a pair U/D that move leftward; when encountering a solid obstacle they split, one particle goes above, the other below, and they recombine after bypassing the obstacle. This mechanism yields two distinct phases:

1. Erosion – any finite configuration eventually erodes all solid cells that do not satisfy Σ_S, leaving only a collection of disjoint rectangles (Lemma 1).
2. Infiltration – particles can be injected arbitrarily far away and, by repeatedly splitting and recombining, can reach any liquid cell inside the obstacle field (Lemma 3).

Using these phases the authors prove two fundamental dynamical properties of F:

• Non‑sensitivity (Proposition 2): For any ε > 0 a sufficiently large, well‑formed obstacle can be placed so that any configuration differing from the obstacle by less than ε/4 never diverges from it by more than ε under iteration. Hence F is not sensitive to initial conditions.

• Absence of equicontinuous points (Proposition 3): Assuming an equicontinuous point x leads to a contradiction. Choose a zero cell z₀ in x and set ε = 2^{−‖z₀‖∞−1}. By inserting a particle far enough away (using Lemma 3) one can create a configuration y′ that stays ε‑close to x initially but later differs at z₀, violating the definition of equicontinuity. Consequently F belongs to a new class N of CA that are neither sensitive nor equicontinuous, refuting the 1‑D dichotomy.

Beyond dynamics, the paper explores computational complexity aspects that arise only in higher dimensions:

• Non‑recursivity of sensitivity constants (Proposition 8): Determining the minimal ε for which a CA is sensitive is undecidable and not even recursively approximable in dimension 2, unlike the 1‑D case where such constants are computable.

• Non‑recursive equicontinuous points (Proposition 9): The authors construct CA whose equicontinuous points, when they exist, are not recursively enumerable; some CA have only countably many equicontinuous points, a phenomenon absent in 1‑D.

• Complexity of classification: The three natural classes—Equ (has equicontinuous points), Sens (sensitive), and N (neither)—are each shown to be neither r.e. nor co‑r.e. in dimension 2, providing examples of natural CA properties that are strictly harder than classic decision problems such as reversibility, surjectivity, or nilpotency (all of which are r.e. or co‑r.e.).

Finally, the authors examine how the descriptive‑set‑theoretic complexity of the set of sensitive CA changes with dimension. In dimension 1 the set is Π₂‑complete (a universal‑existential property). By adapting the undecidability constructions of