On the Dynamical Behaviour of Cellular Automata

In this paper we study the dynamics of 1- and 2- dimensional cellular automata, using a 2-adic representation of the states, we give a simple graphical technique for finding periodic solutions. We also study the continuity properties of the associated 2-adic system and show how to compute the entropy.

💡 Research Summary

The paper investigates the dynamical behavior of one‑ and two‑dimensional cellular automata (CA) by embedding their binary state spaces into the unit interval using a 2‑adic (binary) representation. For a one‑dimensional CA with an odd‑length local rule (\mathcal R:{0,1}^p\to{0,1}), the authors first isolate the subset of (p)-bit patterns that leave the central cell unchanged. Each such pattern becomes a vertex in a directed graph (G); an edge from vertex (i) to vertex (j) is drawn when the rightmost (p-1) bits of (i) match the leftmost (p-1) bits of (j). An infinite path in this graph corresponds to a configuration that is invariant under the CA, i.e., a fixed point. The existence of a cycle in (G) guarantees the existence of a periodic orbit, because a cycle can be repeated indefinitely to produce a spatially periodic configuration. This graphical construction yields Lemma 2.1 and Theorem 2.1, providing a simple visual method for locating fixed points and periodic orbits.

To find periodic solutions of period (k), the authors consider the (k)-step evolution operator (N^k). They show that (N^k) can be represented by a new local rule of length (p_k = kp - (k-1)). The same graph‑building procedure applied to this enlarged rule identifies period‑(k) orbits as cycles in the corresponding graph. An explicit example with a 3‑bit rule demonstrates how a period‑2 orbit is obtained by constructing the associated 5‑bit rule and its graph.

The binary configurations are then mapped to real numbers in (