## Maximal Temporal Period of a Periodic Solution Generated by a One-Dimensional Cellular Automaton

Janko Gravner
Xiaochen Liu

Department of Mathematics
University of California, Davis
1 Shields Avenue
Davis, CA 95616, USA

#### Abstract

One-dimensional cellular automata evolutions with both temporal and spatial periodicity are studied. The main objective is to investigate the longest temporal periods among all two-neighbor rules, with a fixed spatial period σ and number of states n. When σ = 2, 3, 4 or 6, and the rules are restricted to be additive, the longest period can be expressed as the exponent of the multiplicative group of an appropriate ring. Non-additive rules are also constructed with temporal period on the same order as the trivial upper bound ${n}^{\sigma }$. Experimental results, open problems and possible extensions of the results are also discussed.

Keywords: cellular automaton; exponent of a multiplicative group; periodic solution

Cite this publication as:
J. Gravner and X. Liu, “Maximal Temporal Period of a Periodic Solution Generated by a One-Dimensional Cellular Automaton,” Complex Systems, 30(3), 2021 pp. 239–272.
https://doi.org/10.25088/ComplexSystems.30.3.239