Topological transitivity of group cellular automata is decidable

TL;DR

This paper proves that topological transitivity, a key concept in dynamical systems, is decidable for all multi-dimensional group cellular automata over finite groups, extending previous results from one-dimensional cases.

Contribution

It establishes the first decidability result for topological transitivity in higher-dimensional and non-abelian group cellular automata.

Findings

Decidability of topological transitivity for $d$-dimensional group cellular automata over finite groups.

Decidability of related properties like mixing and ergodicity.

Reduction technique for analyzing complex automata by decomposing into simpler components.

Abstract

Topological transitivity is a fundamental notion in topological dynamics and is widely regarded as a basic indicator of global dynamical complexity. For general cellular automata, topological transitivity is known to be undecidable. By contrast, positive decidability results have been established for one-dimensional group cellular automata over abelian groups, while the extension to higher dimensions and to non-abelian groups has remained an open problem. In this work, we settle this problem by proving that topological transitivity is decidable for the class of $d$-dimensional ($d\geq 1$) group cellular automata over arbitrary finite groups. Our approach combines a decomposition technique for group cellular automata, reducing the problem to the analysis of simpler components, with an extension of several results from the existing literature in the one-dimensional setting. As a consequence of our results, and exploiting known equivalences among dynamical properties for group cellular automata, we also obtain the decidability of several related notions, including total transitivity, topological mixing and weak mixing, weak and strong ergodic mixing, and ergodicity.