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.

翻译：拓扑传递性是拓扑动力学中的一个基本概念，被广泛视为全局动力学复杂性的基本指标。对于一般的胞自动机，已知拓扑传递性是不可判定的。相比之下，对于阿贝尔群上的一维群胞自动机，已经建立了正面的可判定性结果，而将其推广到更高维度和非阿贝尔群的情况一直是一个悬而未决的问题。在本工作中，我们通过证明对于任意有限群上的 $d$ 维（$d\geq 1$）群胞自动机类，拓扑传递性是可判定的，从而解决了这个问题。我们的方法结合了群胞自动机的分解技术（将问题简化为对更简单组件的分析）以及对一维背景下现有文献中若干结果的扩展。作为我们结果的推论，并利用群胞自动机中动力学性质之间的已知等价关系，我们还获得了若干相关概念的可判定性，包括完全传递性、拓扑混合与弱混合、弱与强遍历混合，以及遍历性。