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Topological transitivity of group cellular automata is decidable

Niccolò Castronuovo, Alberto Dennunzio, Luciano Margara

TL;DR

The paper resolves the decidability of topological transitivity for $d$-dimensional GCAs over arbitrary finite groups, a longstanding open problem beyond the abelian, one-dimensional case. Its core method combines a transitivity-preserving VerbalDecomposition with reduction to direct products of isomorphic simple groups, enabling separate treatments of the abelian and non-abelian components. In the abelian case, transitivity is characterized by a matrix representation over Laurent polynomials and a gcd condition $\gcd(\chi(t), t^{p^i-1}-1)=1$ for $i=1,\dots,n$, where $\chi(t)$ is the characteristic polynomial; in the non-abelian case, a minimal-component analysis reduces the test to computable orders $o$, $o_i$ and the vectors $\alpha$ and $\beta$. An explicit algorithm combines these criteria to decide transitivity, and the result extends to the equivalent mixing/ergodic properties in GCAs.

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.

Topological transitivity of group cellular automata is decidable

TL;DR

The paper resolves the decidability of topological transitivity for -dimensional GCAs over arbitrary finite groups, a longstanding open problem beyond the abelian, one-dimensional case. Its core method combines a transitivity-preserving VerbalDecomposition with reduction to direct products of isomorphic simple groups, enabling separate treatments of the abelian and non-abelian components. In the abelian case, transitivity is characterized by a matrix representation over Laurent polynomials and a gcd condition for , where is the characteristic polynomial; in the non-abelian case, a minimal-component analysis reduces the test to computable orders , and the vectors and . An explicit algorithm combines these criteria to decide transitivity, and the result extends to the equivalent mixing/ergodic properties in GCAs.

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 -dimensional () 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.
Paper Structure (8 sections, 6 theorems, 29 equations, 2 algorithms)

This paper contains 8 sections, 6 theorems, 29 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. CAs and GCAs
  3. Dynamical properties and their relations
  4. A decomposition technique for GCAs preserving topolological transitivity
  5. Topological transitivity of GCAs over direct products of isomorphic abelian simple groups
  6. Topological transitivity of GCAs over direct products of isomorphic non-abelian simple groups
  7. Decidability of topological transitivity
  8. Conclusions and Further Work

Key Result

Theorem 1

Let $\mathbb{G}$ be a finite group and let $F$ be a GCA on $\mathbb{G}^{\mathbb{Z}^d}.$ Let be the output produced by the function VerbalDecomposition called on $(\mathbb{G},F)$. Then $(\mathbb{G}^{\mathbb{Z}^d},F)$ is topologically transitive if and only if each $(\mathbb{G}_i^{\mathbb{Z}^d},F_i)$ is topologically transitive.

Theorems & Definitions (13)

  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2: DennunzioFGM2020INS
  • Theorem 3
  • proof
  • Definition 1
  • Definition 2
  • Remark 2
  • Lemma 1
  • ...and 3 more