Mathematics > Commutative Algebra
[Submitted on 19 Jul 2019 (v1), last revised 6 Jun 2020 (this version, v3)]
Title:Integrality of matrices, finiteness of matrix semigroups, and dynamics of linear and additive cellular automata
View PDFAbstract:Let $\mathbb{K}$ be a finite commutative ring, and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Let $A$ and $B$ be two $n \times n$-matrices over $\mathbb{L}$ that have the same characteristic polynomial. The main result of this paper states that the set $\left\{ A^0,A^1,A^2,\ldots\right\}$ is finite if and only if the set $\left\{ B^0,B^1,B^2,\ldots\right\}$ is finite. We apply this result to Cellular Automata (CA). Indeed, it gives a complete and easy-to-check characterization of sensitivity to initial conditions and equicontinuity for linear CA over the alphabet $\mathbb{K}^n$ for $\mathbb{K} = \mathbb{Z}/m\mathbb{Z}$ i.e., CA in which the local rule is defined by $n\times n$-matrices with elements in $\mathbb{Z}/m\mathbb{Z}$. To prove our main result, we derive an integrality criterion for matrices that is likely of independent interest. Namely, let $\mathbb{K}$ be any commutative ring (not necessarily finite), and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Consider any $n \times n$-matrix $A$ over $\mathbb{L}$. Then, $A \in \mathbb{L}^{n \times n}$ is integral over $\mathbb{K}$ (that is, there exists a monic polynomial $f \in \mathbb{K}\left[t\right]$ satisfying $f\left(A\right) = 0$) if and only if all coefficients of the characteristic polynomial of $A$ are integral over $\mathbb{K}$. The proof of this fact relies on a strategic use of exterior powers (a trick pioneered by Gert Almkvist). Furthermore, we extend the decidability result concerning sensitivity and equicontinuity to the wider class of additive CA over a finite abelian group. For such CA, we also prove the decidability of injectivity, surjectivity, topological transitivity and all the properties (as, for instance, ergodicity) that are equivalent to the latter.
Submission history
From: Darij Grinberg [view email][v1] Fri, 19 Jul 2019 16:34:57 UTC (34 KB)
[v2] Tue, 7 Jan 2020 17:31:22 UTC (41 KB)
[v3] Sat, 6 Jun 2020 09:09:10 UTC (42 KB)
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