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arXiv 0902.3762math.DS
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可逆元胞自动机的遍历性

Some Ergodic Properties of Invertible Cellular Automata

Hasan Akin

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9 pages
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摘要：

本文考虑定义在有限基数字母表$p^k$上的可逆一维线性元胞自动机（CA），即环$\mathbb{Z}_{p^k}$（$(k \geq 2$和$p$为素数）上由$T_{f[l,r]}(x) = (y_n)_{n=-\infty}^{\infty} $，$y_{n} = f(x_{n+l}, ..., x_{n+r}) =\overset{r}{\underset{i=l}{\sum}}λ_{i}x_{n+i}(\text{mod} p^k)$，$x=(x_n)_{n=-\infty}^{\infty}\in \mathbb{Z}^{\mathbb{Z}}_{p^k}$和$f:\mathbb{Z}^{r-l+1}_{p^k}\to \mathbb{Z}_{p^k}$给出的映射$T_{f[l,r]}:\mathbb{Z}^{\mathbb{Z}}_{p^k}\to\mathbb{Z}^{\mathbb{Z}}_{p^k}$，其中$gcd(p,λ_r)=1$和$p| λ_i$对所有$i \neq r $（或$gcd(p, λ_l)=1$和$p|λ_i$对所有$i\neq l$）.在一定的假设下，证明了任意右（左）置换可逆的一维线性CA $T_{f[l,r]}$及其逆是强混合的.我们还证明了任何右（左）置换，可逆的一维线性CA是伯努利自同构，而不利用以前在文献中使用的自然扩展。

英文摘要：

In this paper we consider invertible one-dimensional linear cellular automata (CA hereafter) defined on a finite alphabet of cardinality $p^k$, i.e. the maps $T_{f[l,r]}:\mathbb{Z}^{\mathbb{Z}}_{p^k}\to\mathbb{Z}^{\mathbb{Z}}_{p^k}$ which are given by $T_{f[l,r]}(x) = (y_n)_{n=-\infty}^{\infty} $, $y_{n} = f(x_{n+l}, ..., x_{n+r}) =\overset{r}{\underset{i=l}{\sum}}λ_{i}x_{n+i}(\text{mod} p^k)$, $x=(x_n)_{n=-\infty}^{\infty}\in \mathbb{Z}^{\mathbb{Z}}_{p^k}$ and $f:\mathbb{Z}^{r-l+1}_{p^k}\to \mathbb{Z}_{p^k}$, over the ring $\mathbb{Z}_{p^k}$ $(k \geq 2$ and $p$ is a prime number), where $gcd(p,λ_r)=1$ and $p| λ_i$ for all $i \neq r $ (or $gcd(p, λ_l)=1$ and $p|λ_i$ for all $i\neq l$). Under some assumptions we prove that any right (left) permutative, invertible one-dimensional linear CA $T_{f[l,r]}$ and its inverse are strong mixing. We also prove that any right(left) permutative, invertible one-dimensional linear CA is Bernoulli automorphism without making use of the natural extension previously used in the literature.

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