We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts $S_{\beta }$ and its relation to direct topological factorizations. We show that any reversible CA $F:S_{\beta }\to S_{\beta }$ has an almost equicontinuous direction whenever $S_{\beta }$ is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime, i.e. they are not conjugate to direct topological factorizations $X\times Y$ of two nontrivial subshifts $X$ and $Y$. We also give a simple criterion to determine whether $S_{n\gamma }$ is conjugate to $S_n\times S_{\gamma }$ for a given integer $n\ge 1$ and a given real $\gamma >1$ when $S_{\gamma }$ is a subshift of finite type. When $S_{\gamma }$ is strictly sofic, we show that such a conjugacy is not possible at least when $\gamma$ is a quadratic Pisot number of degree $2$. We conclude by using direct factorizations to give a new proof for the classification of reversible multiplication automata on beta-shifts with integral base and ask whether nontrivial multiplication automata exist when the base is not an integer.

我们考虑了双向β移位中细胞自动机（CA）可能动力学的范围 ${\mathrm{小号}}_{\beta }$及其与直接拓扑分解的关系。我们证明任何可逆的CA$F：{\mathrm{小号}}_{\beta }\to {\mathrm{小号}}_{\beta }$ 每当有几乎相等的方向 ${\mathrm{小号}}_{\beta }$不是苏菲克式的。这样的推论是，非声波β位移是拓扑上直接的素数，即它们不与直接拓扑因式分解共轭$X\times ÿ$ 两个非平凡的子移位 $X$ 和 $ÿ$。我们还给出了一个简单的标准来确定是否${\mathrm{小号}}_{ñ\gamma }$ 与...共轭 ${\mathrm{小号}}_{ñ}\times {\mathrm{小号}}_{\gamma }$ 对于给定的整数 $ñ\ge \mathrm{1个}$ 和给定的真实 $\gamma >\mathrm{1个}$ 什么时候 ${\mathrm{小号}}_{\gamma }$是有限类型的子移位。什么时候${\mathrm{小号}}_{\gamma }$ 严格来说，我们证明至少在以下情况下这种共轭是不可能的 $\gamma$ 是度数的二次Pisot数 $\mathrm{2个}$。我们通过使用直接因式分解得出结论，从而为具有整数基数的beta移位上的可逆乘法自动机分类提供了新的证据，并询问当基数不是整数时是否存在非平凡乘法自动机。