We show that spacetime diagrams of linear cellular automata $\Phi : {\mathbb F}_p^{\mathbb Z} \to {\mathbb F}_p^{\mathbb Z}$ with $(-p)$-automatic initial conditions are automatic. This extends existing results on initial conditions which are eventually constant. Each automatic spacetime diagram defines a $(\sigma, \Phi)$-invariant subset of ${\mathbb F}_p^{\mathbb Z}$, where $\sigma$ is the left shift map, and if the initial condition is not eventually periodic then this invariant set is nontrivial. For the Ledrappier cellular automaton we construct a family of nontrivial $(\sigma, \Phi)$-invariant measures on ${\mathbb F}_3^{\mathbb Z}$. Finally, given a linear cellular automaton $\Phi$, we construct a nontrivial $(\sigma, \Phi)$-invariant measure on ${\mathbb F}_p^{\mathbb Z}$ for all but finitely many $p$.

中文翻译：

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线性元胞自动机的自动性和不变性测度

我们展示了线性元胞自动机 $\Phi 的时空图： {\mathbb F}_p^{\mathbb Z} \to {\mathbb F}_p^{\mathbb Z}$ 与 $(-p)$-自动初始条件是自动的。这扩展了最终恒定的初始条件下的现有结果。每个自动时空图都定义了 ${\mathbb F}_p^{\mathbb Z}$ 的 $(\sigma, \Phi)$-不变子集，其中 $\sigma$ 是左移映射，如果初始条件最终不是周期性的，那么这个不变集是非平凡的。对于 Ledrappier 元胞自动机，我们在 ${\mathbb F}_3^{\mathbb Z}$ 上构建了一系列非平凡的 $(\sigma, \Phi)$-不变测度。最后，给定一个线性元胞自动机 $\Phi$，我们在 ${\mathbb F}_p^{\mathbb Z}$ 上构造一个非平凡的 $(\sigma, \Phi)$-不变测度，除了有限多个 $p $.