The ring $\mathbb Z_d$ of $d$-adic integers has a natural interpretation as the boundary of a rooted $d$-ary tree $T_d$. Endomorphisms of this tree (i.e. solenoid maps) are in one-to-one correspondence with 1-Lipschitz mappings from $\mathbb Z_d$ to itself and automorphisms of $T_d$ constitute the group $\mathrm{Isom}(\mathbb Z_d)$. In the case when $d=p$ is prime, Anashin showed that $f\in\mathrm{Lip}^1(\mathbb Z_p)$ is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a $p$-automatic sequence over a finite subset of $\mathbb Z_p\cap\mathbb Q$. We generalize this result to arbitrary integer $d\geq 2$, describe the explicit connection between the Moore automaton producing such sequence and the Mealy automaton inducing the corresponding endomorphism. Along the process we produce two algorithms allowing to convert the Mealy automaton of an endomorphism to the corresponding Moore automaton generating the sequence of the reduced van der Put coefficients of the induced map on $\mathbb Z_d$ and vice versa. We demonstrate examples of applications of these algorithms for the case when the sequence of coefficients is Thue-Morse sequence, and also for one of the generators of the standard automaton representation of the lamplighter group.

中文翻译：

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电磁图、自动序列、Van Der Put 系列和 Mealy-Moore 自动机

$d$-adic 整数的环 $\mathbb Z_d$ 自然地解释为有根的 $d$-ary 树 $T_d$ 的边界。这棵树的自同构（即螺线管映射）与从 $\mathbb Z_d$ 到自身的 1-Lipschitz 映射一一对应，$T_d$ 的自同构构成了群 $\mathrm{Isom}(\mathbb Z_d) $. 在 $d=p$ 为素数的情况下，Anashin 证明 $f\in\mathrm{Lip}^1(\mathbb Z_p)$ 由有限 Mealy 自动机定义当且仅当其范德的约简系数Put 系列在 $\mathbb Z_p\cap\mathbb Q$ 的有限子集上构成 $p$-自动序列。我们将这个结果推广到任意整数 $d\geq 2$，描述了产生这种序列的 Moore 自动机和诱导相应自同态的 Mealy 自动机之间的显式联系。在这个过程中，我们生成了两种算法，允许将自同态的 Mealy 自动机转换为相应的 Moore 自动机，从而生成 $\mathbb Z_d$ 上的诱导映射的缩减 van der Put 系数的序列，反之亦然。我们演示了这些算法在系数序列是 Thue-Morse 序列的情况下的应用示例，以及点灯组的标准自动机表示的生成器之一。