论文标题
螺线管图，自动序列，Van der Put系列和Mealy-Moore Automata
Solenoid Maps, Automatic Sequences, Van Der Put Series, and Mealy-Moore Automata
论文作者
论文摘要
$ \ Mathbb z_d $ $ d $ -adic整数具有自然解释，作为根源$ d $ - ary tree $ t_d $的边界。这棵树的内态（即螺线管映射）与从$ \ mathbb z_d $到本身的1- lipschitz映射一对一，而$ t_d $的自动形态构成组$ \ mathrm {isom}（iSOM}（\ mathbb z_d）$。如果$ d = p $是PRIME，则Anashin表明$ f \ in \ Mathrm {Lip}^1（\ Mathbb Z_P）$由有限的MeAly Automaton定义，并且仅当其van der Put系列的降低的系数构成A $ p $ - p $ - p $ - p $ - p $ - autasual-autasual-autaination sequence y math y math的$ $ \ $ \ bbb q____________________________________________________ $ \ $ \ $ \ $ \ bbb。我们将此结果推广到任意整数$ d \ geq 2 $，描述了产生该序列的摩尔自动机与诱导相应内态的Mealy Automaton之间的明确连接。在整个过程中，我们产生了两种算法，允许将内态的Mealy自动机转换为相应的Moore Automaton，生成了还原的Van der的序列，将诱导的MAP的系数放在$ \ Mathbb Z_D $上，而Vicea。我们演示了这些算法在系数序列是thue-morse序列以及同样的标准自动机表示的发电机之一时，这些算法的应用示例。
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.
