Solenoid Maps, Automatic Sequences, Van Der Put Series, and Mealy-Moore Automata

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 $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}({\mathbb{Z}}_d)$. In the case when $d=p$ is prime, Anashin showed that $f\in {\mathrm{L}\mathrm{i}\mathrm{p}}^1({\mathbb{Z}}_p)$ 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 $d\ge 2$, 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.