SOLENOIDAL MAPS, AUTOMATIC SEQUENCES, VAN DER PUT SERIES, AND MEALY AUTOMATA

Abstract

AbstractThe ring$\mathbb Z_{d}$ofd-adic integers has a natural interpretation as the boundary of a rootedd-ary tree$T_{d}$. Endomorphisms of this tree (that is, solenoidal maps) are in one-to-one correspondence with 1-Lipschitz mappings from$\mathbb Z_{d}$to itself. In the case when$d=p$is prime, Anashin [‘Automata finiteness criterion in terms of van der Put series of automata functions’,p-Adic Numbers Ultrametric Anal. Appl.4(2) (2012), 151–160] 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 ap-automatic sequence over a finite subset of$\mathbb Z_{p}\cap \mathbb Q$. We generalize this result to arbitrary integers$d\geq 2$and describe the explicit connection between the Moore automaton producing such a sequence and the Mealy automaton inducing the corresponding endomorphism of a rooted tree. We also produce two algorithms converting one automaton to the other andvice versa. As a demonstration, we apply our algorithms to the Thue–Morse sequence and to one of the generators of the lamplighter group acting on the binary rooted tree.