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

The ring 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 (that is, solenoidal maps) are in one-to-one correspondence with 1-Lipschitz mappings from 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 is an element of Lip(1)(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 Z(p) boolean AND Q. We generalize this result to arbitrary integers d >= 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 and vice 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.