SOLENOIDAL MAPS, AUTOMATIC SEQUENCES, VAN DER PUT SERIES, AND MEALY AUTOMATA Academic Article uri icon

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), 151160] showed that$fin mathrm {Lip}^{1}(mathbb Z_{p})$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$dgeq 2$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 ThueMorse sequence and to one of the generators of the lamplighter group acting on the binary rooted tree.

published proceedings

  • JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY

author list (cited authors)

  • Grigorchuk, R., & Savchuk, D.

citation count

  • 0

complete list of authors

  • Grigorchuk, Rostislav||Savchuk, Dmytro

publication date

  • February 2023