SOLENOIDAL MAPS, AUTOMATIC SEQUENCES, VAN DER PUT SERIES, AND MEALY AUTOMATA
| Autor: | ROSTISLAV GRIGORCHUK, DMYTRO SAVCHUK |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Journal of the Australian Mathematical Society. 114:78-109 |
| ISSN: | 1446-8107 1446-7887 |
| Popis: | The 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. |
| Databáze: | OpenAIRE |
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