Endomorphisms of regular rooted trees induced by the action of polynomials on the ring ℤd of d-adic integers

Ahmed, E.; Savchuk, Dmytro

doi:10.1142/s0219498820501546

Cited by 4 publications

(3 citation statements)

References 22 publications

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“…Namely, if we endow the boundary X ∞ of the tree with some structure, then some of the transformations of X ∞ will induce automorphisms of X * . For example, one can endow X ∞ with the structure of the ring of d-adic numbers [BŠ06,AS17] and study the automorphisms induced by polynomials. Here, we will use two other interpretations of the boundary X ∞ as the ring Z d [[t]] of formal power series over Z d , and as an infinite dimensional free Z d -module Z ∞ d (which is a vector space over Z d in the case of prime d).…”

Section: Preliminariesmentioning

confidence: 99%

The lamplighter group of rank two generated by a bireversible automaton

Ahmed

Savchuk

2019

Communications in Algebra

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Section: Preliminariesmentioning

confidence: 99%

The lamplighter group of rank two generated by a bireversible automaton

Ahmed

Savchuk

2019

Communications in Algebra

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“…The first realization of an affine transformation of by a finite Mealy automaton was constructed by Bartholdi and Šuniḱ in [9]. Ahmed and the second author in [1] described automata defining polynomial functions on , where , and using the language of groups acting on rooted trees, deduced conditions for ergodicity of the action of f on obtained by completely different methods by Larin [27]. In [4], Anashin proved an excellent result relating finiteness of the Mealy automaton generating an endomorphism of the p -ary tree to automaticity of the sequence of reduced van der Put coefficients of the induced functions on , which are discussed below in detail.…”

Section: Introductionmentioning

confidence: 99%

Solenoidal Maps, Automatic Sequences, Van Der Put Series, and Mealy Automata

Grigorchuk¹,

Savchuk

2022

J. Aust. Math. Soc.

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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 (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 a p-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 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.

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“…The first realization of an affine transformation of Z p by a finite Mealy automaton was constructed by Bartholdi and Šunić in [9]. Ahmed and the second author in [1] described automata defining polynomial functions x → f (x) on Z d , where f ∈ Z[x], and using the language of groups acting on rooted trees deduced conditions for ergodicity of the action of f on Z 2 obtained by completely different methods by Larin [27]. In [3] Anashin proved an excellent result relating finiteness of the Mealy automaton generating an endomorphism of the p-ary tree with automaticity of the sequence of reduced van der Put coefficients of the induced functions on Z p , which will be discussed below in details.…”

Section: Introductionmentioning

confidence: 99%

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

Grigorchuk,

Savchuk

2020

Preprint

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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 (i.e. solenoid maps) are in one-to-one correspondence with 1-Lipschitz mappings from Z d to itself and automorphisms of T d constitute the group Isom(Z d ). In the case when d = p is prime, Anashin showed in [3] that f ∈ 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 ∩ Q. We generalize this result to arbitrary integer d ≥ 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 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.

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Endomorphisms of regular rooted trees induced by the action of polynomials on the ring ℤd of d-adic integers

Cited by 4 publications

References 22 publications

The lamplighter group of rank two generated by a bireversible automaton

The lamplighter group of rank two generated by a bireversible automaton

Solenoidal Maps, Automatic Sequences, Van Der Put Series, and Mealy Automata

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

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