Solenoid Maps, Automatic Sequences, Van Der Put Series, and Mealy-Moore Automata
Rostislav Grigorchuk, Dmytro Savchuk

TL;DR
This paper extends the characterization of 1-Lipschitz functions on p-adic integers via automata and automatic sequences to all integer bases, providing algorithms to convert between different automaton representations.
Contribution
It generalizes previous results from prime p to arbitrary integer bases d, and establishes explicit links between Mealy and Moore automata for these functions.
Findings
Provides algorithms for automaton conversion
Demonstrates applications with Thue-Morse sequence
Connects automaton representations with van der Put series
Abstract
The ring of -adic integers has a natural interpretation as the boundary of a rooted -ary tree . Endomorphisms of this tree (i.e. solenoid maps) are in one-to-one correspondence with 1-Lipschitz mappings from to itself and automorphisms of constitute the group . In the case when is prime, Anashin showed that is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a -automatic sequence over a finite subset of . We generalize this result to arbitrary integer , 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…
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Taxonomy
TopicsMathematical Dynamics and Fractals · Geometric and Algebraic Topology · semigroups and automata theory
