English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The ring $\mathbb Z_d$ of $d$-adic integers has a natural interpretation asthe 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$\mathbb Z_d$ to itself and automorphisms of $T_d$ constitute the group$\mathrm{Isom}(\mathbb Z_d)$. In the case when $d=p$ is prime, Anashin showedthat $f\in\mathrm{Lip}^1(\mathbb Z_p)$ is defined by a finite Mealy automatonif and only if the reduced coefficients of its van der Put series constitute a$p$-automatic sequence over a finite subset of $\mathbb Z_p\cap\mathbb Q$. Wegeneralize this result to arbitrary integer $d\geq 2$, describe the explicitconnection between the Moore automaton producing such sequence and the Mealyautomaton inducing the corresponding endomorphism. Along the process we producetwo algorithms allowing to convert the Mealy automaton of an endomorphism tothe corresponding Moore automaton generating the sequence of the reduced vander Put coefficients of the induced map on $\mathbb Z_d$ and vice versa. Wedemonstrate examples of applications of these algorithms for the case when thesequence of coefficients is Thue-Morse sequence, and also for one of thegenerators of the standard automaton representation of the lamplighter group.

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