Automatic sequences: from rational bases to trees

TL;DR

This paper explores a new class of automatic sequences generated from tree-based numeration systems, including rational bases, providing characterizations via periodic block substitutions and subtree structures.

Contribution

It introduces novel characterizations of automatic sequences built on tree-based numeration systems, extending beyond regular numeration languages and including non-morphic examples.

Findings

Characterization via $r$-block substitutions with periodic morphisms

Sequences built on tree-based systems can be non-morphic

Analysis of subtrees and factors in the associated trees

Abstract

The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider those built on languages associated with trees having periodic labeled signatures and, in particular, rational base numeration systems. We obtain two main characterizations of these sequences. The first one is concerned with $r$-block substitutions where $r$ morphisms are applied periodically. In particular, we provide examples of such sequences that are not morphic. The second characterization involves the factors, or subtrees of finite height, of the tree associated with the numeration system and decorated by the terms of the sequence.