Automatic sequences are also non-uniformly morphic

TL;DR

This paper proves that every k-automatic sequence can be represented as the image of a fixed point of a non-uniform morphism, expanding understanding of the relationship between automatic sequences and morphic sequences.

Contribution

It establishes that all k-automatic sequences are obtainable from non-uniform morphic fixed points, bridging a gap in the theory of automatic and morphic sequences.

Findings

Every k-automatic sequence is the image of a non-uniform morphic fixed point.

Non-uniform morphisms can generate all k-automatic sequences.

The result links automatic sequences to non-uniform morphic structures.

Abstract

It is well-known that there exist infinite sequences that are the fixed point of non-uniform morphisms, but not $k$-automatic for any $k$. In this note we show that every $k$-automatic sequence is the image of a fixed point of a {\it non-uniform\/} morphism.