Extremal words in morphic subshifts

TL;DR

This paper investigates extremal words in morphic subshifts, providing conditions under which these extremal words are also morphic, and applies these results to specific well-known infinite words.

Contribution

It establishes simple conditions on morphisms that ensure extremal words in morphic subshifts are morphic, extending understanding of their structure and providing new proofs for classical words.

Findings

All extremal words are morphic under certain morphism conditions.

Characterizations of extremal words for the Period-doubling and Chacon words.

A new proof for the lexicographically least word in the Rudin-Shapiro shift orbit.

Abstract

Given an infinite word X over an alphabet A a letter b occurring in X, and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure of X an extremal word of X. In this paper we consider the extremal words of morphic words. If X = g(f^{\omega}(a)) for some morphisms f and g, we give two simple conditions on f and g that guarantees that all extremal words are morphic. This happens, in particular, when X is a primitive morphic or a binary pure morphic word. Our techniques provide characterizations of the extremal words of the Period-doubling word and the Chacon word and give a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin-Shapiro word.