Infinite Words with Finite Defect

TL;DR

This paper characterizes uniformly recurrent words with finite defect through palindromic and factor complexity relations, and shows they can be obtained via morphisms from a specific class related to rich words.

Contribution

It introduces a new characterization of such words and defines a class of morphisms P_ret that generate these words from rich words.

Findings

Characterization of uniformly recurrent words with finite defect.

Introduction of the morphism class P_ret and its properties.

Connection between P_ret morphisms and the class P by Hof, Knill, and Simon.

Abstract

In this paper, we provide a new characterization of uniformly recurrent words with finite defect based on a relation between the palindromic and factor complexity. Furthermore, we introduce a class of morphisms P_ret closed under composition and we show that a uniformly recurrent word with finite defect is an image of a rich (also called full) word under a morphism of class P_ret. This class is closely related to the well-known class P defined by Hof, Knill, and Simon; every morphism from P_ret is conjugate to a morphism of class P.