On the number of return words in infinite words with complexity 2n+1

TL;DR

This paper investigates the count of return words in certain infinite words with linear complexity, focusing on those derived from rotations and interval exchange transformations, and establishes exact counts for these return words.

Contribution

It provides a precise count of return words in infinite words with complexity 2n+1, especially those generated by rotations and interval exchanges, which was previously unknown.

Findings

Number of return words in specific infinite words is exactly k.

Infinite words from rotations have a predictable return word structure.

The study extends understanding of combinatorial properties of infinite words.

Abstract

In this article, we count the number of return words in some infinite words with complexity 2n+1. We also consider some infinite words given by codings of rotation and interval exchange transformations on k intervals. We prove that the number of return words over a given word w for these infinite words is exactly k.