Palindromic length of words and morphisms in class $\mathcal{P}$

TL;DR

This paper investigates how the palindromic length of factors in infinite words fixed by class P morphisms grows, showing logarithmic bounds and providing specific estimates for Fibonacci and Thue-Morse words.

Contribution

It establishes growth bounds for palindromic length in class P morphic words and constructs a new example with different growth behavior.

Findings

Palindromic length grows at most logarithmically for class P morphic words.

Specific growth constants are estimated for Fibonacci and Thue-Morse words.

A new infinite word with palindromic length growing as √n is constructed.

Abstract

We study the palindromic length of factors of infinite words fixed by morphisms of the so-called class $\mathcal{P}$ introduced by Hof, Knill and Simon. We show that it grows at most logarithmically with the length of the factor. For the Fibonacci word and the Thue-Morse word we provide estimates on the constants of the growth. We also construct an infinite word rich in palindromes for which the palindromic length grows as $\sqrt{n}$.