On the Asymptotic Palindrome Density of Fibonacci Infinite Words

Abstract

In this paper, we investigate the combinatorial and density properties of infinite words generated by Fibonacci-type morphisms, focusing on their subword structure, palindrome density, and extremal statistical behaviors. Using the morphism $0\to 01$, $1\to 0$, we define a derived ternary word $\mathbb{Y}$ and establish new results relating its density components $\mathrm{dens}(\lambda ,n)$, $\mathrm{dens}(\alpha ,n)$, and $\mathrm{dens}(\beta ,n)$, deriving explicit formulae and bounds on their behavior. We further prove a general density theorem for infinite words with paired subwords, showing that the associated palindromic prefix density is bounded above by $\frac{1}{{\phi }_1}$, where ${\phi }_1=(1+\sqrt{5})/2$ is the golden ratio. Our approach connects the structure of Fibonacci and Thue--Morse sequences with precise asymptotic and combinatorial interpretations for the observed densities.

10 pages, 2 figures and 2 tables