On the Asymptotic Palindrome Density of Fibonacci Infinite Words

TL;DR

The paper investigates asymptotic palindrome density and subword densities in Fibonacci-type infinite words generated by the morphism $0 \to 01$, $1 \to 0$, introducing a novel word $\mathbb{Y}$ and dissecting density components $\mathrm{dens}(\lambda,n)$, $\mathrm{dens}(\alpha,n)$, and $\mathrm{dens}(\beta,n)$. It establishes a central binomial identity $\dfrac{2^n (F_k)^n}{n! (F_{n+k-3} F_{n-k-5})} = \binom{2n}{n}$ via singular factorization under the Fibonacci morphism and links these combinatorics to central binomial coefficients. A novel novelty-density framework for subwords is developed using $\mathbb{Y}$, drawing connections to the Thue–Morse sequence and proving bounds such as $\mathrm{dens}(\mathbb{Y}^*) \le \dfrac{2}{φ_1}$ and a general density formula (Theorem ThmDensityTypn1) for infinite words with paired subwords $m$ and $m^{\perp}$. Overall, the work unifies Fibonacci, Sturmian, and Thue–Morse structures to provide precise asymptotic interpretations of density and palindrome phenomena in infinite morphic words, with potential implications for combinatorics on words and symbolic dynamics.

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 the word $\mathbb{Y}$ and establish new results relating its density components $\mathrm{dens}(λ,n)$, $\mathrm{dens}(α,n)$, and $\mathrm{dens}(β,n)$, deriving explicit formulae and bounds on their behavior. We further prove a general density theorem for infinite words with paired subwords, showing that the density is bounded above by $\frac{1}{\varphi_1}$, where $\varphi_1 = (1 + \sqrt{5})/2$ is the golden ratio. Our approach connects the structure of Fibonacci and Thue--Morse sequences to central binomial coefficients, and yields precise asymptotic and combinatorial interpretations for the observed densities.

Figures (2)

• Figure 1: Density of $\mathbb{Y}$ with $y_{12}$.
• Figure 2: The effect density of $\mathbb{Y}$.

Theorems & Definitions (17)

• Definition 1: PalindromeRigo2014
• Definition 2: Sturmian characteristic Adamczewski2007
• Proposition 2.1: Adamczewski2007Allouche2003
• Definition 3: Fibonacci word Adamczewski2007
• Theorem 2.2: Fischler2013
• Proposition 3.1
• Proposition 3.2
• proof
• Theorem 4.1
• proof