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Analyze the Density of Words over Morphism $\{a,b\}$

Jasem Hamoud, Duaa Abdullah

TL;DR

This work addresses the density of the Fibonacci word and its morphism-derived forms by developing a morphism-based framework and Beatty-type representations tied to the golden ratio $\varphi$. It introduces a power operation on Fibonacci words and proves an index-invariant expression Pow$(\mathcal{F}_k)=aabaa+abababaabaaba$ with density $DF(\operatorname{Pow}(\mathcal{F}_k))=\varphi-1$, illustrating remarkable cancellations in noncommutative word combinatorics. The study further analyzes framed words and higher-order Fibonacci constructs, establishing asymptotic letter densities $\mathrm{dens}_a(\mathbb{Q})=1/\varphi$ and $\mathrm{dens}_b(\mathbb{Q})=1/\varphi^2$ and connecting these results to Fibonacci and Lucas number identities. Overall, the paper links combinatorics on words with number-theoretic properties of Fibonacci/Lucas sequences, yielding exact density formulas and structural insights for morphic word families.

Abstract

In this paper, we analyze the density of the Fibonacci word and its derived forms by examining the morphisms associated with each. It offers a comparative analysis of the density of Fibonacci numbers alongside other words derived from Fibonacci word. Fibonacci words over the alphabet $\{a,b\}$, we define a novel \emph{power} operation that yields a formal linear combination in the free abelian group generated by all finite words.

Analyze the Density of Words over Morphism $\{a,b\}$

TL;DR

This work addresses the density of the Fibonacci word and its morphism-derived forms by developing a morphism-based framework and Beatty-type representations tied to the golden ratio . It introduces a power operation on Fibonacci words and proves an index-invariant expression Pow with density , illustrating remarkable cancellations in noncommutative word combinatorics. The study further analyzes framed words and higher-order Fibonacci constructs, establishing asymptotic letter densities and and connecting these results to Fibonacci and Lucas number identities. Overall, the paper links combinatorics on words with number-theoretic properties of Fibonacci/Lucas sequences, yielding exact density formulas and structural insights for morphic word families.

Abstract

In this paper, we analyze the density of the Fibonacci word and its derived forms by examining the morphisms associated with each. It offers a comparative analysis of the density of Fibonacci numbers alongside other words derived from Fibonacci word. Fibonacci words over the alphabet , we define a novel \emph{power} operation that yields a formal linear combination in the free abelian group generated by all finite words.
Paper Structure (8 sections, 13 theorems, 52 equations, 2 figures, 1 table)

This paper contains 8 sections, 13 theorems, 52 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem of search
  4. Main Result
  5. A General Characterization of Fibonacci Words
  6. Characterization Fibonacci and Lucas numbers
  7. Power of Fibonacci word over Alphabeta
  8. Conclusion and Discussion

Key Result

Lemma 2.1

For any $k-$ ary words of length $n$, the number of this words given by $k^n$.

Figures (2)

  • Figure 1: Morphism of the Fibonacci word Berstel2007.
  • Figure 2: Both cases such that discussed among Proposition \ref{['sec2:proposition2']}.

Theorems & Definitions (26)

  • Definition 1: Heubach2009
  • Definition 2: Elzinga2008
  • Lemma 2.1: Heubach2009
  • Lemma 2.2: Rigo2014
  • Proposition 3.1
  • proof
  • Remark 1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 16 more