[Paper Review] Powers in a class of A-strict standard episturmian words

Amy Glen|arXiv (Cornell University)|Aug 31, 2007

semigroups and automata theory3 citations

TL;DR

This paper characterizes all integer powers (squares, cubes, etc.) in a specific class of A-strict standard episturmian words, using canonical decompositions and generalized singular words. It proves that such words are 4-power free and provides explicit formulas for the number of distinct squares and cubes in their building blocks, extending prior results on Sturmian and Fibonacci words to higher-order generalizations like the k-bonacci word.

ABSTRACT

This paper concerns a specific class of strict standard episturmian words whose directive words resemble those of characteristic Sturmian words. In particular, we explicitly determine all integer powers occurring in such infinite words, extending recent results of Damanik and Lenz (2003), who studied powers in Sturmian words. The key tools in our analysis are canonical decompositions and a generalization of singular words, which were originally defined for the ubiquitous Fibonacci word. Our main results are demonstrated via some examples, including the $k$-bonacci word: a generalization of the Fibonacci word to a $k$-letter alphabet ($k\geq2$).

Motivation & Objective

• To extend the characterization of integer powers in Sturmian words to a broader class of episturmian words over larger alphabets.
• To determine all non-trivial integer powers (squares, cubes, etc.) occurring in A-strict standard episturmian words.
• To generalize the concept of singular words to episturmian words and use them as a key tool in analyzing repeated factors.
• To provide explicit formulas for the number of distinct squares and cubes in the building blocks of such words.
• To demonstrate the results on the k-bonacci word, a natural generalization of the Fibonacci word to k letters.

Proposed method

• The analysis relies on canonical decompositions of the infinite word into its fundamental building blocks s_n, which are recursively defined.
• Generalized singular words are introduced as a tool to analyze the structure of factors and their repetitions.
• The index (maximal fractional power) of each building block s_n is computed using properties of their conjugacy classes and associated words D_i.
• Theorems 6.11, 6.12, and 6.17 are derived to determine the number of distinct squares and cubes of given lengths in the word.
• The method is validated on the k-bonacci word, where all d_i = 1, leading to a recursive construction s_n = s_{n-1}s_{n-2}...s_{n-k}.
• The number of distinct powers of length m is determined by counting conjugates of specific words, with explicit formulas involving |D_i|.

Experimental results

Research questions

• RQ1 Which integer powers occur in A-strict standard episturmian words, and how can they be systematically characterized?
• RQ2 How do the number of distinct squares and cubes in the building blocks s_n depend on their structure and conjugacy classes?
• RQ3 Can the concept of singular words be generalized meaningfully to episturmian words beyond the Fibonacci and Sturmian cases?
• RQ4 What is the maximal power (e.g., 4-power) freeness status of such words, and how does it vary with the directive word?
• RQ5 How do the results for the k-bonacci word, a generalization of the Fibonacci word, compare to the classical Sturmian case?

Key findings

• All integer powers in the A-strict standard episturmian word s are determined by analyzing lengths in the set S = { |s_3|, 2|s_3|, |s_4|, |s_5|, 2|s_5|, |s_3 s_2|, |s_3^2 s_2|, |s_3 s_2^2 s_1|, |s_4 s_3|, |s_5 s_4|, |s_5^2 s_4|, |s_5 s_4 s_3| }.
• For length |s_3| = 11, there are 11 distinct squares; for length 2|s_3| = 22, there is exactly 1 distinct square, namely s_3^2.
• For length |s_4| = 32, there are 32 distinct squares, and for length |s_5| = 58, there are 58 distinct squares.
• For cubes, there are 11 distinct cubes of length |s_3| = 11, and 58 distinct cubes of length |s_5| = 58; for length |s_4| = 32, there are 2 distinct cubes, corresponding to the first two conjugates of s_4.
• The k-bonacci word is 4-power free, and all powers are fully characterized: P(|s_n|;2) = C(s_n), P(|s_n|;3) = {C_j(s_n) : 0 ≤ j ≤ |D_{n-k}|}, and P(m;l) = ∅ for all other m and l ≥ 2.
• The number of distinct squares of length |s_n s_{n-1}...s_{n+1-i}| is |D_{n+1-i}| + 1 for each i ∈ [2,k-1], and these are the only non-empty power sets.