Infinite Fibonacci Word Research Articles

Repetition Factorization of Automatic Sequences

Following Inoue et al., we define a word to be a repetition if it is a (fractional) power of exponent at least [Formula: see text]. A word has a repetition factorization if it is the product of repetitions. We study repetition factorizations in several (generalized) automatic sequences, including the infinite Fibonacci word, the Thue-Morse word, paperfolding words, and the Rudin-Shapiro sequence.

Two-dimensional Fibonacci words: Tandem repeats and factor complexity

If x is a non-empty string then the repetition xx is called a tandem repeat. Similarly, a tandem in a two-dimensional array X is a configuration consisting of copies of a same primitive block W that touch each other with one side or a corner. In [6], Apostolico and Brimkov have proved various bounds for the number of tandems in a two-dimensional word of size m×n. Of the two types of tandems considered therein, they also proved that, for one type, the number of occurrences in an m×n Fibonacci array attained the general upper bound, O(m2nlogn). In this paper, we derive an expression for the exact number of tandems in a given finite Fibonacci array fm,n. As a required result, we derive the factor complexities of fm,n, m,n≥0 and that of the infinite Fibonacci word f∞,∞. Generations of f∞,∞ and fm,n, for any given m,n≥1 using a two-dimensional homomorphism are also achieved.

Patterns in the generalized Fibonacci word, applied to games

We analyze a natural generalization, W, of the infinite Fibonacci word over the alphabet Σ={a,b}. We provide tools to represent explicitly the set {s∈Z≥0:W(s)=b,W(s+x)=a} for any fixed positive integer x. We show how this representation can be used to analyze the preservation of P-positions of any game whose P-positions are a pair of complementary Beatty sequences, in particular a certain generalization of Wythoff Nim (Holladay, 1968; Fraenkel, 1982).

Counting Lyndon Factors

In this paper, we determine the maximum number of distinct Lyndon factors that a word of length $n$ can contain. We also derive formulas for the expected total number of Lyndon factors in a word of length $n$ on an alphabet of size $\sigma$, as well as the expected number of distinct Lyndon factors in such a word. The minimum number of distinct Lyndon factors in a word of length $n$ is $1$ and the minimum total number is $n$, with both bounds being achieved by $x^n$ where $x$ is a letter. A more interesting question to ask is what is the minimum number of distinct Lyndon factors in a Lyndon word of length $n$? In this direction, it is known (Saari, 2014) that a lower bound for the number of distinct Lyndon factors in a Lyndon word of length $n$ is $\lceil\log_{\phi}(n) + 1\rceil$, where $\phi$ denotes the golden ratio $(1 + \sqrt{5})/2$. Moreover, this lower bound is sharp when $n$ is a Fibonacci number and is attained by the so-called finite Fibonacci Lyndon words, which are precisely the Lyndon factors of the well-known infinite Fibonacci word $\boldsymbol{f}$ (a special example of an infinite Sturmian word). Saari (2014) conjectured that if $w$ is Lyndon word of length $n$, $n\ne 6$, containing the least number of distinct Lyndon factors over all Lyndon words of the same length, then $w$ is a Christoffel word (i.e., a Lyndon factor of an infinite Sturmian word). We give a counterexample to this conjecture. Furthermore, we generalise Saari's result on the number of distinct Lyndon factors of a Fibonacci Lyndon word by determining the number of distinct Lyndon factors of a given Christoffel word. We end with two open problems.

On the conjugacy class of the Fibonacci dynamical system

We characterize the symbolical dynamical systems which are topologically isomorphic to the Fibonacci dynamical system. We prove that there are infinitely many injective primitive substitutions generating a dynamical system in the Fibonacci conjugacy class. In this class there are infinitely many dynamical systems not generated by a substitution. An example is the system generated by doubling the 0's in the infinite Fibonacci word.

Abelian powers and repetitions in Sturmian words

Richomme, Saari and Zamboni (2011) [39] proved that at every position of a Sturmian word starts an abelian power of exponent k for every k>0. We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words. We give a formula for computing the maximum exponent of an abelian power of abelian period m starting at a given position in any Sturmian word of rotation angle α. By considering all possible abelian periods m, we recover the result of Richomme, Saari and Zamboni.As an analogue of the critical exponent, we introduce the abelian critical exponent A(sα) of a Sturmian word sα of angle α as the quantity A(sα)=limsupkm/m=limsupkm′/m, where km (resp. km′) denotes the maximum exponent of an abelian power (resp. of an abelian repetition) of abelian period m (the superior limits coincide for Sturmian words). We show that A(sα) equals the Lagrange constant of the number α. This yields a formula for computing A(sα) in terms of the partial quotients of the continued fraction expansion of α. Using this formula, we prove that A(sα)≥5 and that the equality holds for the Fibonacci word. We further prove that A(sα) is finite if and only if α has bounded partial quotients, that is, if and only if sα is β-power-free for some real number β.Concerning the infinite Fibonacci word, we prove that: i) The longest prefix that is an abelian repetition of period Fj, j>1, has length Fj(Fj+1+Fj−1+1)−2 if j is even or Fj(Fj+1+Fj−1)−2 if j is odd, where Fj is the jth Fibonacci number; ii) The minimum abelian period of any factor is a Fibonacci number. Further, we derive a formula for the minimum abelian periods of the finite Fibonacci words: we prove that for j≥3 the Fibonacci word fj, of length Fj, has minimum abelian period equal to F⌊j/2⌋ if j=0,1,2mod4 or to F1+⌊j/2⌋ if j=3mod4.

Searching for Zimin patterns

In the area of pattern avoidability the central role is played by special words called Zimin patterns. The symbols of these patterns are treated as variables and the rank of the pattern is its number of variables. Zimin type of a word x is introduced here as the maximum rank of a Zimin pattern matching x. We show how to compute Zimin type of a word on-line in linear time. Consequently we get a quadratic time, linear-space algorithm for searching Zimin patterns in words. Then we demonstrate how the Zimin type of the length n prefix of the infinite Fibonacci word is related to the representation of n in the Fibonacci numeration system. Using this relation, we prove that Zimin types of such prefixes and Zimin patterns inside them can be found in logarithmic time. Finally, we give some upper bounds on the function f(n,k) such that every k-ary word of length at least f(n,k) has a factor that matches the rank n Zimin pattern.

The [formula omitted]-representation of nonnegative integers and the Fibonacci factorization of suffixes of infinite Fibonacci words

For each suffix X of a two-way infinite Fibonacci word, we consider the factorization X=ukuk+1uk+2⋯, where k is a positive integer, and the length of the factor ui is the ith Fibonacci number (i≥k). It is called the Fibonacci factorization of X of order k. We show that in such a factorization, either all ui are singular words, or there exists a positive integer l≥k such that ul,ul+1,ul+2,… are the Fibonacci words along an infinite path in the tree of Fibonacci words and the rest of the uis are singular words. The labels of such infinite paths are determined by the D-representation of nonnegative integers.

Fibonacci word patterns in two-way infinite Fibonacci words

The studies of 1-Fibonacci word patterns and 0-Fibonacci word patterns were initiated by Turner (1988) [18] and Chuan (1993) [2] respectively. It is known that each proper suffix of the infinite Fibonacci word is an r-Fibonacci word pattern, r∈{0,1}. In this paper, we consider the suffixes of the two two-way infinite extensions G and G′ of the infinite Fibonacci word. We obtain necessary and sufficient conditions for suffixes of G and G′ to be r-Fibonacci word patterns. This gives us all the mechanical words with slope α=5−12 which are r-Fibonacci word patterns. All possible r-seed words of each of them are determined. Finally, images of suffixes of G and G′ under the action of certain Sturmian morphisms are computed.

A simple representation of subwords of the Fibonacci word

We introduce a representation of subwords of the infinite Fibonacci word f ∞ by a specific concatenation of finite Fibonacci words. It is unique and easily computable by backward processing of the given word. This provides an efficient recognition algorithm for subwords of f ∞ as well as a full description of their occurrences in f ∞ . Our representation yields a natural notion of rank of a subword, which explains main properties of subwords of f ∞ in a unified way.

Extensions and restrictions of Wythoff's game preserving its [formula omitted] positions

We consider extensions and restrictions of Wythoff's game having exactly the same set of P positions as the original game. No strict subset of rules gives the same set of P positions. On the other hand, we characterize all moves that can be adjoined while preserving the original set of P positions. Testing if a move belongs to such an extended set of rules is shown to be doable in polynomial time. Many arguments rely on the infinite Fibonacci word, automatic sequences and the corresponding numeration system. With these tools, we provide new two-dimensional morphisms generating an infinite picture encoding respectively P positions of Wythoff's game and moves that can be adjoined.

Locating factors of the infinite Fibonacci word

Let τ = ( 5 - 1 ) / 2 . Let a , b be two distinct letters. The infinite Fibonacci word is the infinite word G = babbababbabbababbababbabba ⋯ whose nth letter is a (resp., b) if [ ( n + 1 ) τ ] - [ n τ ] = 0 (resp., 1). For a factor w of G, the location of w is the set of all positions in G at which w occurs. Only the locations of the following factors of G are already known: squares, singular words and those factors of G whose lengths are Fibonacci numbers. The purpose of this paper is to determine the locations of all factors of G. Our results contain all the known ones as consequences. Moreover, using our results, we are able to identify any factor of G whenever its starting position and length are given; also we are able to tell whether two suffixes of G have a common prefix of a certain length.

Smooth words over arbitrary alphabets

Smooth infinite words over Σ = { 1 , 2 } are connected to the Kolakoski word K = 221121 ⋯ , defined as the fixpoint of the function Δ that counts the length of the runs of 1's and 2's. In this paper we extend the notion of smooth words to arbitrary alphabets and study some of their combinatorial properties. Using the run-length encoding Δ , every word is represented by a word obtained from the iterations of Δ . In particular we provide a new representation of the infinite Fibonacci word F as an eventually periodic word. On the other hand, the Thue–Morse word is represented by a finite one.

Conjugates of characteristic Sturmian words generated by morphisms

This article is concerned with characteristic Sturmian words of slope α and 1− α (denoted by c α and c 1− α resp.), where α∈(0,1) is an irrational number such that α=[0;1+d 1, d 2,…,d n ] with d n ≥ d 1≥1. It is known that both c α and c 1− α are fixed points of non-trivial (standard) morphisms σ and σ ̂ , respectively, if and only if α has a continued fraction expansion as above. Accordingly, such words c α and c 1− α are generated by the respective morphisms σ and σ ̂ . For the particular case when α=[0;2, r ̄ ] (r≥1) , we give a decomposition of each conjugate of c α (and hence c 1− α ) into generalized adjoining singular words, by considering conjugates of powers of the standard morphism σ by which it is generated. This extends a recent result of Levé and Séébold on conjugates of the infinite Fibonacci word.

On a combinatorial property of Fibonacci semigroup

We say that a semigroup S has property P ∗ n , n ⩾ 2, if, given elements x i ,…, x n of S, at least two of the n! products of these elements coincide. In a recent paper, Restivo considered the Fibonacci semigroup (i.e. the Rees quotient of { a,b} + by the ideal of nonfactors of the well-known infinite Fibonacci word abaababaabaab … and proved that it has property P ∗ 8 . Aim of this paper is to prove that it has property P ∗ 4 . As it does not have property P ∗ 3 , this is the best possible result.

A generalization of automatic sequences

We generalize the uniform tag sequences of Cobham, which arise as images of fixed points of k-uniform homomorphisms, to the case where the homomorphism ϕ is not necessarily uniform, but rather satisfies the analogue of an algebraic equation. We show that these sequences coincide with 1. (a) the class of sequences accepted by a finite automaton with “generalized digits” as input, and 2. (b) generalizations of the “locally catenative formula” of Rozenberg and Lindenmayer. Examples include the infinite Fibonacci word, which is generated as the fixed point of the homomorphism ϕ( a) = ab, ϕ( b) = a, and sequences of Rauzy and De Bruijn.