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Abstract: The powerful (and so far under-utilized) Goulden-Jackson Cluster method for finding the generating function for the number of words avoiding, as factors, the members of a prescribed set of ‘dirty words’, is tutorialized and extended in various directions. The authors’ Maple implementations, contained in several Maple packages available from this paper’s website

Abstract: We prove that the ‘connective constant ’ for ternary square-free words is at least 2 1/17 = 1.0416..., improving on Brinkhuis and Brandenburg’s lower bounds of 2 1/24 = 1.0293... and 2 1/22 = 1.032... respectively. This is the first improvement since 1983. A word is square-free if it never stutters, i.e. if it cannot be written as axxb for words a,b and nonempty word x. For example, ‘example ’ is square-free, but ‘exampample ’ is not. See Steven Finch’s famous Mathematical Constants site[3] for a thorough discussion and many references. Let a(n) be the number of ternary square-free n-letter words ( A006156, M2550 in the Sloane-Plouffe[4] listing, 1,3,6,12,18,30,42,...). Brinkhuis[2] and Brandenburg[1] showed that a(n) ≥ 2 n/24, and a(n) ≥ 2 n/22 respectively. Here we show, by extending the method of [2], that a(n) ≥ 2 n/17, and hence that µ: = limn→ ∞ a(n) 1/n ≥ 2 1/17 = 1.0416.... Definition: A triple-pair [[U0,V0], [U1,V1], [U2,V2]] where U0,V0,U1,V1,U2,V2 are words in the alphabet {0,1,2} of the same length k, will be called a k-Brinkhuis triple-pair if the following conditions are satisfied. • The 24 words of length 2k,

Abstract. Improved upper and lower bounds on the number of squarefree ternary words are obtained. The upper bound is based on the enumeration of square-free ternary words up to length 110. The lower bound is derived by constructing generalised Brinkhuis triples. The problem of finding such triples can essentially be reduced to a combinatorial problem, which can efficiently be treated by computer. In particular, it is shown that the number of square-free ternary words of length n grows at least as 65 n/40, replacing the previous best lower bound of 2 n/17. 1.

Finite alphabets of at least three letters permit the construction of square-free words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an approximate formula which is asymptotically exact with rapid convergence in the number of letters. Résumé Il est possible de construire des mots de longueur infinie sans carré sur un alphabet ayant au moins trois lettres. Nous démontrons que l’entropie du langage des mots sans carré sur un tel alphabet est strictement positive et l’encadrons par des bornes inférieure et supérieure raisonnables. Enfin, nous donnons pour l’entropie une expression approchée qui est asymptotiquement correcte et converge rapidement lorsque le nombre de lettres de l’alphabet tend vers l’infini.

A new lower bound on the number of n-letter ternary square-free words is presented: 110 , which improves the previous best result of 65 .

We enumerate all ternary length-ℓ square-free words, which are words avoiding squares of words up to length ℓ, for ℓ ≤ 24. We analyse the singular behaviour of the corresponding generating functions. This leads to new upper entropy bounds for ternary square-free words. We then consider ternary square-free words with fixed letter densities, thereby proving exponential growth for certain ensembles with various letter densities. We derive consequences for the free energy and entropy of ternary square-free words.

My aim in this article is to provide an accessible introduction to the notion of topological entropy and (for context) its measure theoretic analogue, and then to present some recent work applying related ideas to the structure of iterated preimages for a continuous (in general non-invertible) map of a compact metric space to itself. These ideas will be illustrated by two classes of examples, from circle maps and symbolic dynamics. My focus is on motivating and explaining definitions; most results are stated with at most a sketch of the proof. The informed reader will recognize imagery from Bowen’s exposition of topological entropy [Bow78] which I have freely adopted for motivation. 1 Measure-theoretic entropy How much can we learn from observations using an instrument with finite resolution? A simple model of a single observation on a ”state space ” X is a finite partition P = {A1,..., AN} of X into atoms, grouping the points (states) in X according to the reading they induce on our instrument. A measure µ on X with total measure µ(X) = 1 defines the probability of a given reading as pi = µ(Ai), i = 1,..., N. Shannon [Sha63] (see also [Khi57]) noted that the ”entropy ” of the partition H(P): = − N� i=0

We propose a new effective method for obtaining lower bounds on the number of repetition-free words over a finite alphabet. 1

We show that Dejean’s conjecture holds for n ≥ 27. Repetitions in words have been studied since the beginning of the previous century [13, 14]. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 5, 6, 7, 9]. For rational 1 < r ≤ 2, a fractional r-power is a non-empty word w = xx ′ such that x ′ is the prefix of x of length (r − 1)|x|. For example, 010 is a 3/2-power. A basic problem is that of identifying the repetitive threshold for each alphabet size n> 1: What is the infimum of r such that an infinite sequence on n letters exists, not containing any factor of exponent greater than r? The infimum is called the repetitive threshold of an n-letter alphabet, denoted by RT(n). Dejean’s conjecture [5] is that ⎨ 7/4, n = 3 RT(n) = 7/5, n = 4 n/(n − 1) n ̸ = 3, 4 Thue, Dejean and Pansiot, respectively [14, 5, 12] established the values RT(2), RT(3), RT(4). Moulin-Ollagnier [11] verified Dejean’s conjecture for 5 ≤ n ≤ 11, and Mohammad-Noori and Currie [10] proved the conjecture for 12 ≤ n ≤ 14.