A characterization of power-free morphisms

Leconte, Michel

doi:10.1016/0304-3975(85)90213-0

Cited by 18 publications

(7 citation statements)

References 2 publications

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“…There is a reasonably large literature about these morphisms, with most investigators concentrating on giving computable characterizations of such morphisms; see, for example, [11,15,21,27,35].…”

Section: Decision Problems About Repetitionsmentioning

confidence: 99%

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Periodicity, repetitions, and orbits of an automatic sequence

Allouche

Rampersad

Shallit

2009

Theoretical Computer Science

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a b s t r a c tWe revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given k-automatic sequence is ultimately periodic. We prove that it is decidable whether a given k-automatic sequence is overlap-free (or squarefree, or cubefree, etc.). We prove that the lexicographically least sequence in the orbit closure of a k-automatic sequence is k-automatic, and use this last result to show that several related quantities, such as the critical exponent, irrationality measure, and recurrence quotient for Sturmian words with slope α, have automatic continued fraction expansions if α does.

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Section: Decision Problems About Repetitionsmentioning

confidence: 99%

“…A morphism h : Σ * → ∆ * is said to be k-power-free if whenever w is k-powerfree, so is h(w). There is a reasonably large literature about these morphisms, with most investigators concentrating on giving computable characterizations of such morphisms; see, for example, [11,15,21,27,35].…”

Section: Decision Problems About Repetitionsmentioning

confidence: 99%

Periodicity, repetitions, and orbits of an automatic sequence

Allouche

Rampersad

Shallit

2009

Theoretical Computer Science

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“…, where 1 ≤ j 0 , j 1 , j 2 ≤ k. As proven in [4,7,12], the ρ are squarefree morphisms mapping each squarefree word of length m to k m squarefree words of length nm. Thus, existence of an n-Brinkhuis k-triple indicates that…”

Section: N-brinkhuis K-triplesmentioning

confidence: 96%

An Improved Lower Bound for $n$-Brinkhuis $k$-Triples

Sollami,

Douglas,

Liebmann

2016

Preprint

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Let s n be the number of words in the ternary alphabet Σ = {0, 1, 2} such that no subword (or factor) is a square (a word concatenated with itself, e.g., 11, 1212, or 102102). From computational evidence, s n grows exponentially at a rate of about 1.317277 n . While known upper bounds are already relatively close to the conjectured rate, effective lower bounds are much more difficult to obtain. In this paper, we construct a 54-Brinkhuis 952-triple, which leads to an improved lower bound on the number of n-letter ternary squarefree words: 952 n/53 ≈ 1.1381531 n .

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“…The iteration of a non-trivial repetition-free endomorphism or a substitution produces repetition-free words of any length. Dealing with substitutions somewhat later, we point out that repetition-free morphisms have been sharply characterised in [2][3][4][5][6][7][8][9]. Those results concern different types of repetitions (k-repetitions for a given integer k ≥ 2) and alphabet sizes.…”

Section: Introductionmentioning

confidence: 98%

A powerful abelian square-free substitution over 4 letters

Keränen

2009

Theoretical Computer Science

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a b s t r a c t In 1961, Paul Erdös posed the question whether abelian squares can be avoided in arbitrarily long words over a finite alphabet. An abelian square is a non-empty word uv, where u and v are permutations (anagrams) of each other. The case of the four letter alphabet Σ 4 = {a, b, c, d} turned out to be the most challenging and remained open until 1992 when the author presented an abelian square-free (a-2-free) endomorphism g 85 of Σ * 4 . The size of this g 85 , i.e., |g 85 (abcd)|, is equal to 4×85 (uniform modulus). Until recently, all known methods for constructing arbitrarily long a-2-free words on Σ 4 have been based on the structure of g 85 and on the endomorphism g 98 of Σ * 4 found in 2002. In this paper, a great many new a-2-free endomorphisms of Σ * 4 are reported. The sizes of these endomorphisms range from 4 × 102 to 4 × 115. Importantly, twelve of the new a-2-free endomorphisms, of modulus m = 109, can be used to construct an a-2-free (commutatively functional) substitution σ 109 of Σ * 4 with 12 image words for each letter. The properties of σ 109 lead to a considerable improvement for the lower bound of the exponential growth of c n , i.e., of the number of a-2-free words over 4 letters of length n. It is obtained that c n > β −50 β n with β = 12 1/m 1.02306. Originally, in 1998, Carpi established the exponential growth of c n by showing that c n > β −t β n with β = 2 19/t = 2 19/(85 3 −85)1.000021, where t = 85 3 − 85 is the modulus of the substitution that he constructs starting from g 85 .

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A characterization of power-free morphisms

Cited by 18 publications

References 2 publications

Periodicity, repetitions, and orbits of an automatic sequence

Periodicity, repetitions, and orbits of an automatic sequence

An Improved Lower Bound for $n$-Brinkhuis $k$-Triples

A powerful abelian square-free substitution over 4 letters

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