There are \(k\)-uniform cubefree binary morphisms for all \(k \geq 0\). (English) Zbl 1211.05006
Summary: A word is cubefree if it contains no non-empty subword of the form \(xxx\). A morphism \(h: \varSigma ^{*} \rightarrow \varSigma ^{*}\) is \(k\)-uniform if \(h(a)\) has length \(k\) for all \(a \in \varSigma \). A morphism is cubefree if it maps cubefree words to cubefree words. We show that for all \(k\geq 0\) there exists a \(k\)-uniform cubefree binary morphism. By a result of Leconte, this implies the following stronger result: for all \(k\geq 0\) and \(n\geq 3\), there exists a \(k\)-uniform \(n\)-power-free binary morphism.
MSC:
| 05A05 | Permutations, words, matrices |
References:
| [1] | Aberkane, A.; Currie, J., There exist binary circular \(5 / 2^+\) power free words of every length, Electron, J. Combin., 11, #R10 (2004) · Zbl 1058.68084 |
| [2] | J.-P. Allouche, J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge, 2003; J.-P. Allouche, J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge, 2003 · Zbl 1086.11015 |
| [3] | Bean, D.; Ehrenfeucht, A.; McNulty, G., Avoidable patterns in strings of symbols, Pacific J. Math., 85, 261-294 (1979) · Zbl 0428.05001 |
| [4] | Berstel, J.; Séébold, P., A characterization of overlap-free morphisms, Discrete Appl. Math., 46, 275-281 (1993) · Zbl 0824.68093 |
| [5] | Brandenburg, F.-J., Uniformly growing \(k\) th power-free homomorphisms, Theoret. Comput. Sci., 23, 69-82 (1983) · Zbl 0508.68051 |
| [6] | Crochemore, M., Sharp characterizations of squarefree morphisms, Theoret. Comput. Sci., 18, 221-226 (1982) · Zbl 0482.68085 |
| [7] | Karhumäki, J., On cube-free \(\omega \)-words generated by binary morphisms, Discrete Appl. Math., 5, 279-297 (1983) · Zbl 0505.03022 |
| [8] | Keränen, V., On \(k\)-repetition freeness of length uniform morphisms over a binary alphabet, Discrete Appl. Math., 9, 301-305 (1984) · Zbl 0547.68082 |
| [9] | M. Leconte, Codes sans répétition, Thèse de 3ème cycle, Université Paris 6, LITP research report 85-56, Paris, 1985; M. Leconte, Codes sans répétition, Thèse de 3ème cycle, Université Paris 6, LITP research report 85-56, Paris, 1985 |
| [10] | Richomme, G., Some non finitely generated monoids of repetition-free endomorphisms, Inform. Process. Lett., 85, 61-66 (2003) · Zbl 1042.68070 |
| [11] | Richomme, G.; Séébold, P., Characterization of test-sets for overlap-free morphisms, Discrete Appl. Math., 98, 151-157 (1999) · Zbl 0946.68114 |
| [12] | Richomme, G.; Séébold, P., Conjectures and results of morphisms generating \(k\)-power-free words, Internat. J. Found. Comput. Sci., 15, 307-316 (2004) · Zbl 1067.68118 |
| [13] | Richomme, G.; Wlazinski, F., Some results on \(k\)-power-free morphisms, Theoret. Comput. Sci., 273, 119-142 (2002) · Zbl 1014.68127 |
| [14] | Richomme, G.; Wlazinski, F., Overlap-free morphisms and finite test-sets, Discrete Appl. Math., 143, 92-109 (2004) · Zbl 1073.68069 |
| [15] | Richomme, G.; Wlazinski, F., Existence of finite test-sets for \(k\)-power-freeness of uniform morphisms, Discrete Appl. Math., 155, 2001-2016 (2007) · Zbl 1129.68053 |
| [16] | Shur, A. M., The structure of the set of cube-free \(Z\)-words in a two-letter alphabet, Izv. Ross. Akad. Nauk Ser. Mat., 64, 201-224 (2000), (in Russian; English translation in Izv. Math. 64 (2000) 847-871) · Zbl 0972.68131 |
| [17] | Thue, A., Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Kra. Vidensk. Selsk. Skrifter. I. Math. Nat. Kl., 1, 1-67 (1912) · JFM 44.0462.01 |
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