The fact that X is an F -maximal bifix code of F -degree d results from Corollary 6.2.3. Let us show that G F (X) = G(Z) Let B = (R, 1, 1) be the minimal automaton of Z * ,
Denote by Im(w) the image of ?(w) with respect to A Thus Im(w) = {t ? Q | s · w = t for some s ? Q}. Let u ? F be a word with d parses with respect to X. Let I = Im(u) By Lemma 7.1.4, the word u has rank d and thus Card(I) = d. Let Y = R F (u) be the set of first return words to u. By Theorem 6.5.2, the set Y is a basis of the free group A ? . For any y ? Y , the restriction of ?(y) to I is a permutation of I. Indeed, uy ? A + u implies Im(uy) ? I. Since uy ? F , the set Im(uy) has d elements by Lemma 7.1.5 The restriction of e to I is the identity, Thus Im(uy) = I. Since Im(u) = I, this proves the claim. Let e be an idempotent in ?(Y + ) ,
be the maximal group contained in ?(A * ) which contains e. It is a permutation group on I which is equivalent to G F (X) ,
let ?(y) be the restriction of ?(y) to the set I. For any y ? Y * , e?(y)e has the same nuclear equivalence and the same image as e. By Proposition 7.1.2 it implies that they are in the same H-class ,
Repr??sentation g??om??trique de suites de complexit?? $2n+1$, Bulletin de la Société mathématique de France, vol.119, issue.2, pp.199-215, 1991. ,
DOI : 10.24033/bsmf.2164
Rational subsets of groups. European science Foundation, 2011. ,
Codes and sofic constraints, Theoretical Computer Science, vol.340, issue.2, pp.381-393, 2005. ,
DOI : 10.1016/j.tcs.2005.03.033
Completing codes in a sofic shift, Theoretical Computer Science, vol.410, issue.43, pp.4423-4431, 2009. ,
DOI : 10.1016/j.tcs.2009.07.023
Codes and Automata, 2009. ,
DOI : 10.1017/CBO9781139195768
URL : https://hal.archives-ouvertes.fr/hal-00620616
Continued fractions and the Markoff tree, Expositiones Mathematicae, vol.25, issue.3, pp.187-213, 2007. ,
DOI : 10.1016/j.exmath.2006.10.002
Codes of central Sturmian words, Theoretical Computer Science, vol.340, issue.2, pp.220-239, 2005. ,
DOI : 10.1016/j.tcs.2005.03.021
Sur un algorithme donnant les codes bipr??fixes finis, Mathematical Systems Theory, vol.6, issue.1-2, pp.221-225, 1972. ,
DOI : 10.1007/BF01706090
Propriétes combinatoires des codes bipréfixes, Théorie des Codes, pp.20-46, 1979. ,
The construction of decimals normal in the scale of ten, J. London Math. Soc, vol.8, pp.254-260, 1933. ,
Observatio Arithmetica, Annali di Matematica Pura ed Applicata, vol.6, issue.1, pp.145-152 ,
DOI : 10.1007/BF02420125
Markoff forms and primitive words, Mathematische Annalen, vol.41, issue.2, pp.8-22, 1972. ,
DOI : 10.1007/BF01419427
Indecomposable permutations, hypermaps and labeled Dyck paths, Journal of Combinatorial Theory, Series A, vol.116, issue.8, pp.1326-1343, 2009. ,
DOI : 10.1016/j.jcta.2009.02.008
URL : https://hal.archives-ouvertes.fr/hal-00344954
Sequences with minimal block growth, Math. Systems Theory, vol.7, pp.138-153, 1973. ,
Two-way string-matching, Journal of the ACM, vol.38, issue.3, pp.651-675, 1991. ,
DOI : 10.1145/116825.116845
URL : https://hal.archives-ouvertes.fr/hal-00619582
The Markoff and Lagrange spectra , volume 30 of Mathematical Surveys and Monographs, 1989. ,
Finite biprefix sets of paths in a graph, Theoret. Comput. Sci, vol.58, issue.1-3, pp.103-128, 1988. ,
Sturmian words: structure, combinatorics, and their arithmetics, Theoret. Comput. Sci, vol.183, issue.1, pp.45-82, 1997. ,
Studies in the theory of numbers, 1930. ,
Parametrizing the subgroups of finite index in a free group and related topics, Bayreuth. Math. Schr, issue.20, pp.1-8, 1985. ,
Episturmian words and some constructions of de Luca and Rauzy, Theoretical Computer Science, vol.255, issue.1-2, pp.539-553, 2001. ,
DOI : 10.1016/S0304-3975(99)00320-5
A characterization of substitutive sequences using return words, Discrete Mathematics, vol.179, issue.1-3, pp.89-101, 1998. ,
DOI : 10.1016/S0012-365X(97)00029-0
URL : https://hal.archives-ouvertes.fr/hal-00303319
¨ Uber die Markoffschen Zahlen Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, pp.598-627, 1913. ,
Variable length binary encodings, Bell System Tech. J, vol.38, pp.933-967, 1959. ,
Episturmian words: a survey, RAIRO - Theoretical Informatics and Applications, vol.43, issue.3, pp.403-442, 2009. ,
DOI : 10.1051/ita/2009003
Subgroups of finite index in free groups, Journal canadien de math??matiques, vol.1, issue.2, pp.187-190, 1949. ,
DOI : 10.4153/CJM-1949-017-2
Descendants of Primitive Substitutions, Theory of Computing Systems, vol.32, issue.2, pp.133-157, 1999. ,
DOI : 10.1007/s002240000114
Cyclic renewal systems, Theoretical Computer Science, vol.410, issue.27-29, pp.27-292675, 2009. ,
DOI : 10.1016/j.tcs.2009.03.033
Episturmian words and episturmian morphisms, Theoretical Computer Science, vol.276, issue.1-2, pp.281-313, 2002. ,
DOI : 10.1016/S0304-3975(01)00207-9
EPISTURMIAN WORDS: SHIFTS, MORPHISMS AND NUMERATION SYSTEMS, International Journal of Foundations of Computer Science, vol.15, issue.02, pp.329-348, 2004. ,
DOI : 10.1142/S0129054104002455
Return words in Sturmian and episturmian words, RAIRO - Theoretical Informatics and Applications, vol.34, issue.5, pp.343-356, 2000. ,
DOI : 10.1051/ita:2000121
Stallings Foldings and Subgroups of Free Groups, Journal of Algebra, vol.248, issue.2, pp.608-668, 2002. ,
DOI : 10.1006/jabr.2001.9033
Sturmian morphisms, the braid group B 4, Christoffel words and bases of F 2, Annali di Matematica Pura ed Applicata, vol.318, issue.2, pp.317-339, 2007. ,
DOI : 10.1007/s10231-006-0008-z
URL : https://hal.archives-ouvertes.fr/hal-00119874
Combinatorics on Words, 1997. ,
DOI : 10.1017/CBO9780511566097
URL : https://hal.archives-ouvertes.fr/hal-00620607
Algebraic Combinatorics on Words, 2002. ,
DOI : 10.1017/CBO9781107326019
URL : https://hal.archives-ouvertes.fr/hal-00620608
Combinatorial group theory, Classics in Mathematics, 2001. ,
Sur les formes quadratiques binaires indéfinies, Math. Ann, vol.15, issue.3, pp.381-406, 1879. ,
Sur les formes quadratiques binaires indéfinies, Math. Ann, vol.17, issue.3, pp.379-399 ,
Morphismes sturmiens et r??gles de Rauzy, Journal de Th??orie des Nombres de Bordeaux, vol.5, issue.2, pp.221-233, 1993. ,
DOI : 10.5802/jtnb.91
Symbolic Dynamics II. Sturmian Trajectories, American Journal of Mathematics, vol.62, issue.1/4, pp.1-42, 1940. ,
DOI : 10.2307/2371431
Primitives in the free group on two generators, Invent. Math, vol.63, issue.1, pp.17-24, 1981. ,
Transitivity and connectivity of permutations, Combinatorica, vol.24, issue.3, pp.487-501, 2004. ,
URL : https://hal.archives-ouvertes.fr/hal-00005618
Sur les groupes dans les mono¨?desmono¨?des finis, Noncommutative Structures in Algebra and Geometric Combinatorics, pp.27-36, 1978. ,
On syntactic groups, Bull. Belg. Math. Soc. Simon Stevin, vol.10, pp.749-759, 2003. ,
URL : https://hal.archives-ouvertes.fr/hal-00790638
Substitution dynamical systems?spectral analysis, Lecture Notes in Mathematics, vol.1294, 2010. ,
Codes and local constraints, Theoretical Computer Science, vol.72, issue.1, pp.55-64, 1990. ,
DOI : 10.1016/0304-3975(90)90046-K
Une Topologie du Mono??de Libre, Semigroup Forum, vol.3, issue.1, pp.33-49, 1979. ,
DOI : 10.1007/BF02574174
Ensembles libres de chemins dans un graphe, Bulletin de la Société mathématique de France, vol.79, issue.2, pp.135-152, 1986. ,
DOI : 10.24033/bsmf.2051
On factorially balanced sets of words, Theoretical Computer Science, vol.412, issue.39, 2010. ,
DOI : 10.1016/j.tcs.2011.06.027
URL : https://hal.archives-ouvertes.fr/lirmm-00507782
Construction d'une famille de codes associes a certains groupes finis, Theoretical Computer Science, vol.54, issue.2-3, pp.165-179, 1987. ,
DOI : 10.1016/0304-3975(87)90127-7
On an application of semigroup methods to some problems in coding, IRE Trans. Inform. Theory, issue.2, pp.47-60, 1956. ,
On a family of submonoids, Publ. Math. Inst. Hungar. Acad. Sci, pp.381-391, 1961. ,
On a Special Class of Recurrent Events, The Annals of Mathematical Statistics, vol.32, issue.4, pp.1201-1213, 1961. ,
DOI : 10.1214/aoms/1177704860
A property of finitely generated submonoids of free monoids, Algebraic theory of semigroups (Proc. Sixth Algebraic Conf, pp.545-576, 1976. ,
A Characterization of Sturmian Words by Return Words, European Journal of Combinatorics, vol.22, issue.2, pp.263-275, 2001. ,
DOI : 10.1006/eujc.2000.0444
Local isomorphisms of invertible substitutions, C. R. Acad. Sci. Paris Sér. I Math, vol.318, issue.4, pp.299-304, 1994. ,