G. Let and ?. ?. , Then ?(tv) is the identity of G and thus tv ? Z. Since S is a tree set, it is acyclic and thus Z * is saturated in S by Theorem 5.12 Thus This implies that tv ? Z * . Since tv ? A * t, we have f (r)uf (s)v = f (r)qf (s) and thus uf (s)v = qf (s) for some q ? S. Since Z * is right unitary, f (r), f (r)uf (s)v ? Z * imply uf (s)v = qf (s) ? Z * . In turn, since Z * is left unitary, qf (s), f (s) ? Z *

. Proof, Since X is a prefix code, W is a prefix code. Since X is v ?1 S-maximal, W is ? ?1 (v ?1 S)-maximal by Proposition 2.9 (ii) and thus H-maximal since

?. Let and ?. W. , Then ?(xy), ?(y) ? X imply ?(x) ?

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, pp.91-192, 1963.
DOI : 10.1007/978-3-642-01742-1_23

P. Arnoux and G. Rauzy, 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
URL : http://www.numdam.org/article/BSMF_1991__119_2_199_0.pdf

J. Berstel, C. De-felice, D. Perrin, C. Reutenauer, and G. Rindone, Bifix codes and Sturmian words, Journal of Algebra, vol.369, pp.146-202, 2012.
DOI : 10.1016/j.jalgebra.2012.07.013
URL : https://hal.archives-ouvertes.fr/hal-00793907

J. Berstel, D. Perrin, and C. Reutenauer, Codes and Automata, volume 129 of Encyclopedia Math. Appl, 2009.

V. Berthé, C. De-felice, F. Dolce, J. Leroy, D. Perrin et al., Bifix codes and interval exchanges, Journal of Pure and Applied Algebra, vol.219, issue.7, pp.2781-2798
DOI : 10.1016/j.jpaa.2014.09.028

V. Berthé, C. De-felice, F. Dolce, J. Leroy, D. Perrin et al., Acyclic, connected and tree sets, Monatshefte f??r Mathematik, vol.18, issue.3???4
DOI : 10.1007/s00605-014-0721-4

V. Berthé, C. De-felice, F. Dolce, J. Leroy, D. Perrin et al., The finite index basis property, Journal of Pure and Applied Algebra, vol.219, issue.7, pp.2521-2537, 2015.
DOI : 10.1016/j.jpaa.2014.09.014

V. Berthé and M. Rigo, Combinatorics, automata and number theory, Encyclopedia Math. Appl. Cambridge Univ. Press, vol.135, 2010.
DOI : 10.1017/CBO9780511777653

J. Cassaigne, Complexité et facteurs spéciaux, Bull. Belg. Math. Soc. Simon Stevin Journées Montoises, vol.4, issue.1, pp.67-88, 1994.

J. Cassaigne, S. Ferenczi, and A. Messaoudi, M??lange faible et valeurs propres pour les suites d???Arnoux-Rauzy, Annales de l???institut Fourier, vol.58, issue.6, pp.1983-2005, 2008.
DOI : 10.5802/aif.2403

M. Paul and . Cohn, Free rings and their relations, 1985.

P. Issai, S. V. Cornfeld, . Fomin, and G. Yakov, Sina? ?. Ergodic theory, 1982.

M. Ethan, G. A. Coven, and . Hedlund, Sequences with minimal block growth, Math. Systems Theory, vol.7, pp.138-153, 1973.

F. Durand, 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

F. Durand, J. Leroy, and G. Richomme, Do the properties of an S-adic representation determine factor complexity?, J. Integer Seq, vol.16, issue.2, 2013.
URL : https://hal.archives-ouvertes.fr/lirmm-00797654

S. Eilenberg, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], Automata, languages, and machines. Pure and Applied Mathematics, vol.58, 1974.

S. Ferenczi, Rank and symbolic complexity Ergodic Theory Dynam, Systems, vol.16, issue.4, pp.663-682, 1996.
DOI : 10.1017/s0143385700009032
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.513.9270

S. Ferenczi and L. Q. Zamboni, -interval exchange transformations, Bulletin of the London Mathematical Society, vol.40, issue.4, pp.705-714, 2008.
DOI : 10.1112/blms/bdn051
URL : https://hal.archives-ouvertes.fr/hal-01263098

N. and P. Fogg, Substitutions in dynamics, arithmetics and combinatorics , volume 1794 of Lecture Notes in Mathematics, 2002.

A. Glen and J. Justin, Episturmian words: a survey, RAIRO - Theoretical Informatics and Applications, vol.43, issue.3, pp.403-442, 2009.
DOI : 10.1051/ita/2009003

J. Justin and L. Vuillon, 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

M. Keane, Interval exchange transformations, Mathematische Zeitschrift, vol.141, issue.1, pp.25-31, 1975.
DOI : 10.1007/BF01236981

K. Klouda, Bispecial factors in circular non-pushy D0L languages, Theoretical Computer Science, vol.445, pp.63-74, 2012.
DOI : 10.1016/j.tcs.2012.05.007

J. Leroy, An S-adic characterization of minimal subshifts with first difference of complexity 1 ? p(n + 1) ? p(n) ? 2, Discrete Math. Theor. Comput. Sci, vol.16, issue.1, pp.233-286, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01179422

J. Leroy, An S-adic characterization of ternary tree sets, 2014.

M. Lothaire, Algebraic Combinatorics on Words, 2002.
DOI : 10.1017/CBO9781107326019
URL : https://hal.archives-ouvertes.fr/hal-00620608

M. Morse and G. A. Hedlund, Symbolic Dynamics II. Sturmian Trajectories, American Journal of Mathematics, vol.62, issue.1/4, pp.1-42, 1940.
DOI : 10.2307/2371431

I. Valery and . Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR, vol.168, pp.1009-1011, 1966.

J. Sakarovitch, Elements of Automata Theory, 2009.
DOI : 10.1017/CBO9781139195218

B. Tan, Z. Wen, and Y. Zhang, The structure of invertible substitutions on??a??three-letter alphabet, Advances in Applied Mathematics, vol.32, issue.4, pp.736-753, 2004.
DOI : 10.1016/S0196-8858(03)00102-7

A. Zorich, Deviation for interval exchange transformations Ergodic Theory Dynam, Systems, vol.17, issue.6, pp.1477-1499, 1997.