An $S$-adic characterization of minimal subshifts with first difference of complexity $1 \leq p(n+1) - p(n) \leq 2$

In [Ergodic Theory Dynam. System, 16 (1996) 663--682], S. Ferenczi proved that any minimal subshift with first difference of complexity bounded by 2 is $S$-adic with $\card S \leq 3^{27}$. In this paper, we improve this result by giving an $S$-adic charaterization of these subshifts with a set $S$ of 5 morphisms, solving by this way the $S$-adic conjecture for this particular case.

[1]  Sébastien Ferenczi,et al.  Complexity of sequences and dynamical systems , 1999, Discret. Math..

[2]  C. Mauduit,et al.  Substitutions in dynamics, arithmetics, and combinatorics , 2002 .

[3]  Gilles Didier.  Combinatoire des codages de rotations , 1998 .

[4]  Julien Leroy,et al.  Do the Properties of an S-adic Representation Determine Factor Complexity? , 2013 .

[5]  Fabien Durand,et al.  Linearly recurrent subshifts have a finite number of non-periodic subshift factors , 2000, Ergodic Theory and Dynamical Systems.

[6]  Fabien Durand,et al.  A characterization of substitutive sequences using return words , 1998, Discret. Math..

[7]  Gérard Rauzy,et al.  Représentation géométrique de suites de complexité $2n+1$ , 1991 .

[8]  I. Putnam,et al.  Ordered Bratteli diagrams, dimension groups and topological dynamics , 1992 .

[9]  Sébastien Ferenczi,et al.  Languages of k-interval exchange transformations , 2008 .

[10]  Alfred J. van der Poorten,et al.  Automatic sequences. Theory, applications, generalizations , 2005, Math. Comput..

[11]  Jean-Paul Allouche,et al.  Sur la complexite des suites in nies , 1994 .

[12]  G. A. Hedlund,et al.  Symbolic Dynamics II. Sturmian Trajectories , 1940 .

[13]  Filippo Mignosi,et al.  Morphismes sturmiens et règles de Rauzy , 1993 .

[14]  Valérie Berthé,et al.  Initial powers of Sturmian sequences , 2006 .

[15]  O. Bratteli.  Inductive limits of finite dimensional C*-algebras , 1972 .

[16]  Christian F. Skau,et al.  Substitutional dynamical systems, Bratteli diagrams and dimension groups , 1999, Ergodic Theory and Dynamical Systems.

[17]  M. Boshernitzan,et al.  A unique ergodicity of minimal symbolic flows with linear block growth , 1984 .

[18]  Julien Cassaigne,et al.  Complexité et facteurs spéciaux , 1997 .

[19]  Jean-Jacques Pansiot,et al.  Complexité des Facteurs des Mots Infinis Engendrés par Morphimes Itérés , 1984, ICALP.

[20]  Günter Rote.  Sequences With Subword Complexity 2n , 1994 .

[21]  Anatoly M. Vershik,et al.  Adic models of ergodic transformations, spectral theory, substitutions, and related topics , 1992 .

[22]  Julien Leroy,et al.  S -adic conjecture and Bratteli diagrams , 2012, 1210.1311.

[23]  Jacques Justin,et al.  Episturmian words: a survey , 2008, RAIRO Theor. Informatics Appl..

[24]  Julien Leroy.  Some improvements of the S-adic conjecture , 2012, Adv. Appl. Math..

[25]  橋本 武久,et al.  Simon Stevin「簿記論」の原型 , 2006 .

[26]  Gérard Rauzy,et al.  Échanges d'intervalles et transformations induites , 1979 .

[27]  Sébastien Ferenczi,et al.  Rank and symbolic complexity , 1996, Ergodic Theory and Dynamical Systems.

[28]  J. Allouche.  Algebraic Combinatorics on Words , 2005 .

[29]  Julien Cassaigne.  Special Factors of Sequences with Linear Subword Complexity , 1995, Developments in Language Theory.