Extremal words in morphic subshifts

Given an infinite word x over an alphabet A, a letter b occurring in x, and a total order @s on A, we call the smallest word with respect to @s starting with b in the shift orbit closure of x an extremal word of x. In this paper we consider the extremal words of morphic words. If x=g(f^@w(a)) for some morphisms f and g, we give two simple conditions on f and g that guarantee that all extremal words are morphic. This happens, in particular, when x is a primitive morphic or a binary pure morphic word. Our techniques provide characterizations of the extremal words of the period-doubling word and the Chacon word and a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin-Shapiro word.

[1]  David Damanik,et al.  Local symmetries in the period-doubling sequence , 2000, Discret. Appl. Math..

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

[3]  Jeffrey Shallit,et al.  Automatic Sequences: Theory, Applications, Generalizations , 2003 .

[4]  Sébastien Ferenczi,et al.  Les transformations de Chacon : combinatoire, structure géométrique, lien avec les systèmes de complexité $2n+1$ , 1995 .

[5]  Dalia Krieger.  On stabilizers of infinite words , 2008, Theor. Comput. Sci..

[6]  Giuseppe Pirillo,et al.  Characterizations of finite and infinite episturmian words via lexicographic orderings , 2008, Eur. J. Comb..

[7]  Giuseppe Pirillo,et al.  Inequalities characterizing standard Sturmian and episturmian words , 2005, Theor. Comput. Sci..

[8]  Luca Q. Zamboni,et al.  Descendants of Primitive Substitutions , 1999, Theory of Computing Systems.

[9]  Jean-Paul Allouche.  Théorie des nombres et automates , 1983 .

[10]  M. Lothaire.  Algebraic Combinatorics on Words , 2002 .

[11]  James D. Currie.  Lexicographically least words in the orbit closure of the Rudin-Shapiro word , 2011, Theor. Comput. Sci..

[12]  Shaobo Gan.  Sturmian sequences and the lexicographic world , 2000 .

[13]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[14]  M. Makarov.  On the infinite permutation generated by the period doubling word , 2010, Eur. J. Comb..

[15]  Tom Head,et al.  Fixed and Stationary ω —Words and ω —Languages , 1986 .

[16]  Jean-Paul Allouche,et al.  Extremal properties of (epi)Sturmian sequences and distribution modulo 1 , 2009, 0907.2430.

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

[18]  Jeffrey Shallit,et al.  Periodicity, repetitions, and orbits of an automatic sequence , 2008, Theor. Comput. Sci..

[19]  Jeffrey Shallit,et al.  Extremal Infinite Overlap-Free Binary Words , 1998, Electron. J. Comb..

[20]  J. Cassaigne,et al.  Combinatorics, Automata and Number Theory: Factor complexity , 2010 .