Recurrent word

In mathematics, a recurrent word or sequence is an infinite word over a finite alphabet in which every factor occurs infinitely often.^[1][2][3] An infinite word is recurrent if and only if it is a sesquipower.^[4][5]

A uniformly recurrent word is a recurrent word in which for any given factor X in the sequence, there is some length n_X (often much longer than the length of X) such that X appears in every block of length n_X.^[1][6][7] The term minimal sequence^[8] or almost periodic sequence(Muchnik, Semenov, Ushakov 2003) is also used.

Examples

• The easiest way to make a recurrent sequence is to form a periodic sequence, one where the sequence repeats entirely after a given number m of steps. Such a sequence is in then uniformly recurrent and n_X can be set to any multiple of m that is larger than twice the length of X. A recurrent sequence that is ultimately periodic is purely periodic.^[2]
• The Thue–Morse sequence is uniformly recurrent without being periodic, nor even eventually periodic (meaning periodic after some nonperiodic initial segment).^[9]
• More generally, all Sturmian words are recurrent.^[10]

References

1. Lothaire (2011) p. 30
2. Allouche & Shallit (2003) p.325
3. Pytheas Fogg (2002) p.2
4. Lothaire (2011) p. 141
5. Berstel et al (2009) p.133
6. Berthé & Rigo (2010) p.7
7. Allouche & Shallit (2003) p.328
8. Pytheas Fogg (2002) p.6
9. Lothaire (2011) p.31
10. Berthé & Rigo (2010) p.177

• Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
• Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words. CRM Monograph Series. 27. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4480-9. Zbl 1161.68043.
• Berthé, Valérie; Rigo, Michel, eds. (2010). Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. 135. Cambridge: Cambridge University Press. ISBN 978-0-521-51597-9. Zbl 1197.68006.
• Lothaire, M. (2011). Algebraic combinatorics on words. Encyclopedia of Mathematics and Its Applications. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. ISBN 978-0-521-18071-9. Zbl 1221.68183.
• Pytheas Fogg, N. (2002). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. 1794. Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.
• An. Muchnik, A. Semenov, M. Ushakov, Almost periodic sequences, Theoret. Comput. Sci. vol.304 no.1-3 (2003), 1-33.