Toeplitz Words, Generalized Periodicity and Periodically Iterated Morphisms

We consider so-called Toeplitz words which can be viewed as generalizations of one-way infinite periodic words. We compute their subword complexity, and show that they can always be generated by iterating periodically a finite number of morphisms. Moreover, we define a structural classification of Toeplitz words which is reflected in the way in which they can be generated by iterated morphisms.

[1]  Helmut Prodinger,et al.  Infinite 0-1-sequences without long adjacent identical blocks , 1979, Discret. Math..

[2]  Michel Koskas.  About the p-Paperfolding Words , 1996, Theor. Comput. Sci..

[3]  M. Lothaire.  Combinatorics on words: Bibliography , 1997 .

[4]  Michel Koskas,et al.  Complexités de suites de Toeplitz , 1998, Discret. Math..

[5]  Arto Lepistö,et al.  On the Power of Periodic Iteration of Morphisms , 1993, ICALP.

[6]  H. Wilf,et al.  Uniqueness theorems for periodic functions , 1965 .

[7]  Michael A. Harrison,et al.  Introduction to formal language theory , 1978 .

[8]  A. V. D. Poorten.  Arithmetic and analytic properties of paper folding sequences , 1981, Bulletin of the Australian Mathematical Society.

[9]  Juhani Karhumäki,et al.  Alternating Iteration of Morphisms and the Kolakovski Sequence , 1992 .

[10]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[11]  Andrzej Ehrenfeucht,et al.  Subword Complexities of Various Classes of Deterministic Developmental Languages without Interactions , 1975, Theor. Comput. Sci..

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

[13]  Karel Culik,et al.  Iterative Devices Generating Infinite Words , 1992, STACS.

[14]  O. Toeplitz,et al.  Ein Beispiel zur Theorie der fastperiodischen Funktionen , 1928 .

[15]  L. N. Vaserstein,et al.  RECURRENCE IN ERGODIC THEORY AND COMBINATORIAL NUMBER THEORY , 1982 .

[16]  Michael A. Arbib,et al.  An Introduction to Formal Language Theory , 1988, Texts and Monographs in Computer Science.

[17]  M. Keane,et al.  0-1-sequences of Toeplitz type , 1969 .

[18]  Helmut Prodinger,et al.  Mellin Transforms and Asymptotics: Digital Sums , 1994, Theor. Comput. Sci..

[19]  Mireille Bousquet-Mélou,et al.  Canonical Positions for the Factors in Paperfolding Sequences , 1994, Theor. Comput. Sci..

[20]  Yu. I. Lyubich.  18.5. Uniqueness theorem for mean-periodic functions , 1984 .

[21]  J. Allouche,et al.  Toeplitz sequences, paperfolding, towers of Hanoi and progression-free sequence of integers , 1992 .