Automata and automatic sequences

In the following pages we discuss infinite sequences defined on a finite alphabet, and more specially those which are generated by finite automata. We have divided our paper into seven parts which are more or less self-contained. Needless to say, we feel that the order we propose is the most natural one. References appear at the end of each one of the parts which implies some redundancy.

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

[2]  J. Eckmann,et al.  Iterated maps on the interval as dynamical systems , 1980 .

[3]  Jean-Jacques Pansiot,et al.  Bornes inferieures sur la complexite des facteurs des mots infinis engendres par morphimes iteres , 1984, STACS.

[4]  Jean-Paul Allouche,et al.  Finite automata and morphisms in assisted musical composition , 1995 .

[5]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[6]  Aldo de Luca,et al.  Some Combinatorial Properties of the Thue-Morse Sequence and a Problem in Semigroups , 1989, Theor. Comput. Sci..

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

[8]  Jorge Stolfi,et al.  Two methods for generating fractals , 1989, Comput. Graph..

[9]  Pascal Hubert.  Complexit?e de suites d?efinies par des billards rationnels , 1995 .

[10]  Filippo Mignosi,et al.  Repetitions in the Fibonacci infinite word , 1992, RAIRO Theor. Informatics Appl..

[11]  J. Allouche,et al.  Quasicrystal Ising chain and automata theory , 1986 .

[12]  Jean-Paul Allouche,et al.  Note on the Cyclic Towers of Hanoi , 1994, Theor. Comput. Sci..

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

[14]  T. Kamae.  A continuous family of automata : the ising automata , 1996 .

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

[16]  L. Jonker,et al.  Periodic Orbits and Kneading Invariants , 1979 .

[17]  J. Shallit,et al.  Morphisms, Squarefree Strings, and the Tower of Hanoi Puzzle , 1994 .

[18]  F. Dekking,et al.  Uniform distribution modulo one: a geometrical viewpoint. , 1981 .

[19]  Jean-Paul Allouche,et al.  Complexité des suites de Rudin-Shapiro généralisées , 1993 .

[20]  Pierre Arnoux,et al.  Rauzy's Conjecture on Billiards in the Cube , 1994 .

[21]  J.-P. Bertrandias.  Espaces de fonctions bornées et continues en moyenne asymptotique d’ordre $p$ , 1966 .

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

[23]  William Douglas Weakley.  On the number of Cinfinity-words of each length , 1989, J. Comb. Theory, Ser. A.

[24]  K. Mahler,et al.  The Spectrum of an Array and its Application to the Study of the Translation Properties of a Simple Class of Arithmetical Functions: Part Two On the Translation Properties of a Simple Class of Arithmetical Functions , 1927 .

[25]  Yves Meyer.  Le spectre de Wiener , 1966 .

[26]  R. Baxter.  Exactly solved models in statistical mechanics , 1982 .

[27]  Walter Rudin,et al.  Some theorems on Fourier coefficients , 1959 .

[28]  J.-P. Allouche,et al.  Linear Cellular Automata, Finite Automata and Pascal's Triangle , 1996, Discret. Appl. Math..

[29]  Aldo de Luca,et al.  On the Factors of the Thue-Morse Word on Three Symbols , 1988, Inf. Process. Lett..

[30]  C. Michaux,et al.  LOGIC AND p-RECOGNIZABLE SETS OF INTEGERS , 1994 .

[31]  F. Axel,et al.  VIBRATIONAL MODES IN A ONE DIMENSIONAL "QUASI-ALLOY" : THE MORSE CASE , 1986 .

[32]  J. Cahn,et al.  Metallic Phase with Long-Range Orientational Order and No Translational Symmetry , 1984 .

[33]  Bernard Derrida,et al.  Products of random matrices and one dimensional disordered systems , 1985 .

[34]  N. Wiener.  Generalized harmonic analysis , 1930 .

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

[36]  Günter Rote,et al.  Sequences With Subword Complexity 2n , 1994 .

[37]  Jean-Paul Allouche,et al.  A relative of the Thue-Morse sequence , 1995, Discret. Math..

[38]  H. Shapiro,et al.  Extremal problems for polynomials and power series , 1951 .

[39]  P. Shiue,et al.  Substitution invariant cutting sequences , 1993 .

[40]  Jeffrey Shallit.  A Generalization of Automatic Sequences , 1988, Theor. Comput. Sci..

[41]  B. Cipra.  An introduction to the Ising model , 1987 .

[42]  Anton Bovier,et al.  Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions , 1993 .

[43]  Arturo Capri.  Repetitions in the Kolakovski sequence. , 1993 .

[44]  Shizuo Kakutani,et al.  Strictly ergodic symbolic dynamical systems , 1972 .

[45]  Jean-Paul Allouche,et al.  Facteurs des suites de Rudin-Shapiro généralisées , 1994 .

[46]  E. Ising.  Beitrag zur Theorie des Ferromagnetismus , 1925 .

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

[48]  Jean-Paul Allouche,et al.  The number of factors in a paperfolding sequence , 1992, Bulletin of the Australian Mathematical Society.

[49]  Jean Coquet,et al.  Sur la mesure spectrale de certaines suites arithmétiques , 1977 .

[50]  Pierre Arnoux,et al.  Complexity of sequences defined by billiard in the cube , 1994 .

[51]  Jeffrey Shallit,et al.  Sur Des Points Fixes De Morphismes D'Un Monoïde Libre , 1989, RAIRO Theor. Informatics Appl..

[52]  Ming Li,et al.  Kolmogorov Complexity and its Applications , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[53]  Wen Zhixiong,et al.  Remarques sur la suite engendrée par des substitutions composées , 1988 .

[54]  Gérald Tenenbaum,et al.  Dimension des courbes planes, papiers plies et suites de Rudin-Shapiro , 1981 .

[55]  Julien Cassaigne.  Motifs evitables et regularites dans les mots , 1994 .

[56]  F. Michel Dekking,et al.  Iteration of maps by an automaton , 1994, Discret. Math..

[57]  Jean Bass.  Suites uniformément denses, moyennes trigonométriques, fonctions pseudo-aléatoires , 1959 .

[58]  Jean-Paul Allouche,et al.  Oscillations spatio-temporelles engendrees par un automate cellulaire , 1984, Discret. Appl. Math..

[59]  Jeffrey Shallit,et al.  Simple continued fractions for some irrational numbers, II , 1979 .

[60]  Harold Marston Morse.  Recurrent geodesics on a surface of negative curvature , 1921 .

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

[62]  Colin J. Thompson,et al.  Mathematical Statistical Mechanics , 1972 .

[63]  Françoise Axel,et al.  Spectrum and extended states in a harmonic chain with controlled disorder: Effects of the Thue-Morse symmetry , 1989 .

[64]  Srecko Brlek,et al.  Enumeration of factors in the Thue-Morse word , 1989, Discret. Appl. Math..