Palindrome complexity

We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points of primitive morphisms of constant length belonging to "class P" of Hof-Knill-Simon. We also give an upper bound for the palindrome complexity of a sequence in terms of its (block-)complexity.

[1]  D. Damanik.  Gordon-type arguments in the spectral theory of one-dimensional quasicrystals , 1999 .

[2]  P. Michel.  Stricte ergodicite d’ensembles minimaux de substitution , 1976 .

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

[4]  Michael Baake.  A Note on Palindromicity , 1999 .

[5]  Andrzej Ehrenfeucht,et al.  Subword complexities of various classes of deterministic developmental languages with interactions , 2005, International Journal of Computer & Information Sciences.

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

[7]  Xavier Droubay,et al.  Palindromes in the Fibonacci Word , 1995, Inf. Process. Lett..

[8]  F. M. Dekking.  What is the Long Range Order in the Kolakoski Sequence , 1997 .

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

[10]  Robert V. Moody,et al.  The Mathematics of Long-Range Aperiodic Order , 1997 .

[11]  Giuseppe Pirillo,et al.  Episturmian words and episturmian morphisms , 2002, Theor. Comput. Sci..

[12]  Eric Goodstein.  On Sums of Digits , 1941 .

[13]  Jeffrey Shallit,et al.  Sums of Digits, Overlaps, and Palindromes , 2000, Discret. Math. Theor. Comput. Sci..

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

[15]  B. Mossé.  Reconnaissabilité des substitutions et complexité des suites automatiques , 1996 .

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

[17]  Brigitte Mosse,et al.  Properties of words and recognizability of fixed points of a substitution , 1992 .

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

[19]  P. A. B. Pleasants,et al.  Characterization of two-distance sequences , 1992 .

[20]  Ethan M. Coven,et al.  Sequences with minimal block growth II , 1973, Mathematical systems theory.

[21]  Jeffrey Shallit,et al.  The Ring of k-Regular Sequences , 1990, Theor. Comput. Sci..

[22]  Giuseppe Pirillo,et al.  Palindromes and Sturmian Words , 1999, Theor. Comput. Sci..

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

[24]  Alan Cobham,et al.  Uniform tag sequences , 1972, Mathematical systems theory.

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

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

[27]  Ethan M. Coven,et al.  Sequences with minimal block growth , 2005, Mathematical systems theory.

[28]  Barry Simon,et al.  Singular continuous spectrum for palindromic Schrödinger operators , 1995 .

[29]  Brigitte Mossé,et al.  Puissances de mots et reconnaissabilité des point fixes d'une substitution , 1992, Theor. Comput. Sci..

[30]  M. Queffélec,et al.  Une nouvelle propriété des suites de Rudin-Shapiro , 1987 .

[31]  M. Baake,et al.  Generalized model sets and dynamical systems , 2000 .

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

[33]  Jean Berstel,et al.  Recent Results on Sturmian Words , 1995, Developments in Language Theory.

[34]  Laurent Vuillon,et al.  Palindromes and Two-Dimensional Sturmian Sequences , 2001, J. Autom. Lang. Comb..

[35]  Jean-Paul Allouche,et al.  Generalized Rudin-Shapiro sequences , 1991 .

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

[37]  Sébastien Ferenczi,et al.  Complexity for Finite Factors of Infinite Sequences , 1999, Theor. Comput. Sci..

[38]  David Damanik,et al.  Palindrome complexity bounds for primitive substitution sequences , 2000, Discret. Math..

[39]  J.-P. Allouche,et al.  Schrödinger operators with Rudin-Shapiro potentials are not palindromic , 1997 .

[40]  Jeffrey Shallit,et al.  The ring of k-regular sequences, II , 2003, Theor. Comput. Sci..

[41]  Giuseppe Pirillo,et al.  Episturmian words and some constructions of de Luca and Rauzy , 2001, Theor. Comput. Sci..

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

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

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