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Palindrome complexity
 To appear, Theoret. Comput. Sci
, 2002
"... 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 o ..."
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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 HofKnillSimon. We also give an upper bound for the palindrome complexity of a sequence in terms of its (block)complexity. 1
OverlapFree Symmetric D0L words
, 2001
"... Introduction In his classical 1912 paper [15] (see also [3]), A. Thue gave the first example of an overlapfree infinite word, i. e., of a word which contains no subword of the form axaxa for any symbol a and word x. Thue's example is known now as the ThueMorse word w TM = 011010011001011010010110 ..."
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Cited by 4 (0 self)
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Introduction In his classical 1912 paper [15] (see also [3]), A. Thue gave the first example of an overlapfree infinite word, i. e., of a word which contains no subword of the form axaxa for any symbol a and word x. Thue's example is known now as the ThueMorse word w TM = 01101001100101101001011001101001 : : :: It was rediscovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey [1]). The set of all overlapfree words was studied e. g. by Fife [8] who described all binary overlapfree infinite words and Seebold [13] who proved that the ThueMorse word is essentially the only binary overlapfree word which is a fixed point of a morphism. Nowadays the theory of overlapfree words is a part of a more general theory of pattern avoidance [5]. J.P. Allouche and J. Shallit [2] asked if the initial Thue's construction of an overlapfree wo
Counting ordered patterns in words generated by morphisms
, 2007
"... We start a general study of counting the number of occurrences of ordered patterns in words generated by morphisms. We consider certain patterns with gaps (classical patterns) and that with no gaps (consecutive patterns). Occurrences of the patterns are known, in the literature, as rises, descents, ..."
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Cited by 2 (1 self)
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We start a general study of counting the number of occurrences of ordered patterns in words generated by morphisms. We consider certain patterns with gaps (classical patterns) and that with no gaps (consecutive patterns). Occurrences of the patterns are known, in the literature, as rises, descents, (non)inversions, squares and prepetitions. We give recurrence formulas in the general case, then deducing exact formulas for particular families of morphisms. Many (classical or new) examples are given illustrating the techniques and showing their interest.
Transcendence of certain kary continued fraction expansions
, 2005
"... Let ξ ∈ (0, 1) be an irrational with aperiodic continued fraction expansion: ξ = [0; u0, u1, u2,...], and suppose the sequence (un)n≥0 of partial quotients takes only values from the finite set {a1, a2,..., ak} with 1 ≤ a1 < a2 < · · · < ak, k ≥ 2. We prove that if the frequency of a1 (or ak) in ( ..."
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Let ξ ∈ (0, 1) be an irrational with aperiodic continued fraction expansion: ξ = [0; u0, u1, u2,...], and suppose the sequence (un)n≥0 of partial quotients takes only values from the finite set {a1, a2,..., ak} with 1 ≤ a1 < a2 < · · · < ak, k ≥ 2. We prove that if the frequency of a1 (or ak) in (un)n≥0 is at least 1/2, and (un)n≥0 begins with arbitrarily long blocks that are almost squares, then ξ is transcendental. This extends a result of Allouche et al (2001), who studied the binary case k = 2. Consideration is also given to some examples of such transcendental continued fractions [0; u0, u1, u2,...], including the case when (un)n≥0 is an aperiodic strict standard episturmian word, or a certain generalised ThueMorse word. Keywords: transcendence; continued fraction; episturmian word; ThueMorse word; morphism. MSC (2000): primary 11J81; secondary 68R15, 11B50. 1