Sequences with constant number of return words

Balkova, Lubomira; Pelantová, Edita; Steiner, Wolfgang

doi:10.1007/s00605-008-0001-2

Cited by 30 publications

(42 citation statements)

References 15 publications

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“…easily checked that μ(a 2 ) = 1 and μ(a n ) = 2 for n ≥ 3. The word u = 00101100 is palindromically generated by S = { (1,2), (2,4), (3,5), (4,7), (7,8)}. One can check that S is the smallest palindromic generating set whence μ(00101100) = 5.…”

Section: Introductionmentioning

confidence: 99%

“…We conclude by remarking that it is unlikely that there is a direct link between the quantity ψ(x) and the number of return words in x, or the palindromic complexity of x. For instance, it is known that each factor of the Thue-Morse word T has either 3 or 4 return words (see for instance [2]), and that T contains at most 4 distinct palindromes of each given length n (see [5]). On the other hand, following Proposition 2.8, the number of palindromic relations for the Thue-Morse word is unbounded.…”

Section: Introductionmentioning

confidence: 99%

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On generating binary words palindromically

Harju¹,

Huova²,

Zamboni³

2015

Journal of Combinatorial Theory, Series A

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We regard a finite word u = u 1 u 2 · · · u n up to word isomorphism as an equivalence relation on {1, 2, . . . , n} where i is equivalent to j if and only if u i = u j . Some finite words (in particular all binary words) are generated by palindromic relations of the form k ∼ j + i − k for some choice of 1 ≤ i ≤ j ≤ n and k ∈ {i, i +1, . . . , j}. That is to say, some finite words u are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. In this paper we study the function μ(u) defined as the least number of palindromic relations required to generate u. We show that if x is an infinite word such that μ(u) ≤ 2 for each factor u of x, then x is ultimately periodic. On the other hand, we establish the existence of non-ultimately periodic words for which μ(u) ≤ 3 for each factor u of x, and obtain a complete classification of such words on a binary alphabet (which includes the well known class of Sturmian words). In contrast, for the Thue-Morse word, we show that the function μ is unbounded.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On generating binary words palindromically

Harju¹,

Huova²,

Zamboni³

2015

Journal of Combinatorial Theory, Series A

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“…The return words in the sequences coding the rotations were studied in [10] in particular for palindromic factors. To compute the exact number of return words to a factor of a given Rote sequence, we use the following results from [4] (Lemmas 4.2 and 4.4):…”

Section: Return Words To Prefixes Of Complementary Symmetric Rote Seqmentioning

confidence: 99%

“…The third author characterizes Sturmian sequences as sequences with two return words to each their factor in [31]. Similarly the paper [4] is dedicated to investigation of sequences with a fixed number of return words to any factors, in particular, Arnoux-Rauzy sequences and sequences coding interval exchange transformations are of this type, see [32]. Besides, return words in episturmian sequences were described in [21] while the description of return words in the coding of rotations was used to show their fullness in [10].…”

Section: Introductionmentioning

confidence: 99%

Derived sequences of complementary symmetric Rote sequences

Medková

Pelantová

Vuillon

2019

RAIRO-Theor. Inf. Appl.

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Complementary symmetric Rote sequences are binary sequences which have factor complexity C(n) = 2n for all integers n ≥ 1 and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derivated sequences to the prefixes of any complementary symmetric Rote sequence v which is associated with a standard Sturmian sequence u. We show that any non-empty prefix of v has three return words. We prove that any derivated sequence of v is coding of three interval exchange transformation and we determine the parameters of this transformation. We also prove that v is primitive substitutive if and only if u is primitive substitutive. Moreover, if the sequence u is a fixed point of a primitive morphism, then all derivated sequences of v are also fixed by primitive morphisms. In that case we provide an algorithm for finding these fixing morphisms.

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“…3 2 What we call return words is sometimes in the literature designated first return words, as is the case of the article [13], which is cited later in this paper. The terminology that we adopt appears for instance in [20,21,12].…”

Section: Return Words In the Study Of G(x ) In The Minimal Casementioning

confidence: 99%

A geometric interpretation of the Schützenberger group of a minimal subshift

Almeida

Costa

2016

Ark. Mat.

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The first author has associated in a natural way a profinite group to each irreducible subshift. The group in question was initially obtained as a maximal subgroup of a free profinite semigroup. In the case of minimal subshifts, the same group is shown in the present paper to also arise from geometric considerations involving the Rauzy graphs of the subshift. Indeed, the group is shown to be isomorphic to the inverse limit of the profinite completions of the fundamental groups of the Rauzy graphs of the subshift. A further result involving geometric arguments on Rauzy graphs is a criterion for freeness of the profinite group of a minimal subshift based on the Return Theorem of Berthé et. al.

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Sequences with constant number of return words

Cited by 30 publications

References 15 publications

On generating binary words palindromically

On generating binary words palindromically

Derived sequences of complementary symmetric Rote sequences

A geometric interpretation of the Schützenberger group of a minimal subshift

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