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ON THE STRUCTURE OF (−β)INTEGERS
"... Abstract. The (−β)integers are natural generalisations of the βintegers, and thus of the integers, for negative real bases. When β is the analogue of a Parry number, we describe the structure of the set of (−β)integers by a fixed point of an antimorphism. 1. ..."
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Abstract. The (−β)integers are natural generalisations of the βintegers, and thus of the integers, for negative real bases. When β is the analogue of a Parry number, we describe the structure of the set of (−β)integers by a fixed point of an antimorphism. 1.
E.: Relation between powers of factors and recurrence function characterizing Sturmian words, submitted, arXiv 0809.0603v2[math.CO
"... In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slop ..."
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In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved in [7] and [9]. 1
Factor complexity of infinite words associated with nonsimple Parry
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Palindromes in infinite ternary words
, 2009
"... We study infinite words u over an alphabet A satisfying the property P: P(n) + P(n + 1) = 1 + #A for any n ∈ N, where P(n) denotes the number of palindromic factors of length n occurring in the language of u. We study also infinite words satisfying a stronger property PE: every palindrome of u has ..."
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We study infinite words u over an alphabet A satisfying the property P: P(n) + P(n + 1) = 1 + #A for any n ∈ N, where P(n) denotes the number of palindromic factors of length n occurring in the language of u. We study also infinite words satisfying a stronger property PE: every palindrome of u has exactly one palindromic extension in u. For binary words, the properties P and PE coincide and these properties characterize Sturmian words, i.e., words with the complexity C(n) = n+1 for any n ∈ N. In this paper, we focus on ternary infinite words with the language closed under reversal. For such words u, we prove that if C(n) = 2n + 1 for any n ∈ N then u satisfies the property P and moreover u is rich in palindromes. Also a sufficient condition for the property PE is given. We construct a word demonstrating that P on a ternary alphabet does not imply PE. 1
PRESENTATIONS OF SCHÜTZENBERGER GROUPS OF MINIMAL SUBSHIFTS
"... Abstract. In previous work, the first author established a natural bijection between minimal subshifts and maximal regularJclasses of free profinite semigroups. In this paper, the Schützenberger groups of such Jclasses are investigated in particular in respect to a conjecture proposed by the first ..."
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Abstract. In previous work, the first author established a natural bijection between minimal subshifts and maximal regularJclasses of free profinite semigroups. In this paper, the Schützenberger groups of such Jclasses are investigated in particular in respect to a conjecture proposed by the first author concerning their profinite presentation. The conjecture is established for several types of minimal subshifts associated with substitutions. The Schützenberger subgroup of the Jclass corresponding to the ProuhetThueMorse subshift is shown to admit a somewhat simpler presentation, from which it follows that it satisfies the conjecture, that it has rank three, and that it is nonfree relatively to any pseudovariety of groups. 1.
Preprint Number 15–35 A GEOMETRIC INTERPRETATION OF THE SCHÜTZENBERGER GROUP OF A MINIMAL SUBSHIFT
"... Abstract: 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 fr ..."
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Abstract: 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 Berthe ́ et. al.
Preprint Number 10–01 PRESENTATIONS OF SCHÜTZENBERGER GROUPS OF MINIMAL SUBSHIFTS
"... Abstract: In previous work, the first author established a natural bijection between minimal subshifts and maximal regular Jclasses of free profinite semigroups. In this paper, the Schützenberger groups of such Jclasses are investigated in particular in respect to a conjecture proposed by the f ..."
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Abstract: In previous work, the first author established a natural bijection between minimal subshifts and maximal regular Jclasses of free profinite semigroups. In this paper, the Schützenberger groups of such Jclasses are investigated in particular in respect to a conjecture proposed by the first author concerning their profinite presentation. The conjecture is established for several types of minimal subshifts associated with substitutions. The Schützenberger subgroup of the Jclass corresponding to the ProuhetThueMorse subshift is shown to admit a somewhat simpler presentation, from which it follows that it satisfies the conjecture, that it has rank three, and that it is nonfree relatively to any pseudovariety of groups.