344 research outputs found
The Critical Exponent is Computable for Automatic Sequences
The critical exponent of an infinite word is defined to be the supremum of
the exponent of each of its factors. For k-automatic sequences, we show that
this critical exponent is always either a rational number or infinite, and its
value is computable. Our results also apply to variants of the critical
exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni
and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes
or recovers previous results of Krieger and others, and is applicable to other
situations; e.g., the computation of the optimal recurrence constant for a
linearly recurrent k-automatic sequence.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Critical Exponents and Stabilizers of Infinite Words
This thesis concerns infinite words over finite alphabets. It contributes to two topics in this area: critical exponents and stabilizers.
Let w be a right-infinite word defined over a finite alphabet. The critical exponent of w is the supremum of the set of exponents r such that w contains an r-power as a subword. Most of the thesis (Chapters 3 through 7) is devoted to critical exponents.
Chapter 3 is a survey of previous research on critical exponents and repetitions in morphic words. In Chapter 4 we prove that every real number greater than 1 is the critical exponent of some right-infinite word over some finite alphabet. Our proof is constructive. In Chapter 5 we characterize critical exponents of pure morphic words generated by uniform binary morphisms. We also give an explicit formula to compute these critical exponents, based on a well-defined prefix of the infinite word. In Chapter 6 we generalize our results to pure morphic words generated by non-erasing morphisms over any finite alphabet. We prove that critical exponents of such words are algebraic, of a degree bounded by the alphabet size. Under certain conditions, our proof implies an algorithm for computing the critical exponent. We demonstrate our method by computing the critical exponent of some families of infinite words. In particular, in Chapter 7 we
compute the critical exponent of the Arshon word of order n for n ≥ 3.
The stabilizer of an infinite word w defined over a finite alphabet Σ is the set of morphisms f: Σ*→Σ* that fix w. In Chapter 8 we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements.
We conclude with a list of open problems, including a new problem that has not been addressed yet: the D0L repetition threshold
Periodicity, repetitions, and orbits of an automatic sequence
We revisit a technique of S. Lehr on automata and use it to prove old and new
results in a simple way. We give a very simple proof of the 1986 theorem of
Honkala that it is decidable whether a given k-automatic sequence is ultimately
periodic. We prove that it is decidable whether a given k-automatic sequence is
overlap-free (or squareefree, or cubefree, etc.) We prove that the
lexicographically least sequence in the orbit closure of a k-automatic sequence
is k-automatic, and use this last result to show that several related
quantities, such as the critical exponent, irrationality measure, and
recurrence quotient for Sturmian words with slope alpha, have automatic
continued fraction expansions if alpha does.Comment: preliminary versio
The repetition threshold of episturmian sequences
The repetition threshold of a class of infinite -ary sequences is the
smallest real number such that in the class there exists a sequence
that avoids -powers for all . This notion was introduced by Dejean in
1972 for the class of all sequences over a -letter alphabet. Thanks to the
effort of many authors over more than 30 years, the precise value of the
repetition threshold in this class is known for every . The
repetition threshold for the class of Sturmian sequences was determined by
Carpi and de Luca in 2000. Sturmian sequences may be equivalently defined in
various ways, therefore there exist many generalizations to larger alphabets.
Rampersad, Shallit and Vandome in 2020 initiated a study of the repetition
threshold for the class of balanced sequences -- one of the possible
generalizations of Sturmian sequences. Here, we focus on the class of -ary
episturmian sequences -- another generalization of Sturmian sequences
introduced by Droubay, Justin and Pirillo in 2001. We show that the repetition
threshold of this class is reached by the -bonacci sequence and its value
equals , where is the unique positive root of the
polynomial
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