scholarly journals Complexity and fractal dimensions for infinite sequences with positive entropy

2019 ◽  
Vol 21 (06) ◽  
pp. 1850068
Author(s):  
Christian Mauduit ◽  
Carlos Gustavo Moreira
Keyword(s):  
Exponential Growth ◽  
Fractal Dimensions ◽  
Combinatorial Structure ◽  
Finite Alphabet ◽  
Combinatorial Proof ◽  
Infinite Words ◽  
Entropy Formula ◽  
Complexity Function ◽  
Infinite Word ◽  
Infinite Sequences

The complexity function of an infinite word [Formula: see text] on a finite alphabet [Formula: see text] is the sequence counting, for each non-negative [Formula: see text], the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of the infinite word [Formula: see text]. The goal of this work is to estimate the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of an infinite word [Formula: see text] with a complexity function bounded by a given function [Formula: see text] with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the word entropy [Formula: see text] associated to a given function [Formula: see text] and we determine the fractal dimensions of sets of infinite sequences with complexity function bounded by [Formula: see text] in terms of its word entropy. We present a combinatorial proof of the fact that [Formula: see text] is equal to the topological entropy of the subshift of infinite words whose complexity is bounded by [Formula: see text] and we give several examples showing that even under strong conditions on [Formula: see text], the word entropy [Formula: see text] can be strictly smaller than the limiting lower exponential growth rate of [Formula: see text].

scholarly journals Entropy ratio for infinite sequences with positive entropy

2018 ◽  
Vol 40 (3) ◽  
pp. 751-762 ◽  
Author(s):  
CHRISTIAN MAUDUIT ◽  
CARLOS GUSTAVO MOREIRA
Keyword(s):  
Exponential Growth ◽  
Fractal Dimensions ◽  
Combinatorial Structure ◽  
Finite Alphabet ◽  
Exponential Growth Rate ◽  
Positive Entropy ◽  
Infinite Words ◽  
Complexity Function ◽  
Infinite Word ◽  
Infinite Sequences

The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. For any given function $f$ with exponential growth, we introduced in [Complexity and fractal dimensions for infinite sequences with positive entropy. Commun. Contemp. Math. to appear] the notion of word entropy$E_{W}(f)$ associated to $f$ and we described the combinatorial structure of sets of infinite words with a complexity function bounded by $f$. The goal of this work is to give estimates on the word entropy $E_{W}(f)$ in terms of the limiting lower exponential growth rate of $f$.

scholarly journals Defect Effect of Bi-infinite Words in the Two-element Case

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Ján Maňuch
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Alphabet ◽  
Infinite Words ◽  
Infinite Word ◽  
The Family ◽  
International Audience ◽  
Necessary And Sufficient ◽  
Primitive Roots ◽  
Element Set

International audience Let X be a two-element set of words over a finite alphabet. If a bi-infinite word possesses two X-factorizations which are not shiftequivalent, then the primitive roots of the words in X are conjugates. Note, that this is a strict sharpening of a defect theorem for bi-infinite words stated in \emphKMP. Moreover, we prove that there is at most one bi-infinite word possessing two different X-factorizations and give a necessary and sufficient conditions on X for the existence of such a word. Finally, we prove that the family of sets X for which such a word exists is parameterizable.

Generalized Hausdorff dimensions of sets of real numbers with zero entropy expansion

2011 ◽  
Vol 32 (3) ◽  
pp. 1073-1089 ◽  
Author(s):  
CHRISTIAN MAUDUIT ◽  
CARLOS GUSTAVO MOREIRA
Keyword(s):  
Continued Fraction ◽  
Negative Integer ◽  
Finite Alphabet ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Hausdorff Dimensions ◽  
Subexponential Growth ◽  
Complexity Function ◽  
Infinite Word ◽  
Zero Entropy

AbstractThe complexity function of an infinite wordwon a finite alphabetAis the sequence counting, for each non-negative integern, the number of words of lengthnon the alphabetAthat are factors of the infinite wordw. Letfbe a given function with subexponential growth. The goal of this work is to estimate the generalized Hausdorff dimensions of the set of real numbers whoseq-adic expansion has a complexity function bounded byfand the set of real numbers whose continued fraction expansion is bounded byqand has a complexity function bounded byf.

A NEW COMPLEXITY FUNCTION FOR WORDS BASED ON PERIODICITY

2013 ◽  
Vol 23 (04) ◽  
pp. 963-987 ◽  
Author(s):  
FILIPPO MIGNOSI ◽  
ANTONIO RESTIVO
Keyword(s):  
The Other ◽  
Factorization Theorem ◽  
Complexity Measures ◽  
Infinite Words ◽  
Average Value ◽  
Complexity Function ◽  
Binary Word ◽  
Infinite Word

Motivated by the extension of the critical factorization theorem to infinite words, we study the (local) periodicity function, i.e. the function that, for any position in a word, gives the size of the shortest square centered in that position. We prove that this function characterizes any binary word up to exchange of letters. We then introduce a new complexity function for words (the periodicity complexity) that, for any position in the word, gives the average value of the periodicity function up to that position. The new complexity function is independent from the other commonly used complexity measures as, for instance, the factor complexity. Indeed, whereas any infinite word with bounded factor complexity is periodic, we will show a recurrent non-periodic word with bounded periodicity complexity. Further, we will prove that the periodicity complexity function grows as Θ( log n) in the case of the Fibonacci infinite word and that it grows as Θ(n) in the case of the Thue–Morse word. Finally, we will show examples of infinite recurrent words with arbitrary high periodicity complexity.

scholarly journals Towards sharp Bohnenblust–Hille constants

2018 ◽  
Vol 20 (03) ◽  
pp. 1750029 ◽  
Author(s):  
Daniel Pellegrino ◽  
Eduardo V. Teixeira
Keyword(s):  
Exponential Growth ◽  
Optimization Problems ◽  
Best Constants ◽  
Sharp Constants ◽  
Linear Forms ◽  
Entropy Formula ◽  
Optimal Constants

We investigate the optimality problem associated with the best constants in a class of Bohnenblust–Hille-type inequalities for [Formula: see text]-linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust–Hille inequality are universally bounded, irrespectively of the value of [Formula: see text]; hereafter referred as the Universality Conjecture. In our approach, we introduce the notions of entropy and complexity, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to [Formula: see text], then the optimal constants of the [Formula: see text]-linear Bohnenblust–Hille inequality for real scalars are indeed bounded universally with respect to [Formula: see text]. It is likely that indeed the entropy grows as [Formula: see text], and in this scenario, we show that the optimal constants are precisely [Formula: see text]. In the bilinear case, [Formula: see text], we show that any extremum of the Littlewood’s [Formula: see text] inequality has entropy [Formula: see text] and complexity [Formula: see text], and thus we are able to classify all extrema of the problem. We also prove that, for any mixed [Formula: see text]-Littlewood inequality, the entropy do grow exponentially and the sharp constants for such a class of inequalities are precisely [Formula: see text]. In addition to the notions of entropy and complexity, the approach we develop in this work makes decisive use of a family of strongly non-symmetric [Formula: see text]-linear forms, which has further consequences to the theory, as we explain herein.

scholarly journals THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY

2015 ◽  
Vol 3 ◽  
Author(s):  
VAN CYR ◽  
BRYNA KRA
Keyword(s):  
Automorphism Group ◽  
Finite Groups ◽  
Linear Growth ◽  
Finite Alphabet ◽  
Finitely Generated ◽  
Complexity Function ◽  
Algebraic Properties

For a finite alphabet ${\mathcal{A}}$ and shift $X\subseteq {\mathcal{A}}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group $\text{Aut}(X)$. For such systems, we show that every finitely generated subgroup of $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if $X$ is transitive, then $\text{Aut}(X)$ is virtually $\mathbb{Z}$; if $X$ has dense aperiodic points, then $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$. We also classify all finite groups that arise as the automorphism group of a shift.

scholarly journals The infinite word problem and limit sets in Fuchsian groups

1981 ◽  
Vol 1 (3) ◽  
pp. 337-360 ◽  
Author(s):  
Caroline Series
Keyword(s):  
Gibbs Measures ◽  
Fuchsian Groups ◽  
Countable Number ◽  
Exponent Of Convergence ◽  
Limit Sets ◽  
Infinite Word ◽  
Dimensional Measure ◽  
Fixed Set ◽  
Infinite Sequences ◽  
Natural Way

AbstractLet Г be a finitely generated non-elementary Fuchsian group acting in the disk. With the exception of a small number of co-compact Г, we give a representation of g ∈ Г as a product of a fixed set of generators Гo in a unique shortest ‘admissible form’. Words in this form satisfy rules which after a suitable coding are of finite type. The space of infinite sequences Σ of generators satisfying the same rules is identified in a natural way with the limit set Λ of Г by a map which is bijective except at a countable number of points where it is two to one. We use the theory of Gibbs measures onΣ to construct the so-called Patterson measure on Λ [8], [9]. This measure is, in fact, Hausdorff 5-dimensional measure on Λ, where S is the exponent of convergence of Г.

scholarly journals Lyndon factorization of generalized words of Thue

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Anton Černý
Keyword(s):  
Infinite Words ◽  
Infinite Word ◽  
International Audience

International audience The i-th symbol of the well-known infinite word of Thue on the alphabet \ 0,1\ can be characterized as the parity of the number of occurrences of the digit 1 in the binary notation of i. Generalized words of Thue are based on counting the parity of occurrences of an arbitrary word w∈\ 0,1\^+-0^* in the binary notation of i. We provide here the standard Lyndon factorization of some subclasses of this class of infinite words.

scholarly journals The Overlap Gap Between Left-Infinite and Right-Infinite Words

2021 ◽  
pp. 1-9
Author(s):  
José Carlos Costa ◽  
Conceição Nogueira ◽  
Maria Lurdes Teixeira
Keyword(s):  
Infinite Words ◽  
Infinite Word ◽  
Positive Integers

We study ultimate periodicity properties related to overlaps between the suffixes of a left-infinite word [Formula: see text] and the prefixes of a right-infinite word [Formula: see text]. The main theorem states that the set of minimum lengths of words [Formula: see text] and [Formula: see text] such that [Formula: see text] or [Formula: see text] is finite, where [Formula: see text] runs over positive integers and [Formula: see text] and [Formula: see text] are respectively the suffix of [Formula: see text] and the prefix of [Formula: see text] of length [Formula: see text], if and only if [Formula: see text] and [Formula: see text] are ultimately periodic words of the form [Formula: see text] and [Formula: see text] for some finite words [Formula: see text], [Formula: see text] and [Formula: see text].

The Markoff Property

2018 ◽  
pp. 29-34
Author(s):  
Christophe Reutenauer
Keyword(s):  
Combinatorial Property ◽  
Periodic Pattern ◽  
Infinite Words ◽  
Infinite Word ◽  
Markoff Property

The Markoff property is a combinatorial property of infinite words on the alphabet {a,b}, and of bi-infinite words. Such a word has this property if whenever there is a factor xy in the word,with x,y equal to the letters a,b (in some order), then itmay be extended into a factor of the formym’xymx, wherem’ is the reversal ofm, and where the length ofmis bounded (the bound depends only on the infinite word). As discussed in this chapter, the main theorem, due toMarkoff, is that this property implies periodicity, with a periodic pattern which must be a Christoffel word. It is one of the crucial results inMarkoff’s theory.

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