Sturmian words, β-shifts, and transcendence

Pyo, Dong; Kwon, DoYong

doi:10.1016/j.tcs.2004.03.035

Cited by 16 publications

(26 citation statements)

References 15 publications

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“…Explicit natural examples of transcendental numbers in C 3 were given by Allouche and Cosnard [5], and by Chi and Kwon [16]. Both results are a consequence of the so-called Mahler's method introduced in [27].…”

Section: Blanchard's Classification Of β-Shifts and Transcendental Numentioning

confidence: 93%

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Dynamics forβ-shifts and Diophantine approximation

Adamczewski

Bugeaud²

2007

Ergod. Th. Dynam. Sys.

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Section: Blanchard's Classification Of β-Shifts and Transcendental Numentioning

confidence: 93%

“…This expansion and related objects have been studied for many reasons and in many areas, including ergodic theory and symbolic dynamics [10,22,30,32,33], tilings [9,36], quasi-crystals and mathematical physics [28,29], number theory [5,10,16,34,37], and formal languages and theoretical computer science [26,Ch. 7].…”

Section: Introductionmentioning

confidence: 99%

Dynamics forβ-shifts and Diophantine approximation

Adamczewski

Bugeaud²

2007

Ergod. Th. Dynam. Sys.

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“…-Self-Sturmian numbers correspond to Sturmian sequences of the form u = 1v, where v is any characteristic Sturmian sequence (see [9,Remark p. 399]). All self-Sturmian numbers are transcendental [9].…”

Section: Propositionmentioning

confidence: 99%

A note on univoque self-Sturmian numbers

Allouche

2008

RAIRO-Theor. Inf. Appl.

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We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of a unimodal continuous map from the unit interval into itself, but it also characterizes univoque real numbers; the other is an equivalent definition of characteristic Sturmian sequences. As a corollary to our study we obtain that a real number β in (1, 2) is univoque and self-Sturmian if and only if the β-expansion of 1 is of the form 1v, where v is a characteristic Sturmian sequence beginning itself in 1.

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“…Together with Parry's result, Proposition 2.2 guarantees the existence of a unique β > 1 such that d β (1) = 1z p,q 1. This fact and [9] enable us to define the next function. …”

Section: Introductionmentioning

confidence: 99%

“…In [9], β > 1 is called a self-Sturmian number if d β (1) is a Sturmian word. Now we consider its counterpart for Christoffel words.…”

Section: Introductionmentioning

confidence: 99%

A devil's staircase from rotations and irrationality measures for Liouville numbers

Kwon

2008

Math. Proc. Camb. Phil. Soc.

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From Sturmian and Christoffel words we derive a strictly increasing function ∆ : [0, ∞) → R. This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous. Moreover, it assumes algebraic integers at rationals, and transcendental numbers at irrationals. We also see that the differentiation of ∆ distinguishes some irrationality measures of real numbers. IntroductionFor any real number α ∈ [0, 1], we consider the dynamics of rotationUsing an alphabet A = {a, b} with some integers 0 ≤ a < b, the itinerary (R n α (ρ)) n≥0 of some ρ is recorded according to a partitionIn other words, if ρ ∈ [0, 1] has traveled over P, then its infinite history (s α,ρ (n)) n≥0 of rotations is determined by the following rule:In case that we adopt the partition P ′ , a sequence (s ′ α,ρ (n)) n≥0 is also defined in a similar fashion. One can observe that if A = {0, 1}, then these infinite words are obtained by the formulaewhere ⌊t⌋ is the largest integer not greater than t, and ⌈t⌉ is the smallest integer not less than t. The infinite words s α,ρ , s ′ α,ρ are called lower and upper * Keywords: devil's staircase, Sturmian word, Christoffel word, irrationality measure, Liouville number.

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Sturmian words, β-shifts, and transcendence

Cited by 16 publications

References 15 publications

Dynamics forβ-shifts and Diophantine approximation

Dynamics forβ-shifts and Diophantine approximation

A note on univoque self-Sturmian numbers

A devil's staircase from rotations and irrationality measures for Liouville numbers

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