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Periodicity problem of substitutions over ternary alphabets

Bo Tan; Zhi-Ying Wen — 2008

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

In this paper, we characterize the substitutions over a three-letter alphabet which generate a ultimately periodic sequence.

Univoque sets for real numbers

Fan Lü; Bo Tan; Jun Wu — 2014

Fundamenta Mathematicae

For x ∈ (0,1), the univoque set for x, denoted (x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with $x_{i} \in 0, 1$ . We prove that for any x ∈ (0,1), (x) contains a sequence ${β_{k}}_{k \geq 1}$ increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure $\bar{(x)}$ are nowhere dense.

Periodicity Problem of Substitutions over Ternary Alphabets

Bo Tan; Zhi-Ying Wen — 2010

RAIRO - Theoretical Informatics and Applications

In this paper, we characterize the substitutions over a three-letter alphabet which generate a ultimately periodic sequence.

Substitutions with Cofinal Fixed Points

Bo TAN; Zhi-Xiong WEN; Jun WU; Zhi-Ying WEN — 2006

Annales de l’institut Fourier

Let $ϕ$ be a substitution over a 2-letter alphabet, say ${a, b}$ . If $ϕ (a)$ and $ϕ (b)$ begin with $a$ and $b$ respectively, $ϕ$ has two fixed points beginning with $a$ and $b$ respectively. We characterize substitutions with two cofinal fixed points (i.e., which differ only by prefixes). The proof is a combinatorial one, based on the study of repetitions of words in the fixed points.

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