Periodicity problem of substitutions over ternary alphabets
In this paper, we characterize the substitutions over a three-letter alphabet which generate a ultimately periodic sequence.
Univoque sets for real numbers
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 . We prove that for any x ∈ (0,1), (x) contains a sequence increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure are nowhere dense.
Periodicity Problem of Substitutions over Ternary Alphabets
In this paper, we characterize the substitutions over a three-letter alphabet which generate a ultimately periodic sequence.
Substitutions with Cofinal Fixed Points
Let be a substitution over a 2-letter alphabet, say . If and begin with and respectively, has two fixed points beginning with and 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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