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- W2170830187 abstract "Abstract Let β ∈(1,2). Each x ∈[0,1/( β −1)] can be represented in the form where ε k ∈{0,1} for all k (a β -expansion of x ). If $beta >{(1+sqrt 5)}/{2}$ , then, as is well known, there always exist x ∈(0,1/( β −1)) which have a unique β -expansion. We study (purely) periodic unique β -expansions and show that for each n ≥2 there exists $beta _nin [{(1+sqrt 5)}/{2},2)$ such that there are no unique periodic β -expansions of smallest period n for β ≤ β n and at least one such expansion for β > β n . Furthermore, we prove that β k < β m if and only if k is less than m in the sense of the Sharkovskiĭ ordering. We give two proofs of this result, one of which is independent, and the other one links it to the dynamics of a family of trapezoidal maps." @default.
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- W2170830187 date "2009-08-01" @default.
- W2170830187 modified "2023-09-25" @default.
- W2170830187 title "Periodic unique beta-expansions: the Sharkovskiĭ ordering" @default.
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- W2170830187 doi "https://doi.org/10.1017/s0143385708000746" @default.
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