Periodic unique beta-expansions: the Sharkovskii ordering

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Let β∈(1,2). Each x∈[0,1/β-1] can be represented in the form x=∑ₖ₌₁∞ εₖβ⁻ᵏ, where εₖ∈{0,1} for all k (a β-expansion of x). If β>1+√5/2, then, as is well known, there always exist x∈(0,1β-1) which have a unique -expansion. In the present paper we study (purely) periodic unique β-expansions and show that for each n≥2 there exists βₙ∈[1+√5/2,2) such that there are no unique periodic β-expansions of smallest period n for β≤βₙ and at least one such expansion for β>βₙ. Furthermore, we prove that βₖ<βₘ if and only if k is less than m in the sense of the Sharkovskiı 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.