On the points without universal expansions

الأصل · EN

Let 1<β<2. Given any x∈[0, (β-1)⁻¹], a sequence (aₙ)∈{0,1}ⁿ is called a β-expansion of x if x=∑ₙ₌₁∞aₙβ⁻ⁿ. For any k≥ 1 and any (b₁b₂ bₖ)∈{0,1}ᵏ, if there exists some k₀ such that aₖ₀₊₁aₖ₀₊₂ aₖ₀₊ₖ=b₁b₂ bₖ, then we call (aₙ) a universal β-expansion of x. Sidorov Sidorov2003, Dajani and de Vries DajaniDeVrie proved that given any 1<β<2, then Lebesgue almost every point has uncountably many universal expansions. In this paper we consider the set Vβ of points without universal expansions. For any n≥ 2, let βₙ be the n-bonacci number satisfying the following equation: βⁿ=βⁿ⁻¹+βⁿ⁻²+ +β+1. Then we have ₕ(Vᵦₙ)=1, where ₕ denotes the Hausdorff dimension. Similar results are still available for some other algebraic numbers. As a corollary, we give some results of the Hausdorff dimension of the survivor set generated by some open dynamical systems. This note is another application of our paper KarmaKan.