Let (tn)n≥0 be the well-known ± 1 Thue–Morse sequence +1,−1,−1, +1,−1, +1, +1,−1,…. Since the 1982–1983 work of Coquet and Dekking, it is known that ∑k<ntke2kπi 3 is strongly related to the famous Koch curve. As a natural generalization, for m ∈ ℕ, we use ∑k<nδke2kπi m to define the generalized Koch curve, where (δn)n≥0 is the generalized Thue–Morse sequence defined to be the unique fixed point of the morphism +1↦ + 1, +δ1,…, +δm, −1↦ − 1,−δ1,…,−δm beginning with δ0 = +1 and δ1,…,δm ∈{+1,−1}, and we prove that generalized Koch curves are the attractors of the corresponding iterated function systems. For the case that m ≥ 2, δ0 = ⋯ = δ⌊m 4 ⌋ = +1, δ⌊m 4 ⌋+1 = ⋯ = δm−⌊m 4 ⌋−1 = −1 and δm−⌊m 4 ⌋ = ⋯ = δm = +1, the open set condition holds, and then the corresponding generalized Koch curve has Hausdorff, packing and box dimension log(m + 1)/log |∑k=0mδ ke2kπi m|, where taking m = 3 and then δ0 = +1,δ1 = δ2 = −1,δ3 = +1 will recover the result on the classical Koch curve.

中文翻译：

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广义柯赫曲线和 Thue-Morse 序列

让(吨n)n≥0出名 ± 1Thue-Morse 序列 +1,-1,-1, +1,-1, +1, +1,-1,…. 自从 Coquet 和 Dekking 1982-1983 年的工作以来，众​​所周知，∑ķ<n吨ķe2ķπ一世 3与著名的科赫曲线密切相关。作为一个自然的概括，对于米 ∈ ℕ， 我们用∑ķ<nδķe2ķπ一世 米定义广义科赫曲线，其中(δn)n≥0是广义 Thue-Morse 序列，定义为态射的唯一不动点 +1↦ + 1, +δ1,…, +δ米, -1↦ - 1,-δ1,…,-δ米 以。。。开始δ0 = +1和δ1,…,δ米 ∈{+1,-1}，我们证明了广义科赫曲线是相应迭代函数系统的吸引子。对于这种情况米 ≥ 2,δ0 = ⋯ = δ⌊米 4 ⌋ = +1,δ⌊米 4 ⌋+1 = ⋯ = δ米-⌊米 4 ⌋-1 = -1和δ米-⌊米 4 ⌋ = ⋯ = δ米 = +1, 开集条件成立, 则对应的广义 Koch 曲线有 Hausdorff、packing 和 box 维数日志(米 + 1)/日志 |∑ķ=0米δ ķe2ķπ一世 米|, 在哪里取米 = 3然后δ0 = +1,δ1 = δ2 = -1,δ3 = +1将恢复经典科赫曲线上的结果。