论文标题
广义的Koch曲线和Thue-Morse序列
Generalized Koch curves and Thue-Morse sequences
论文作者
论文摘要
令$（t_n）_ {n \ ge0} $成为井Konwn $ \ pm1 $ thue-morse sequence $$+1，-1，-1，-1，-1，-1，-1，-1，+1，+1，-1，-1，-1，-1，-1，-1，\cdots。$ $ $ \ sum_ {k <n} t_ke^\ frac {2kπi} {3} $与著名的科赫曲线密切相关。作为自然概括，对于整数$ m \ ge1 $，我们使用$ \ sum_ {k <n}δ_ke^\ frac {2kπi} {m} {m} {m} {m $$+1 \ mapsto+1，+δ_1，\ cdots，+δ_m$$ $$ $$ - 1 \ mapsto-1，-Δ_1，\ cdots，-Δ_m$$以$δ_0=+1 $ =+1 $和$Δ_1，$δ_1，\ cdots，\ cdots，Δ_m\ in \ in \ in \ cl {相应的迭代功能系统。对于$ m \ ge2 $，$δ_0= \ cdots =δ_ {\ lfloor \ frac {m} {4} {4} \ rfloor} =+1 $， $δ_ {\ lfloor \ frac {m} {4} \ rfloor+1} = \ cdots =δ__{m- \ lfloor \ frac {m} {4} {4} \ rfloor-rfloor-rfloor-rfloor-rfloor-n-1} = - 1 $ and and $Δ_{m- \ lfloor \ frac {m} {4} \ rfloor} = \ cdots =Δ_m=+1 $，开放式条件保持，然后相应的广义Koch Curve具有Hausdorff，包装和盒子维度 $\log(m+1)/\log|\sum_{k=0}^mδ_ke^{\frac{2kπi}{m}}|$, where taking $m=3$ and then $δ_0=+1,δ_1=δ_2=-1,δ_3=+1$ will recover the result on the classical Koch curve.
Let $(t_n)_{n\ge0}$ be the well konwn $\pm1$ Thue-Morse sequence $$+1,-1,-1,+1,-1,+1,+1,-1,\cdots.$$ Since the 1982-1983 work of Coquet and Dekking, it is known that $\sum_{k<n}t_ke^\frac{2kπi}{3}$ is strongly related to the famous Koch curve. As a natural generalization, for integer $m\ge1$, we use $\sum_{k<n}δ_ke^\frac{2kπi}{m}$ to define generalized Koch curve, where $(δ_n)_{n\ge0}$ is the generalized Thue-Morse sequence defined to be the unique fixed point of the morphism $$+1\mapsto+1,+δ_1,\cdots,+δ_m$$ $$-1\mapsto-1,-δ_1,\cdots,-δ_m$$ beginning with $δ_0=+1$ and $δ_1,\cdots,δ_m\in\{+1,-1\}$, and we prove that generalized Koch curves are the attractors of corresponding iterated function systems. For the case that $m\ge2$, $δ_0=\cdots=δ_{\lfloor\frac{m}{4}\rfloor}=+1$, $δ_{\lfloor\frac{m}{4}\rfloor+1}=\cdots=δ_{m-\lfloor\frac{m}{4}\rfloor-1}=-1$ and $δ_{m-\lfloor\frac{m}{4}\rfloor}=\cdots=δ_m=+1$, the open set condition holds, and then the corresponding generalized Koch curve has Hausdorff, packing and box dimension $\log(m+1)/\log|\sum_{k=0}^mδ_ke^{\frac{2kπi}{m}}|$, where taking $m=3$ and then $δ_0=+1,δ_1=δ_2=-1,δ_3=+1$ will recover the result on the classical Koch curve.
