Generalized Koch curves and Thue-Morse sequences

Abstract

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 $\sum_{k<n}t_ke^\frac{2k\pi i}{3}$ is strongly related to the famous Koch curve. As a natural generalization, for integer $m\ge1$, we use $\sum_{k<n}\delta_ke^\frac{2k\pi i}{m}$ to define generalized Koch curve, where $(\delta_n)_{n\ge0}$ is the generalized Thue-Morse sequence defined to be the unique fixed point of the morphism $$+1\mapsto+1,+\delta_1,\cdots,+\delta_m$$ $$-1\mapsto-1,-\delta_1,\cdots,-\delta_m$$ beginning with $\delta_0=+1$ and $\delta_1,\cdots,\delta_m\in\{+1,-1\}$, and we prove that generalized Koch curves are the attractors of corresponding iterated function systems. For the case that $m\ge2$, $\delta_0=\cdots=\delta_{\lfloor\frac{m}{4}\rfloor}=+1$, $\delta_{\lfloor\frac{m}{4}\rfloor+1}=\cdots=\delta_{m-\lfloor\frac{m}{4}\rfloor-1}=-1$ and $\delta_{m-\lfloor\frac{m}{4}\rfloor}=\cdots=\delta_m=+1$, the open set condition holds, and then the corresponding generalized Koch curve has Hausdorff, packing and box dimension $\log(m+1)/\log|\sum_{k=0}^m\delta_ke^{\frac{2k\pi i}{m}}|$, where taking $m=3$ and then $\delta_0=+1,\delta_1=\delta_2=-1,\delta_3=+1$ will recover the result on the classical Koch curve.