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.
Categories: math.DS math.MG