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 $\sum_{k<n}t_ke^\frac{2kπi}{3}$ 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.