Generalized Koch curves and Thue-Morse sequences

TL;DR

This paper generalizes the classical Koch curve using Thue-Morse sequences, establishing their fractal properties and dimensions through iterated function systems and specific sequence conditions.

Contribution

It introduces a new class of generalized Koch curves based on Thue-Morse sequences and proves their fractal dimensions and attractor properties.

Findings

Generalized Koch curves are attractors of iterated function systems.

Under certain sequence conditions, the open set condition holds.

The Hausdorff, packing, and box dimensions are explicitly calculated.

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.