On hybrid fractal curves of the Heighway and Lévy dragon curves

Abstract

We introduce an automaton $M$ with some conditions. The automatic sequence $\{a(n)\}_{n=0}^{\infty}$ for $M$ gives a quantity $\mu_{a}$, which is similar to a measure on the unit interval. $M$ also gives an iterated function system, hence a fractal $A$ for $M$ is determined. Fractals treated in this paper are hybrids of the Heighway dragon curve and the Lévy dragon curve. We introduce a modified iterated function system to approximate $A$ by directed piecewise linear curves $A_{k}$. We will study a relation between $\mu_{a}$ and $A_{k}$.