$\Delta$-revolving sequences and self-similar sets in the plane

TL;DR

This paper extends the representation of self-similar sets in the plane using complex power series parametrized by $ abla$-revolving sequences, covering more complex fractals like the Heighway dragon and Twindragon.

Contribution

It generalizes previous work to include self-similar sets with multiple contractions, introducing $ abla$-revolving sequences for their complex power series representation.

Findings

Self-similar sets with more than two contractions have a natural complex power series representation.

The method applies to well-known fractals such as the Heighway dragon, Twindragon, and Fudgeflake.

The representation is parametrized by $ abla$-revolving sequences, broadening the class of analyzable fractals.

Abstract

Initiated by Mizutani and Ito's work in 1987, Kawamura and Allen recently showed that certain self-similar sets generalized by two similar contractions have a natural complex power series representation, which is parametrized by past-dependent revolving sequences. In this paper, we generalize the work of Kawamura and Allen to include a wider collection of self-similar sets. We show that certain self-similar sets consisting of more than two similar contractions also have a natural complex power series representation, which is parametrized by {\it $\Delta$-revolving sequences}. This result applies to several other famous self-similar sets such as the Heighway dragon, Twindragon, and Fudgeflake.