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$Δ$-revolving sequences and self-similar sets in the plane

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 $Δ$-revolving sequences}. This result applies to several other famous self-similar sets such as the Heighway dragon, Twindragon, and Fudgeflake.