Abstract

The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets which includes Cantor sets, Koch curves, Lévy curves, Sierpinski gaskets, Rauzy fractals, and fractal dendrites. We note a fundamental dichotomy for n-ary complex trees that allows us to study topological changes in regions where one-parameter families of connected self-similar sets are defined. Moreover, we show how to obtain these families from systems of equations encoded by tip-to-tip equivalence relations. As far as we know, these families and the Mandelbrot-like sets that we introduce to study them are new. These sets are related to a set discussed in Bill Thurston’s last paper, though in many ways easier to understand.