Complex trees and their families of connected self-similar sets

Abstract

[en] 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, Sierpiński 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 $\mathcal{R}$ 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 sets $M , M_{0}$, and $\mathcal{K}$ that we introduce to study $\mathcal{R}$ are new.

We provide a theorem, and a necessary condition, for certifying if a given tipset (self-similar set associated to a complex tree) is a fractal dendrite. We highlight a special class of totally connected tipsets that we call root-connected. And we provide a pair of theorems related to them. For a given one-parameter family we also define the set of root-connected trees $M_{0}$ which presents an asymptotic similarity between its boundary and their associated tipsets.

By adapting the notion of post-critically finite self-similar set (p.c.f. for short), the open set condition, and the Hausdorff dimension, we arrive to an upper bound for the existence of p.c.f. trees in a given one-parameter family. We also provide a theorem that allows us to discard non-p.c.f. trees just by looking at some local properties. In relation to this theorem, we set a conjecture of an interesting observation that has been consistent in numerous computational experiments.

The space of one-parameter families of tipset-connected complex trees has just begun to be explored. For the family $TS(z) := T \{z, 1/2, 1/4z\}$ we prove that there is a pair of regions contained in the set $\mathcal{K}$ with a piece-wise smooth boundary. We show that this piece-wise smooth boundary is a rather exceptional case by considering a closely related family, $T S(z) := T {z, -1/2, 1/4z}$. Finally we indicate how the general framework works for one-parameter families with non-fixed mirror-symmetric trees.