Families of connected self-similar sets generated by complex trees

TL;DR

This paper introduces complex trees as a novel framework for analyzing connected self-similar sets, defining new parameter sets to understand their topological changes and connectivity properties.

Contribution

It develops a unified approach using complex trees to study families of self-similar sets and their connectivity, extending traditional methods.

Findings

Defined a new set al for connectivity analysis

Established a theorem for special connectivity types

Unified analysis of disconnected self-similar sets

Abstract

The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets. Systems of equations encoded by complex trees tip-to-tip equivalence relations are used to obtain one-parameter families of connected self-similar sets $F_\mathcal{A}(z)$. In order to study topological changes of $F_\mathcal{A}(z)$ in regions $\mathcal{R}\subset\mathbb{C}$ where these families are defined, we introduce a new kind of set $\mathcal{M}\subseteq\mathcal{R}$ which extends the usual notion of connectivity locus for a parameter space. Moreover we consider another set $\mathcal{M}_0\subseteq\mathcal{M}$ related to a special type of connectivity for which we provide a theorem. Among other things, the present theory provides a unified framework to families of self-similar sets traditionally studied as separate with elements $F_\mathcal{A}(z)$ disconnected for parameters $z\in\mathcal{R}\backslash\mathcal{M}$.