Plane trees, Shabat-Zapponi polynomials and Julia sets

TL;DR

This paper explores the relationship between embedded trees, their associated Shabat-Zapponi polynomials, and the properties of their Julia sets, focusing on form, connectedness, and Hausdorff dimension.

Contribution

It introduces a novel experimental approach linking tree structures to Julia set properties via unique polynomial forms.

Findings

Correlations between tree structure and Julia set form

Insights into connectedness of Julia sets based on tree embedding

Analysis of Hausdorff dimension variations in Julia sets

Abstract

A tree, embedded into plane, is a dessin d'enfant and its Belyi function is a polynomial --- Shabat polynomial. The Zapponi form of this polynomial is unique, so we can correspond to an embedded tree the Julia set of its Shabat-Zapponi polynomial. In this purely experimental work we study relations between the form of a tree and properties (form, connectedness, Hausdorff dimension) of its Julia set.