On the geometry of Julia sets

TL;DR

This paper presents a transseries formula for Julia sets of quadratic maps within hyperbolic components, revealing their self-similarity, structure, and Hausdorff dimension, with extensions to polynomial maps.

Contribution

It introduces a novel transseries-based approach to describe Julia sets, connecting their geometry to rescaled elementary pieces and providing dimension estimates.

Findings

Julia sets are described by rapidly convergent transseries formulas.

Self-similarity of Julia sets is explicitly demonstrated through the formulas.

Hausdorff dimension estimates are derived from the transseries expressions.

Abstract

We show that the Julia set of quadratic maps with parameters in hyperbolic components of the Mandelbrot set is given by a transseries formula, rapidly convergent at any repelling periodic point. Up to conformal transformations, we obtain $J$ from a smoother curve of lower Hausdorff dimension, by replacing pieces of the more regular curve by increasingly rescaled elementary "bricks" obtained from the transseries expression. Self-similarity of $J$, up to conformal transformation, is manifest in the formulas. The Hausdorff dimension of $J$ is estimated by the transseries formula. The analysis extends to polynomial maps.