On the geometry of Julia sets

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.