On the geometry of Julia sets
Dynamical Systems
2009-10-29 v1
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 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 , up to conformal transformation, is manifest in the formulas. The Hausdorff dimension of is estimated by the transseries formula. The analysis extends to polynomial maps.
Cite
@article{arxiv.0910.5278,
title = {On the geometry of Julia sets},
author = {O. Costin and M. Huang},
journal= {arXiv preprint arXiv:0910.5278},
year = {2009}
}