Geometric construction and analytic representation of Julia sets of polynomial maps

Abstract

We show that in the hyperbolic components of the Mandelbrot set, the Julia set J of a quadratic map is given by transseries formulae, rapidly convergent as repelling periodic points are approached.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 rescalings and conformal transformations in a neighbourhood of J, is manifest in the formulae.The Hausdorff dimension of J is estimated by the transseries formula. We discuss how the analysis extends to more general polynomial maps.