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.