Lebesgue measure of Feigenbaum Julia sets

We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family $P_c: z \mapsto z^2+c$ corresponding parameters $c$ is shown to have Hausdorff dimension. Our examples include renormalization fixed points, and polynomials their stable manifold are first known rational for which hyperbolic dimension different from set.