Let F(z) be the fixed point of the period doubling renormalization (known as the Feigenbaum map). Recently, jointly with Scott Sutherland we showed that the Julia set of F(z) has Hausdorff dimension strictly less than two (withoug obtaining any more accurate information). In this talk I will recall the latter result and will present new estimates of this Hausdorff dimension from below. The approach used is applicable to other renormalization fixed points.