  We calculate rigorous bounds on the Hausdorff dimension of the attractor at
the accumulation of the period-doubling cascade for families of maps with
quadratic, cubic, and quartic critical point. To do this, we express the
attractors as the limit sets of appropriate Iterated Function Systems
constructed using rigorous bounds on the corresponding renormalisation fixed
point functions. We use interval arithmetic with rigorous directed rounding
modes to show that the respective dimensions lie in subintervals of the
intervals (0.5370,0.5392), (0.6040,0.6091), and (0.6395,0.6474).
