Measure and dimension of some Julia sets and parameter sets

While studying Fatou's memoirs in the graduate school, I got interested in the problem of the area of Julia sets:
Paper [1] contains some observations in this direction -- it is one of the earliest applications of the Koebe Distortion Theorem to dynamics.
It became the starting point of the series of papers with Blokh on the Smooth Ergodic Theory on the interval.