On an analytic approach to the Fatou conjecture

Let f be a quadratic map (more generally, f(z) = z^d + c, d > 1) of the complex plane. We give sufficient conditions for f to have no measurable invariant linefields on its Julia set. We also prove that if the series ∑_n≥0 1/(fⁿ)′(c) converges absolutely, then its sum is non-zero. In the proof we use analytic tools, such as integral and transfer (Ruelle-type) operators and approximation theorems.