On an analytic approach to the Fatou conjecture
Volume 171 / 2002
Fundamenta Mathematicae 171 (2002), 177-196
MSC: 37F10, 30D05, 37C30.
DOI: 10.4064/fm171-2-5
Abstract
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 $\sum _{n\ge 0} {1/(f^n)'(c)}$ 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.
