Typical Orbits of Quadratic Polynomials with a Neutral Fixed Point: Brjuno Type

We describe the topological behavior of typical orbits of complex quadratic polynomials ${P_{\alpha}(z) = e^{2 \pi \alpha {\bf i}} z + z^{2}}$ P α ( z ) = e 2 π α i z + z 2 , with α of high return type. Here we prove that for such Brjuno values of α the closure of the critical orbit, which is the measure theoretic attractor of the map, has zero area. Then we show that the limit set of the orbit of a typical point in the Julia set of P _α is equal to the closure of the critical orbit. Our method is based on the near parabolic renormalization of Inou-Shishikura, and a uniform optimal estimate on the derivative of the Fatou coordinate that we prove here.