Typical orbits of quadratic polynomials with a neutral fixed point: non-Brjuno type

We investigate the quantitative aspects of the near-parabolic renor- malization scheme introduced by Inou and Shishikura. This is used to study the dynamics of an infinite dimensional class of holomorphic maps of the form $f(z)=e^{2\pi i \alpha} z+ O(z^2)$, including the quadratic polynomials $f(z)=e^{2\pi i \alpha} z+ z^2$, for some irrational values of $\alpha$. We prove an optimal upper bound on the size of their max- imal linearization domain in terms of the Siegel-Brjuno series of $\alpha$. In particular, in the special case of quadratic polynomials, we obtain a topological description of the orbits of typical points, a fine-scale feature of the post-critical set, as well as a semi-continuity property of the post-critical set.