We investigate the quantitative and analytic aspects of the near-parabolic renormalization scheme introduced by Inou and Shishikura in 2006. These provide techniques to study the
dynamics of some holomorphic maps of the form f .z/ D e
2 i˛z C O.z2
/, including the quadratic
polynomials e
2 i˛z C z
2
, for some irrational values of ˛. The main results of the paper concern finescale features of the measure-theoretic attractors of these maps, and their dependence on the data. As
a bi-product, we establish an optimal upper bound on the size of the maximal linearization domain in
terms of the Siegel-Brjuno-Yoccoz series of ˛.