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 ˛
