We prove the uniform hyperbolicity of the near-parabolic renormalization
operators acting on an infinite-dimensional space of holomorphic
transformations. This implies the universality of the scaling laws, conjectured
by physicists in the 70's, for a combinatorial class of bifurcations. Through
near-parabolic renormalizations the polynomial-like renormalizations of
satellite type are successfully studied here for the first time, and new
techniques are introduced to analyze the fine-scale dynamical features of maps
with such infinite renormalization structures. In particular, we confirm the
rigidity conjecture under a quadratic growth condition on the combinatorics.
The class of maps addressed in the paper includes infinitely-renormalizable
maps with degenerating geometries at small scales (lack of a priori bounds).