2015/09/25 by Cheraghi, Davoud, Shishikura, Mitsuhiro
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1509.07843
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).