Uniform bounds for bubbles of neutral quadratic polynomials

Given a quadratic polynomial with an irrationally indifferent fixed point of bounded type, a bubble is an iterated preimage of its Siegel disk. Similar to pseudo-Siegel disks, one constructs pseudo-bubbles from bubbles by filling in parabolic fjords. We introduce the near-degenerate regime for the $α'$-Dynamics controlling the geometry of these pseudo-bubbles in deep scale. Consequently, we obtain uniform bounds on the size and regularity of pseudo-bubbles independent of the rotation number. In [DLL], jointly with Lyubich, this result serves as a necessary ingredient for establishing uniform butterfly bounds and ultimately the combinatorial rigidity of the full attractor of neutral renormalization.