Uniform a priori bounds for neutral renormalization. Variation II: ${\psi }^{\bullet }$-ql Siegel maps

Abstract

We extend uniform pseudo-Siegel bounds for neutral quadratic polynomials to ${\psi }^{\bullet }$-quadratic-like Siegel maps. In this form, the bounds are compatible with the $\psi$-quadratic-like renormalization theory and are easily transferable to various families of rational maps. The main theorem states that the degeneration of a Siegel disk is equidistributed among combinatorial intervals. This provides a precise description of how the ${\psi }^{\bullet }$-quadratic-like structure degenerates around the Siegel disk on all geometric scales except on the ``transitional scales'' between two specific combinatorial levels.

56 pages, 10 figures