Hyperbolicity of Renormalization of Critical Quasicircle Maps

The paper constructs a compact analytic corona renormalization operator R on a Banach analytic manifold of (d0,d∞)-critical coronas, proves existence/uniqueness of a fixed point f∗ for periodic-type θ, identifies the local stable manifold with rotational coronas of angle θ, and shows the local unstable manifold is one-dimensional. These steps are explicitly developed via (i) a precise prime/iterate renormalization scheme and a Banach neighborhood framework (Theorem 4.6), (ii) construction of f∗ from commuting-pair renormalization (Theorem 6.4 and Corollary 6.5), (iii) a stability/hyperbolicity analysis that rules out neutral eigenvalues and guarantees repelling directions (Theorem 6.12), and (iv) a transcendental “cascade” rigidity argument yielding dim Wu_loc = 1 (Theorem 13.1). All four items (1)–(4) of Theorem A are thereby established in the paper . By contrast, the candidate solution replaces the core compactness and hyperbolicity arguments with unproven transfers from pacman renormalization and appeals to uniform complex a priori bounds not established for the corona setting. It does not justify the claimed surgery-equivalence that would carry over pacman periodic points and their spectra to coronas, nor does it reproduce the cascade-based rigidity needed for the 1D unstable manifold. Consequently, the model argument is incomplete/non-rigorous where it matters most, even though its conclusions align with the paper’s.