Hyperbolicity of renormalization of critical quasicircle maps

TL;DR

The paper extends hyperbolicity results from critical circle maps to a broader class of critical quasicircle maps by introducing Corona Renormalization, a compact analytic operator on a Banach space of $(d_0,d_ty)$-critical coronas with a hyperbolic fixed point $f_*$. The local stable manifold corresponds to rotational coronas with a periodic rotation number, while the local unstable manifold is one-dimensional, established via a combination of small-orbits theory and quasiconformal rigidity. A central novelty is passing from coronas to cascades: each corona on the unstable manifold yields a maximal $\sigma$-proper transcendental extension $\Fbold$, enabling a global, renormalization-based analysis of escaping dynamics and Julia sets, and allowing a rigidity result that excludes invariant line fields on the escaping set. The results yield a robust hyperbolic structure for corona renormalization, with profound implications for universality and rigidity in renormalization theory beyond the classical circle/Siegel settings. The work blends Pacman renormalization ideas, transcendental dynamics, and detailed combinatorics of wakes and alpha-points to realize a comprehensive hyperbolic framework for critical quasicircle renormalization.

Abstract

There is a well developed renormalization theory of real analytic critical circle maps by de Faria, de Melo, and Yampolsky. In this paper, we extend Yampolsky's result on hyperbolicity of renormalization periodic points to a larger class of dynamical objects, namely critical quasicircle maps, i.e. analytic self homeomorphisms of a quasicircle with a single critical point. Unlike critical circle maps, the inner and outer criticalities of critical quasicircle maps can be distinct. We develop a compact analytic renormalization operator called Corona Renormalization with a hyperbolic fixed point whose stable manifold has codimension one and consists of critical quasicircle maps of the same criticality and periodic type rotation number. Our proof is an adaptation of Pacman Renormalization Theory for Siegel disks as well as rigidity results on the escaping dynamics of transcendental entire functions.

Figures (14)

• Figure 1: The Julia sets of f_{3,2}(z) = bz^3\dfrac{4-z}{1-4z+6z^2} \quad \text{and} \quad f_{2,2}(z) = cz^2 \dfrac{z-3}{1-3z}at the top and the bottom respectively. The critical values $b\approx -1.144208-0.964454i$ and $c \approx -0.755700-0.654917i$ are picked such that $f_{3,2}:\Hq \to \Hq$ is a $(3,2)$-critical quasicircle map on some quasicircle $\Hq$, $f_{2,2}: \T \to \T$ is a $(2,2)$-critical circle map, and both have the golden mean rotation number $\theta = \frac{\sqrt{5}-1}{2}$. Both $\Hq$ and $\T$ are colored red, and their preimages are colored green.
• Figure 2: A $(d_0,d_\infty)$-critical corona $f: U \to V$ with $d_0=2$ and $d_\infty=3$. There are $2d_0-2$ preimages of $\gamma_1$ on the inner boundary of $U$, and $2d_\infty-2$ preimages of $\gamma_1$ on the outer boundary of $U$.
• Figure 3: A (2,3)-critical pre-corona. It projects to the corona in Figure \ref{['fig:corona']} after gluing $\beta_+$ and $\beta_-$
• Figure 4: The construction of the pre-corona in the proof of Lemma \ref{['lem:construction-of-pre-corona']} when $(d_0,d_\infty)=(3,2)$. The triangle defined by the dotted line in $S_+$ and its preimages are to be removed.
• Figure 5: The construction of the map $f^p: D_{-1} \to D_0$ in the first part of the proof of Lemma \ref{['lem:construction-of-periodic-bubble']}.

Theorems & Definitions (229)

• definition 1
• theorem 1: Hyperbolicity of the renormalization horseshoe Y01Y03aY03b
• lemma 1: DLS
• lemma 2: DLS
• lemma 3
• lemma 4
• definition 2
• lemma 5: DL23
• lemma 6: Proper discontinuity
• lemma 7: DL23