Renormalization of Quadratic Polynomials

• Renormalization theory of quadratic polynomials is a framework that uses quadratic-like maps and analytic operators to uncover self-similarity, scaling laws, and universality in dynamics.
• It employs rigorous hyperbolicity and rigidity results, alongside parabolic and satellite renormalization schemes, to ensure exponential convergence in infinitely renormalizable maps.
• The framework integrates analytic, combinatorial, and geometric tools to address the Mandelbrot set’s local connectivity and establish scaling relationships across diverse dynamical regimes.

Renormalization theory of quadratic polynomials provides a unified analytic and geometric framework to study the small-scale structure, universality, and rigidity properties of iterations $p_c(z) = z^2 + c$. This theory systematically explains the phenomena of universality, scaling laws, and the parameter geometry of the Mandelbrot set $\mathfrak{M}$, revealing the interplay between analytic geometry, combinatorics, and holomorphic dynamics through renormalization operators acting on spaces of quadratic-like maps, parabolic/cylinder classes, and puzzle configurations.

1. Quadratic-Like Maps and Fundamental Renormalization Constructions

A quadratic-like map $f: U \to V$ is a proper, holomorphic degree-2 map with $U, V \subset \mathbb{C}$ topological disks, $U \Subset V$, and unique critical point at $0 \in U$. The filled Julia set $K_f = \{z \in U: f^n(z) \in U, \forall n\}$ and Julia set $J_f = \partial K_f$ encode the recurrent dynamics.

By the Douady–Hubbard straightening theorem, if $J_f$ is connected, $f$ is hybrid-equivalent to a unique quadratic polynomial $p_c$. Two modulus notions, $$\operatorname{mod}^-(f) = \operatorname{mod}(V \setminus U)$$ and $$\operatorname{mod}^+(f)= \sup \operatorname{mod}(A)$$ for annuli $A \subset V \setminus K_f$, are asymptotically equivalent and provide geometric control.

Banach chart neighborhoods $B_\epsilon(f|_U)$ enable the definition of analytic structures on the class $\mathfrak{QL}$ of all quadratic-like maps. The quadratic-like renormalization operator $\mathcal{R}_{QL}$ acts by restriction to "little copies," i.e., if $f^p: U_1 \to V_1$ is quadratic-like with a controlled postcritical orbit and collar, then $\mathcal{R}_{QL}$ is a compact analytic map between Banach balls.

At a fixed point $f_*$, the renormalization equation $f_*(z) = \Lambda^{-1} f_*(f_*(\Lambda z))$ yields the universal Feigenbaum functional equation in the normalized form $g(z) = -\lambda^{-1}g(g(\lambda z))$ where $\lambda \approx 2.5029\ldots$ is the scaling constant. The operator admits compact horseshoe attractors for infinitely renormalizable maps with bounded modulus (Dudko, 30 Dec 2025).

2. Hyperbolicity and Rigidity in Renormalization

Central hyperbolicity results establish that for bounded-type combinatorics and renormalization periods $p_n \leq P < \infty$, the operator $\mathcal{R}_{QL}$ is hyperbolic with a one-dimensional unstable manifold; all infinitely renormalizable quadratic-like maps of bounded type converge exponentially fast to this hyperbolic horseshoe (Lyubich, Dudko–Lyubich).

In the near-parabolic regime (rotation numbers $\theta = [0;a_1,a_2,\ldots]$ with $a_i \geq N \gg 1$), the cylinder renormalization $\mathcal{R}_{cyl}$ is hyperbolic with a single unstable direction (Inou–Shishikura), generalizing to high type (Dudko, 30 Dec 2025, Cheraghi et al., 2015, Cheraghi, 2010, Kapiamba, 2022). For bounded-type Siegel rotation numbers, the pacman/Siegel renormalization operator is again hyperbolic with a one-dimensional unstable direction (Dudko et al., 27 Sep 2025).

The hyperbolicity is underpinned by uniform a priori bounds on moduli, Teichmüller contraction on hybrid leaves, compactness in Banach charts, and explicit control on combinatorial data.

3. Combinatorial, Sectorial, and Parabolic Tools in Renormalization

Several key combinatorial and geometric devices serve as the analytic backbone for renormalization:

• Yoccoz puzzles and principal nests: External rays landing at periodic points partition the filled Julia set into pieces; nested puzzle pieces around the critical point encode the combinatorial return structure (Dudko, 30 Dec 2025).
• Parabolic implosion/Lavaurs maps: As parameters approach parabolic points, the dynamic pairs $(p_{c_n}, p_{c_n}^{q_n})$ converge to $(p_{1/4}, \text{Lavaurs map})$, realizing cylinder renormalization and organizing the sectorial degeneration (Dudko, 30 Dec 2025, Kapiamba, 2022).
• Thin–thick decomposition, covering lemma, quasi-additivity law: These provide control of modulus and extremal width in regimes of positive entropy and satellite combinatorics (Kahn–Lyubich). Pseudo-Siegel disks encode nearly-invariant sets for near-neutral maps (Dudko, 30 Dec 2025, Dudko et al., 27 Sep 2025).

These tools facilitate the construction of almost-invariant domains, provide modulus bounds, and determine scaling relationships critical for universality and rigidity.

4. Satellite and Near-Parabolic Renormalization Schemes

Beyond period-doubling, satellite and near-parabolic renormalization address quadratic polynomials with rotation numbers $p/q \in (-1/2,1/2]$, governed by modified continued fraction expansions and cascades of satellite combinatorics.

Near-parabolic renormalization operators ($\mathcal{R}_{NP-t}$, $\mathcal{R}_{NP-b}$) act on Banach–Teichmüller spaces of germs and realize uniformly hyperbolic dynamics with invariant cones, spectral gaps, and explicit scaling laws. Universality holds: parameter scaling in infinite satellite cascades follows an exponential rule governed by the unstable multiplier $\chi_u(\Theta)$ (Cheraghi et al., 2015, Kapiamba, 2022, Chéritat, 2014).

Rigidity results confirm combinatorial rigidity—local connectivity in $\mathfrak{M}$—for parameters with quadratic-growth combinatorics, established via a tower of renormalization with exponential scaling of diameters in the parameter space. The satellite regime generalizes the Feigenbaum scenario to all satellite cascades, including fine-scale features and degenerating geometries.

5. A Priori Bounds, Modulus Control, and Siegel Disk Degeneration

Uniform a priori bounds play a crucial role, controlling the moduli of fundamental annuli and external degeneration at every combinatorial scale. In the neutral (Siegel) case, $\psi^\bullet$-quadratic-like Siegel maps admit a compatible renormalization theory with equidistributed degeneration among combinatorial intervals defined by continued fraction data.

The degeneration of the Siegel disk is precisely described: deep scales display uniform modulus bounds; shallow scales equidistribute the degeneration proportional to arc length; transitional scales exhibit square-root scaling in modulus. Pseudo-Siegel disks are constructed by welding parabolic fjords, and width-modulus duality and amplification lemmas yield bounded degeneration at all but transitional scales (Dudko et al., 27 Sep 2025).

Existence and uniqueness of Siegel fixed points in renormalization are established by compactness induced by the a priori bounds. Rigidity follows from infinitesimal contraction in the relevant analytic metrics.

6. Applications to the Mandelbrot Local Connectivity Conjecture

The renormalization framework directly addresses the MLC (Rigidity) Conjecture: that $\mathfrak{M}$ is locally connected. By Yoccoz's reduction, one controls the shrinkage of infinite nests of little copies via renormalization operators. Uniform lower bounds on modulus guarantee bounded geometry and exponential shrinkage in parameter space (Dudko, 30 Dec 2025).

Complete proofs exist for MLC at all bounded-type quadratic-like combinatorics, Feigenbaum parameters, and many satellite regimes with a priori bounds. Open problems include pseudo-Siegel bounds in unbounded satellite combinatorics and synthesizing "virtual molecule" scales for the degenerate regime.

7. Functional Equations, Scaling Laws, and Universality

Renormalization operators exhibit classical functional equations at fixed points—particularly, the universal Feigenbaum equation for quadratic polynomials, characterizing scaling constants and universality classes (Dudko, 30 Dec 2025, Cruz et al., 2010).

Explicit renormalization formulas govern scaling at small dynamical scales:

$$(_\eta p_c)(z) = \frac{1}{\eta}\left(p_c^q(\eta z + c) - c\right), \quad q = \min \{n>0: p_c^n(\eta z+c) \in D(c,\eta)\},$$

and the scaling of the Siegel disk's conformal radius is governed by the Brjuno series:

$$\operatorname{dist}(\partial\Delta(f), 0) \lesssim \exp(-B(\alpha)), \quad B(\alpha) = \sum_j q_j^{-1} \log q_{j+1}.$$

Universality of scaling, exponential convergence, and rigidity tie together the architectural principles of renormalization theory (Cheraghi, 2010, Dudko, 30 Dec 2025, Cruz et al., 2010, Cheraghi et al., 2015).

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The renormalization theory of quadratic polynomials systematically organizes their analytic, combinatorial, and geometric features, yielding deep results on universality, parameter scaling, rigidity, and local connectivity. Contemporary research continues to refine the understanding of the unbounded and near-degenerate regimes, seeking a unified hyperbolic operator encompassing all major renormalization schemes (Dudko, 30 Dec 2025, Cheraghi et al., 2015, Kapiamba, 2022, Dudko et al., 27 Sep 2025, Cruz et al., 2010, Cheraghi, 2010, Chéritat, 2014).