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

Dzmitry Dudko, Yusheng Luo, Mikhail Lyubich

TL;DR

The paper develops a rigorous framework for uniform a priori bounds in neutral dynamics by extending pseudo-Siegel bounds to ψ^bullet-ql Siegel maps and placing them within a ψ^bullet renormalization scheme. It introduces ψ^bullet-ql maps, their inflations, and a detailed degeneration toolbox including coastlines, full/outer/vertical/peripheral/external/diving families, parabolic fjords, and a calibration machinery. The central contributions are equidistribution results for the degeneration around Siegel disks across combinatorial levels, with explicit thresholds for regularization and a localized combinatorial localization principle. The results enable transferring uniform pseudo-Siegel bounds to broad rational-map families, including matings of neutral polynomials and boundaries of hyperbolic components, and provide a robust toolkit for renormalization in rational dynamics with Siegel disks.

Abstract

We extend uniform pseudo-Siegel bounds for neutral quadratic polynomials to $ψ^\bullet$-quadratic-like Siegel maps. In this form, the bounds are compatible with the $ψ$-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 $ψ^\bullet$-quadratic-like structure degenerates around the Siegel disk on all geometric scales except on the ``transitional scales'' between two specific combinatorial levels.

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

TL;DR

The paper develops a rigorous framework for uniform a priori bounds in neutral dynamics by extending pseudo-Siegel bounds to ψ^bullet-ql Siegel maps and placing them within a ψ^bullet renormalization scheme. It introduces ψ^bullet-ql maps, their inflations, and a detailed degeneration toolbox including coastlines, full/outer/vertical/peripheral/external/diving families, parabolic fjords, and a calibration machinery. The central contributions are equidistribution results for the degeneration around Siegel disks across combinatorial levels, with explicit thresholds for regularization and a localized combinatorial localization principle. The results enable transferring uniform pseudo-Siegel bounds to broad rational-map families, including matings of neutral polynomials and boundaries of hyperbolic components, and provide a robust toolkit for renormalization in rational dynamics with Siegel disks.

Abstract

We extend uniform pseudo-Siegel bounds for neutral quadratic polynomials to -quadratic-like Siegel maps. In this form, the bounds are compatible with the -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 -quadratic-like structure degenerates around the Siegel disk on all geometric scales except on the ``transitional scales'' between two specific combinatorial levels.

Paper Structure

This paper contains 78 sections, 21 theorems, 151 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Pseudo-Siegel bounds for quadratic polynomials
  3. Quadratic-like maps
  4. $\psi^\bullet$-ql maps
  5. Outline of the paper
  6. $\psi^\bullet$-ql Siegel maps
  7. Definition of $\psi^\bullet$-ql maps
  8. Iteration ${F^k\colon U^k\rightrightarrows V}$
  9. Inflation of rational maps
  10. Inflation: detailed description
  11. Comments to Figure \ref{['Fig:UnwindingSiegelFull']}
  12. Examples
  13. Matings of neutral quadratic polynomials
  14. Neutral Dynamics at boundaries of hyperbolic components
  15. Disjoint type hyperbolic components
  16. ...and 63 more sections

Key Result

Theorem 1.1

Let $f: X \longrightarrow Y, \ X\Subset Y$ be a quadratic-like map with a bounded-type Siegel disk $Z_f$ at its $\alpha$-fixed point. Let $K_f = W(f)$ be the degeneration of $f$ around its Siegel disk. Then for any level $m$ combinatorial interval $I$, we have Moreover, there is an absolute constant ${\mathbf K}\gg 1$ such that if ${\mathfrak l}_m K_f \ge {\mathbf K}$, then for all $f$ and $m$ a

Figures (11)

  • Figure 1: An example of an immersion $\iota$, see also §\ref{['ss:push forw curves']}. A vertical lamination ${\mathcal{L}}$ always lifts to a vertical lamination $\iota^*{\mathcal{L}}$. The lift of a peripheral lamination $\mathcal{R}$ can have vertical and peripheral components (§ \ref{['subsec:familyofarcs']}).
  • Figure 2: Item \ref{['dfn:psib-ql:4']} in §\ref{['ss:psi bullet:defn']} prevents overlapping of immersed limbs as illustrated on this figure.
  • Figure 3: Illustration to Proposition \ref{['prop:g:unwind']}: the inflation of $f\colon A\to B$ in $\widehat{\mathbb{C}}$ around a Siegel disk $\overline Z$ assuming that ${\mathbf \Upsilon}$ controls the postcritical set; see §\ref{['sss:Fig:UnwindingSiegelFull']} for details regarding the figure.
  • Figure 4: An illustration of the vertical and peripheral families of curves. The peripheral families are further decompose into external and diving components. The diving family may intersect either $\partial U^{\mathfrak{q}_{m+1}}$ or $\mathcal{K}_m$.
  • Figure 5: Pushforward of ${\mathcal{F}}_\tau(I)$ under the branched covering. Observe that $Y\coloneqq V\setminus [\gamma \cup (\tau I_k)^c ]$ is a disk.
  • ...and 6 more figures

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2: Equidistribution, see Theorems \ref{['thm:main:psi-ql maps']} and \ref{['thm:main:psi-ql maps:extra']}
  • Theorem 1.3: $\psi^\bullet$-case, see Theorems \ref{['thm:main:psi-ql maps']} and \ref{['thm:main:psi-ql maps:extra']}
  • Definition 2.1
  • Definition 2.2: Siegel prerenormalization
  • Proposition 2.3: Inflation of \ref{['eq:f:BtoA']}, see Figure \ref{['Fig:UnwindingSiegelFull']}
  • Remark 2.4: Variations of Proposition \ref{['prop:g:unwind']}
  • proof : Proof of Proposition \ref{['prop:g:unwind']}
  • Conjecture 2.5: Mother Hedgehog is well defined
  • Lemma 3.1: $\iota\colon {\mathcal{C}}^k_s \hookrightarrow {\mathcal{C}}^{k-1}_s$
  • ...and 52 more