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Sierpinski carpet hyperbolic components of disjoint type are bounded

Dzimitry Dudko, Yusheng Luo

TL;DR

This work proves that Sierpinski carpet hyperbolic components of disjoint type are bounded in the fixed-point marked moduli space, and extends uniform dynamical control to maps on the component boundary. The authors develop a priori bounds for geometrically infinite degenerations via eventually-golden-mean maps, pseudo-Siegel disks, and valuable-attracting domains, tying local degeneration to a global linear relation and a pulled-off constant N_Siegel. Central to the argument is a novel combination of renormalization techniques and Thurston theory: the pulled-off constant bounds arc and loop degenerations, enabling a limiting map on a finite tree and a Thurston obstruction that rules out unbounded degeneration. They also establish a semiconjugacy to an expanding topological model, which yields a quadratic-like restriction around every non-repelling cycle and connects the parameter boundary to a refined Mandelbrot-like dynamics. The results provide a robust framework for bounding disjoint-type components and lay groundwork for extending the approach beyond the Sierpinski case, with explicit illustrations on the hyperbolic component of z^2 as a prototype.

Abstract

We establish certain uniform a priori bounds for hyperbolic components of disjoint type. As an application, we will prove that Sierpinski carpet hyperbolic components of disjoint type are bounded. Furthermore, we show that for each map $f$ on the closure of such a hyperbolic component, there exists a quadratic-like restriction around every non-repelling periodic point. Extensions of these results to non-Sierpinski configurations are underway. As a prototype example, we describe the post-critical set of any map on the boundary of the hyperbolic component of $z^2$.

Sierpinski carpet hyperbolic components of disjoint type are bounded

TL;DR

This work proves that Sierpinski carpet hyperbolic components of disjoint type are bounded in the fixed-point marked moduli space, and extends uniform dynamical control to maps on the component boundary. The authors develop a priori bounds for geometrically infinite degenerations via eventually-golden-mean maps, pseudo-Siegel disks, and valuable-attracting domains, tying local degeneration to a global linear relation and a pulled-off constant N_Siegel. Central to the argument is a novel combination of renormalization techniques and Thurston theory: the pulled-off constant bounds arc and loop degenerations, enabling a limiting map on a finite tree and a Thurston obstruction that rules out unbounded degeneration. They also establish a semiconjugacy to an expanding topological model, which yields a quadratic-like restriction around every non-repelling cycle and connects the parameter boundary to a refined Mandelbrot-like dynamics. The results provide a robust framework for bounding disjoint-type components and lay groundwork for extending the approach beyond the Sierpinski case, with explicit illustrations on the hyperbolic component of z^2 as a prototype.

Abstract

We establish certain uniform a priori bounds for hyperbolic components of disjoint type. As an application, we will prove that Sierpinski carpet hyperbolic components of disjoint type are bounded. Furthermore, we show that for each map on the closure of such a hyperbolic component, there exists a quadratic-like restriction around every non-repelling periodic point. Extensions of these results to non-Sierpinski configurations are underway. As a prototype example, we describe the post-critical set of any map on the boundary of the hyperbolic component of .

Paper Structure

This paper contains 73 sections, 42 theorems, 195 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Historical background
  3. A priori bounds
  4. Renormalization theory
  5. Eventually-golden-mean maps
  6. Degenerations of compact Riemann surfaces
  7. Degenerations of eventually-golden-mean maps
  8. The pulled-off constant
  9. Key technical theorem
  10. Sketch of the proof of Theorems \ref{['thm:A']} and \ref{['thm:B']} from Theorem \ref{['thm:cd']}
  11. Boundaries of hyperbolic components of disjoint type
  12. The example of $\mathcal{H}_{z^2}$
  13. Outline of the proof of Theorem \ref{['thm:cd']}
  14. Non-uniform construction of pseudo-Siegel disks.
  15. Pulled-off Argument and Localization.
  16. ...and 58 more sections

Key Result

Theorem A

A Sierpinski carpet hyperbolic component $\mathcal{H} \subseteq \mathop{\mathrm{\mathcal{M}}}\nolimits_{d, \text{fm}}$ of disjoint type is bounded in $\mathop{\mathrm{\mathcal{M}}}\nolimits_d$. Moreover, eq:H is D naturally extends to In particular, $\partial{\mathcal{H}}$ is locally connected.

Figures (8)

  • Figure 1.1: The Julia set of a Sierpinski carpet hyperbolic rational map.
  • Figure 3.1: The first return and combinatorial intervals.
  • Figure 3.2: An illustration of psuedo-Siegel disk. The intersection patterns of the protecting annulus $A_I$, the inner buffer $S^{{\operatorname{inn}}}(I)$, extra outer protection $\mathcal{X}_I$ are indicated on the graph.
  • Figure 5.1: The curve $\gamma_1$ and $\gamma_2$ are the left and right most arcs in $\mathcal{F}_1$. Most of the arcs in $\mathcal{F}_1$ passes through $I_1$ or $I_2$.
  • Figure 6.1: The configuration of the families $\mathcal{F}_{1, new}$ and $\mathcal{G}$.
  • ...and 3 more figures

Theorems & Definitions (93)

  • Conjecture 1.1
  • Theorem A: Parameter Space
  • Theorem B: Dynamical Plane
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4: Precompactness Condition
  • Definition 1.5: Pulled-off constant
  • Theorem C: A priori bounds
  • Remark 1.6
  • Definition 1.7
  • ...and 83 more