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On quasiconformal non-equivalence of gasket Julia sets and limit sets

Yusheng Luo, Yongquan Zhang

TL;DR

The paper addresses whether Julia sets can be quasiconformally equivalent to gasket limit sets, establishing that no Julia set is QC equivalent to the Apollonian gasket and, in the quadratic case, no Julia set is QC equivalent to a gasket limit set of a geometrically finite Kleinian group. It develops a framework combining Kleinian geometry, Fatou-graph dynamics, and Thurston theory to derive obstructions to QC equivalence, culminating in a rigidity result for fat gasket Julia sets. The core technical advances include a characterization of gasket limit sets via circle packings, the construction and analysis of fat gasket Julia sets through finite cores and Thurston realizations, and a detailed combinatorial study of Fatou graphs in the Per$_2(0)$ captured-type locus. Together, these results constrain when a Julia set can be quasiconformally related to a gasket limit set and illuminate the rigid, bipartite structure of Fatou graphs in this setting, with implications for the broader classification of Julia sets and Kleinian limit sets.

Abstract

This paper studies quasiconformal non-equivalence of Julia sets and limit sets. We proved that any Julia set is quasiconformally different from the Apollonian gasket. We also proved that any Julia set of a quadratic rational map is quasiconformally different from the gasket limit set of a geometrically finite Kleinian group.

On quasiconformal non-equivalence of gasket Julia sets and limit sets

TL;DR

The paper addresses whether Julia sets can be quasiconformally equivalent to gasket limit sets, establishing that no Julia set is QC equivalent to the Apollonian gasket and, in the quadratic case, no Julia set is QC equivalent to a gasket limit set of a geometrically finite Kleinian group. It develops a framework combining Kleinian geometry, Fatou-graph dynamics, and Thurston theory to derive obstructions to QC equivalence, culminating in a rigidity result for fat gasket Julia sets. The core technical advances include a characterization of gasket limit sets via circle packings, the construction and analysis of fat gasket Julia sets through finite cores and Thurston realizations, and a detailed combinatorial study of Fatou graphs in the Per captured-type locus. Together, these results constrain when a Julia set can be quasiconformally related to a gasket limit set and illuminate the rigid, bipartite structure of Fatou graphs in this setting, with implications for the broader classification of Julia sets and Kleinian limit sets.

Abstract

This paper studies quasiconformal non-equivalence of Julia sets and limit sets. We proved that any Julia set is quasiconformally different from the Apollonian gasket. We also proved that any Julia set of a quadratic rational map is quasiconformally different from the gasket limit set of a geometrically finite Kleinian group.
Paper Structure (18 sections, 30 theorems, 19 equations, 7 figures)

This paper contains 18 sections, 30 theorems, 19 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Historical Background
  3. Strategy and techniques
  4. Notes and discussions
  5. Structure of the paper
  6. Kleinian groups with gasket limit set
  7. Geometric finiteness and acylindricity
  8. Gasket limit sets
  9. Induced dynamics on the Fatou graph
  10. Construction of fat gasket Julia set
  11. Finite core
  12. Thurston theory of rational maps
  13. The quadratic case
  14. Captured type in Per2(0)
  15. The canonical finite core and critical loop
  16. ...and 3 more sections

Key Result

Theorem 1.2

No Julia set of a rational map is quasiconformally homeomorphic to the Apollonian gasket.

Figures (7)

  • Figure 1.1: The Apollonian gasket on the left, and the Julia set that is homeomorphic to an Apollonian gasket on the right. The two sets are not quasiconformally homeomorphic as Fatou components touch at an angle on the right.
  • Figure 1.2: An example of a fat gasket Julia set. The Fatou graph is bipartite as one can see by the coloring of Fatou set.
  • Figure 5.1: The bifurcation locus of $\mathop{\mathrm{Per}}\nolimits_2(0)$.
  • Figure 5.2: The shortest anchored simple closed curves for a Type I map.
  • Figure 5.3: The shortest anchored simple closed curves for a Type IIA map.
  • ...and 2 more figures

Theorems & Definitions (57)

  • Conjecture 1.1: LLMM19
  • Theorem 1.2
  • Remark
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark
  • Proposition 2.1: Characterization of geom. finite acy. Kleinian groups
  • Lemma 2.2
  • ...and 47 more