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A priori bounds for some infinitely renormalizable quadratic: IV. Elephant Eyes

Jeremy Kahn, Misha Lyubich

TL;DR

The work tackles MLC by deriving a priori bounds for a broad class of infinitely renormalizable quadratic-like maps with elephant eye combinatorics, including unbounded combinatorial limbs. It introduces uniform thin-thick decompositions for bordered Riemann surfaces and a detailed canonical-lamination framework to control geometric and combinatorial data under pullbacks. A central contribution is the Amplification Theorem, providing a dichotomy between amplification (dominance of vertical weight) and confinement (weight localized on short arcs), yielding bounded or controlled unbounded behavior. These tools culminate in local connectivity results for the Julia sets and, via puzzle arguments, MLC at the corresponding parameter values, with broader potential applications to degenerations near the Mandelbrot cusp. The methodology blends uniform-width estimates, Hubbard-tree dynamics, and a sophisticated weight-tracking scheme to tame infinite renormalization.

Abstract

In this paper we prove a priori bounds for an ``elephant eye'' combinatorics. Little $M$-copies specifying these combinatorics are allowed to converge to the cusp of the Mandelbrot set. To handle it, we develope a new geometric tool: uniform thin-thick decompositions for bordered Riemann surfaces.

A priori bounds for some infinitely renormalizable quadratic: IV. Elephant Eyes

TL;DR

The work tackles MLC by deriving a priori bounds for a broad class of infinitely renormalizable quadratic-like maps with elephant eye combinatorics, including unbounded combinatorial limbs. It introduces uniform thin-thick decompositions for bordered Riemann surfaces and a detailed canonical-lamination framework to control geometric and combinatorial data under pullbacks. A central contribution is the Amplification Theorem, providing a dichotomy between amplification (dominance of vertical weight) and confinement (weight localized on short arcs), yielding bounded or controlled unbounded behavior. These tools culminate in local connectivity results for the Julia sets and, via puzzle arguments, MLC at the corresponding parameter values, with broader potential applications to degenerations near the Mandelbrot cusp. The methodology blends uniform-width estimates, Hubbard-tree dynamics, and a sophisticated weight-tracking scheme to tame infinite renormalization.

Abstract

In this paper we prove a priori bounds for an ``elephant eye'' combinatorics. Little -copies specifying these combinatorics are allowed to converge to the cusp of the Mandelbrot set. To handle it, we develope a new geometric tool: uniform thin-thick decompositions for bordered Riemann surfaces.
Paper Structure (41 sections, 39 theorems, 67 equations)

This paper contains 41 sections, 39 theorems, 67 equations.

Table of Contents

  1. Preamble
  2. Introduction
  3. Canonical laminations
  4. Ideally marked hyperbolic plane
  5. Quadrilaterals
  6. Canonical lamination
  7. Conformal geometry vs hyperbolic
  8. Thick part
  9. Thin-thick Decomposition
  10. Local weights and fluxes for a partially marked plane
  11. Riemann surfaces with boundary
  12. Universal covering
  13. Local weghts
  14. Canonical arc diagram
  15. Thin-thick decomposition
  16. ...and 26 more sections

Key Result

Lemma 3.1

Let ${\epsilon}>0$ and let ${\boldsymbol{\gamma}}^{\epsilon}=\{ z\in{\boldsymbol{\gamma}}: \ \operatorname{dist}_{\mathrm{hyp}} (z, {\boldsymbol{\delta}}) < {\epsilon}\}$. Then

Theorems & Definitions (64)

  • Remark 3.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Lemma 3.5
  • Proposition 3.6
  • ...and 54 more