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A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics

Jeremy Kahn

Abstract

We prove the a priori bounds for infinitely renormalizable quadratic polynomials for which we can find an infinite sequence of primitive renormalizations such that the ratios of the periods of successive renormalizations is bounded. This implies the local connectivity of the Mandelbrot set at the corresponding points.

A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics

Abstract

We prove the a priori bounds for infinitely renormalizable quadratic polynomials for which we can find an infinite sequence of primitive renormalizations such that the ratios of the periods of successive renormalizations is bounded. This implies the local connectivity of the Mandelbrot set at the corresponding points.

Paper Structure

This paper contains 74 sections, 80 theorems, 130 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation for renormalization
  3. Statement of the main result
  4. Consequences
  5. Outline of the proof
  6. General strategy: induction on the lengths of the hyperbolic geodesics
  7. Localization "in principle"
  8. Canonical weighted arc diagrams
  9. Terminology and Notation
  10. Acknowledgments
  11. The canonical weighted arc diagram
  12. Proper paths and proper homotopies
  13. Arcs, arc diagrams, and weighted arc-diagrams
  14. Valid WAD's
  15. The canonical foliation and weighted arc-diagram
  16. ...and 59 more sections

Key Result

Theorem 1.2

There exists $\epsilon > 0$, such that for all $p$ there exists $M$: Suppose that $g\mathpunct: U \to V$ is a quadratic-like map that is primitively $p$-renormalizable. Then (with $\gamma \equiv \gamma(g)$ and $\gamma_p = \gamma_p(g)$),

Figures (5)

  • Figure 1.1: On the left we imagine $f$ is renormalizable of periods $3^k$, for $k = 1, 2, 3$. The solid red geodesic is $\gamma_{(3)}$ while the dotted red geodesic is shorter than $\gamma_{(2)}$. On the right we have formed a domain ${\mathbf{V}}_{(2)}$ on which there is a conformal dynamical system that is renormalizable just of period 3 (the ratio of the last two periods). This dynamical system, formed by the canonical renormalization, has three small Julia sets and two geodesics with the same length as their analogs on the left. To first approximation we can think of it as a quadratic-like renormalization $f_{(2)}$ of $f$.
  • Figure 1.2: Here the gray blobs represent ${\mathcal{K}}_p(f)$. The length of the red geodesic $\gamma_p$ going around $K_p(f)$ is roughly the sum of the conformal widths of the thin rectangles, shown in blue.
  • Figure 2.1: The canonical foliation
  • Figure 2.2: Here a leaf of ${\mathcal{F}}_{\mathrm{can}}(I, J)$ crosses a leaf of ${\mathcal{F}}_{\mathrm{can}}(I, J')$. (These are the solid red and blue paths.) This forces all paths in one buffer $B$ crossing all the paths of another buffer $B'$. (There are the dashed red and blue paths.)
  • Figure 4.1: A: We are given a disked tree. B: We can form the associated tree of complete graphs. C: Any path (of red arcs) between the two hollow disks must include at least one each of the dotted, dashed, and dashed-and-dotted arcs. This illustrates Lemma \ref{['lem:into-flags']}.

Theorems & Definitions (149)

  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 139 more