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A simple proof of Sullivan's complex bounds

Genadi Levin

TL;DR

This work gives a conceptually streamlined proof of Sullivan's complex bounds for infinitely renormalizable real quadratic maps with bounded combinatorics by employing towers of maps from the Epstein class. The core idea is to pass from Epstein-class dynamics to a polynomial-like restriction near a completely invariant limit set, leveraging Poincaré-neighborhood geometry and stability under perturbations to obtain a uniform modulus bound. Two main results are established: a tower-to-PL transition theorem (Theorem $\text{t-m}$) and its consequence for complex bounds of bounded-type renormalizations (Theorem $\text{t-su}$). The approach unifies real bounds, Epstein-class theory, and slit-plane hyperbolic geometry to yield complex bounds with a simpler, soft argument relative to prior, highly technical proofs.

Abstract

About 35 years ago Dennis Sullivan proved a precompactness property ("complex bounds") for infinitely renormalizable real quadratic polynomials with bounded combinatorics. We present a simple "soft" proof of this remarkable result.

A simple proof of Sullivan's complex bounds

TL;DR

This work gives a conceptually streamlined proof of Sullivan's complex bounds for infinitely renormalizable real quadratic maps with bounded combinatorics by employing towers of maps from the Epstein class. The core idea is to pass from Epstein-class dynamics to a polynomial-like restriction near a completely invariant limit set, leveraging Poincaré-neighborhood geometry and stability under perturbations to obtain a uniform modulus bound. Two main results are established: a tower-to-PL transition theorem (Theorem ) and its consequence for complex bounds of bounded-type renormalizations (Theorem ). The approach unifies real bounds, Epstein-class theory, and slit-plane hyperbolic geometry to yield complex bounds with a simpler, soft argument relative to prior, highly technical proofs.

Abstract

About 35 years ago Dennis Sullivan proved a precompactness property ("complex bounds") for infinitely renormalizable real quadratic polynomials with bounded combinatorics. We present a simple "soft" proof of this remarkable result.
Paper Structure (6 sections, 14 theorems, 17 equations)

This paper contains 6 sections, 14 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: the Epstein class
  3. Towers of maps from the Epstein class
  4. Preliminaries: Poincare's neighborhoods
  5. Proof of Theorem \ref{['t-m']}
  6. Proof of Theorem \ref{['t-su']}

Key Result

Theorem 1.1

(Sullivan Su1, MS) Given $N$ there exists $m=m(N)>0$ as follows. Let $f$ be an infinitely renormalizable real quadratic polynomial with combinatorics bounded by $N$. Then, for every $n\ge n(f)$, the corresponding renormalization $R^{q_n}(f):[-1,1]\to[-1,1]$ admits a polynomial-like extension $R^{q_n

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • ...and 19 more