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Cubic Siegel polynomials and the bifurcation measure

Matthieu Astorg, Davoud Cheraghi, Arnaud Chéritat

Abstract

We prove that cubic polynomial maps with a fixed Siegel disk and a critical orbit eventually landing inside that Siegel disk lie in the support of the bifurcation measure. This answers a question of Dujardin in positive. Our result implies the existence of holomorphic disks in the support of the bifurcation measure, and also implies that the set of rigid parameters is not closed in the moduli space of cubic polynomials.

Cubic Siegel polynomials and the bifurcation measure

Abstract

We prove that cubic polynomial maps with a fixed Siegel disk and a critical orbit eventually landing inside that Siegel disk lie in the support of the bifurcation measure. This answers a question of Dujardin in positive. Our result implies the existence of holomorphic disks in the support of the bifurcation measure, and also implies that the set of rigid parameters is not closed in the moduli space of cubic polynomials.

Paper Structure

This paper contains 6 sections, 15 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. From elliptic to parabolic
  3. Capture components
  4. Asymptotic size and conformal radius
  5. Parabolic rays converge to Siegel internal rays
  6. Proof of Theorem \ref{['th:supp']}

Key Result

Theorem 1.1

Any Siegel polynomial of capture type $f_{\lambda,a}$ is contained in the support of $\mu_{\mathrm{bif}}$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Theorem 2.2: Cheri2020
  • Lemma 2.3: Jellouli
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 17 more