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Non-autonomous parabolic implosion

Matthieu Astorg, Fabrizio Bianchi

Abstract

We study parabolic implosion in a general non-autonomous setting. Let $f(w)=w+w^2+O(w^3)$ be a holomorphic germ tangent to the identity. We consider the iteration of non-autonomous perturbations of the form \[ w_{j+1}=f(w_j)+\varepsilon_{j,n}^2. \] We show that, when the $\varepsilon_{j,n}^2$'s satisfy a Lavaurs-type condition, the element $w_n$ can be described by means of a suitable Lavaurs map $L_{u_n}$, whose phase $u_n$ is an explicit function of the perturbation parameters. In particular, whenever $u_n\to u\in \mathbb C$, the non-autonomous dynamics converges locally uniformly on compact subsets of the parabolic basin to the corresponding Lavaurs map $L_u$. Our study provides a general description of additive non-autonomous parabolic implosion and yields several deterministic and random convergence results as corollaries, as well as a unified proof of several previous results. As an application, we also obtain strong discontinuity results for the Julia sets of fibered holomorphic endomorphisms of $\mathbb P^2(\mathbb C)$.

Non-autonomous parabolic implosion

Abstract

We study parabolic implosion in a general non-autonomous setting. Let be a holomorphic germ tangent to the identity. We consider the iteration of non-autonomous perturbations of the form We show that, when the 's satisfy a Lavaurs-type condition, the element can be described by means of a suitable Lavaurs map , whose phase is an explicit function of the perturbation parameters. In particular, whenever , the non-autonomous dynamics converges locally uniformly on compact subsets of the parabolic basin to the corresponding Lavaurs map . Our study provides a general description of additive non-autonomous parabolic implosion and yields several deterministic and random convergence results as corollaries, as well as a unified proof of several previous results. As an application, we also obtain strong discontinuity results for the Julia sets of fibered holomorphic endomorphisms of .

Paper Structure

This paper contains 18 sections, 22 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Parabolic points and implosion
  3. Non-autonomous perturbations and main result
  4. Consequences
  5. Structure of the paper
  6. Preliminaries and notations
  7. Parabolic germs and Fatou coordinates
  8. Lavaurs maps
  9. Approximate Fatou coordinates for non-autonomous perturbations
  10. The coordinate $\psi$
  11. The coordinate $\phi$
  12. Proof of Theorem \ref{['th:main']}
  13. Entering the eggbeater
  14. Passing through the eggbeater
  15. Exiting the eggbeater
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $f$, $\{w_k^{(n)}\}$, and $\varepsilon_{k,n}$ be as in eq:int:f, eq:int:wj, and eq:ejn. Assume that the $(\sigma_{k,n})$ as in eq:ejn are uniformly bounded. Then locally uniformly on the parabolic basin $\mathcal{B}_f$, where the phase $u_n$ is given by

Theorems & Definitions (51)

  • Theorem 1.1
  • Corollary 1.2: Proposition A, astorg2014two
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • ...and 41 more