Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parabolic implosion in dimension 2

Matthieu Astorg, Lorena López-Hernanz, Jasmin Raissy

Abstract

In this paper, we extend the theory of parabolic implosion in complex dimension 2 to the case of holomorphic maps tangent to the identity at order 2. We investigate the bifurcation phenomena that occur when a fully parabolic fixed point is perturbed. Under the assumption of a non-degenerate characteristic direction with a formal invariant curve and director $α$ satisfying $\reα> 2$, we establish the existence of Lavaurs maps as limits of iterates $f_{ε_n}^n$ for specific sequences of the perturbation parameter $ε_n$. Finally, we apply these results to prove the discontinuity of the Julia sets $J_1$ and $J_2$ for holomorphic endomorphisms of $\mathbb{P}^2$, generalizing classical one-dimensional results to this higher-dimensional setting.

Parabolic implosion in dimension 2

Abstract

In this paper, we extend the theory of parabolic implosion in complex dimension 2 to the case of holomorphic maps tangent to the identity at order 2. We investigate the bifurcation phenomena that occur when a fully parabolic fixed point is perturbed. Under the assumption of a non-degenerate characteristic direction with a formal invariant curve and director satisfying , we establish the existence of Lavaurs maps as limits of iterates for specific sequences of the perturbation parameter . Finally, we apply these results to prove the discontinuity of the Julia sets and for holomorphic endomorphisms of , generalizing classical one-dimensional results to this higher-dimensional setting.

Paper Structure

This paper contains 11 sections, 34 theorems, 304 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['th:v1']} and \ref{['th:v2']} from Theorem \ref{['th:explicit']}
  3. Asymptotics of Fatou coordinates
  4. Approximate Fatou coordinates and estimation of the error terms
  5. Estimation of the error terms
  6. Controlling the orbit
  7. Approaching the eggbeater
  8. In the eggbeater
  9. After the eggbeater
  10. Conclusion
  11. Proof of Corollaries \ref{['coro:discbigjulia']} and \ref{['coro:endo']}

Key Result

Theorem 1

Let $(\varepsilon_n)_{n \in \mathbb{N}}$ be a sequence of complex numbers, and $\sigma \in \mathbb{C}$. Assume that $\lim_{n \rightarrow \infty} \left(n-\pi/\varepsilon_n\right)= \sigma$. Then $f_{\varepsilon_n}^n \to \mathcal{L}_\sigma$ locally uniformly on $\mathcal{B}$.

Theorems & Definitions (73)

  • Theorem : Lavaurs, lavaurs1989systemes
  • Theorem 1: Non-technical version
  • Proposition 1
  • Remark 1
  • Theorem 2: Coordinate-free version
  • Theorem 3: Coordinate version
  • Corollary 1: Compare to bianchi19parabolic
  • Corollary 2
  • Lemma 2.1
  • proof
  • ...and 63 more