Parabolic implosion in dimension 2

TL;DR

This paper extends parabolic implosion theory to complex dimension 2, analyzing bifurcations at fixed points and demonstrating Julia set discontinuities in higher-dimensional holomorphic maps.

Contribution

It introduces the existence of Lavaurs maps in 2D and generalizes classical bifurcation and Julia set discontinuity results to higher dimensions.

Findings

Existence of Lavaurs maps as limits of iterates under perturbation.

Discontinuity of Julia sets $J_1$ and $J_2$ in $ ext{P}^2$ holomorphic endomorphisms.

Extension of parabolic implosion theory to dimension 2 with specific bifurcation phenomena.

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 $\alpha$ satisfying $\re\alpha> 2$, we establish the existence of Lavaurs maps as limits of iterates $f_{\epsilon_n}^n$ for specific sequences of the perturbation parameter $\epsilon_n$. 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.