Parabolic implosion for endomorphisms of $\mathbb{C}^2$

TL;DR

This paper investigates how small changes in certain complex maps cause abrupt changes in their Julia sets, using advanced theorems and adapting existing strategies to two-dimensional cases.

Contribution

It extends the understanding of Julia set discontinuities to two-dimensional endomorphisms tangent to the identity, applying a Lavaurs Theorem and adapting Bedford, Smillie, and Ueda's approach.

Findings

Discontinuity of the large Julia set under perturbations.

Discontinuity of the filled Julia set for perturbed regular polynomials.

Application of a two-dimensional Lavaurs Theorem.

Abstract

We give an estimate of the discontinuity of the large Julia set for a perturbation of a class of maps tangent to the identity, by means of a two-dimensional Lavaurs Theorem. We adapt to our situation a strategy due to Bedford, Smillie and Ueda in the semiattracting setting. We also prove the discontinuity of the filled Julia set for such perturbations of regular polynomials.