Non-autonomous parabolic implosion

TL;DR

This paper investigates non-autonomous parabolic implosion in complex dynamics, describing how perturbations influence the convergence to Lavaurs maps and analyzing the resulting Julia sets.

Contribution

It provides a general framework for understanding additive non-autonomous parabolic implosion, unifying previous results and extending to deterministic and random cases.

Findings

Non-autonomous dynamics converge to Lavaurs maps under Lavaurs-type conditions.

The phase of Lavaurs maps is explicitly determined by perturbation parameters.

Results include strong discontinuity of Julia sets for fibered holomorphic endomorphisms.

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 $\mathbb P^2(\mathbb C)$.