Non-autonomous parabolic implosion

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