Non-autonomous parabolic implosion
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arXiv
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| Format: | Preprint |
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2026
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| author | Astorg, Matthieu Bianchi, Fabrizio |
| author_facet | Astorg, Matthieu Bianchi, Fabrizio |
| contents | 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 $\mathbb P^2(\mathbb C)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27686 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-autonomous parabolic implosion Astorg, Matthieu Bianchi, Fabrizio Dynamical Systems Complex Variables 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 $\mathbb P^2(\mathbb C)$. |
| title | Non-autonomous parabolic implosion |
| topic | Dynamical Systems Complex Variables |
| url | https://arxiv.org/abs/2603.27686 |