Non-autonomous parabolic implosion
Abstract
We study parabolic implosion in a general non-autonomous setting. Let f(w)=w+w2+O(w3) be a holomorphic germ tangent to the identity. We consider the iteration of non-autonomous perturbations of the form wj+1=f(wj)+εj,n2. We show that, when the εj,n2's satisfy a Lavaurs-type condition, the element wn can be described by means of a suitable Lavaurs map Lun, whose phase un is an explicit function of the perturbation parameters. In particular, whenever un→ u∈ ℂ, the non-autonomous dynamics converges locally uniformly on compact subsets of the parabolic basin to the corresponding Lavaurs map Lu. 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 ℙ2(ℂ).
Related
- Formation of quasi-singularities of shock and implosion type for compressible Euler flows
- Parabolic Bogomolov--Gieseker Inequality and Pluriharmonic Metrics on Regular Parabolic Higgs Sheaves over Compact Kähler Manifolds
- Well-posedness of 2D stochastic parabolic-parabolic Keller-Segel system with logistic source
- Completely parabolic surfaces are Veech
- Geometric analysis of a mixed elliptic-parabolic conformally invariant boundary value problem
BibTeX
@misc{astorg2026nonautonomous,
title = {{Non-autonomous parabolic implosion}},
author = {Matthieu Astorg and Fabrizio Bianchi},
howpublished = {HAL},
year = {2026},
month = aug,
eprint = {2603.27686},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05733139v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Non-autonomous parabolic implosion
AU - Matthieu Astorg
AU - Fabrizio Bianchi
PY - 2026
DA - 2026/08/31
PB - HAL
UR - https://hal.science/hal-05733139v1
AB - 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)+ε_j,n^2. We show that, when the ε_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→ u∈ 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 ℙ^2(ℂ).
ER -