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Complex structures of non-autonomous basins
Abstract
We study non-autonomous basins of Fj automorphisms of ℂ2 satisfying aj|p|≤|Fj(p)|≤ bj|p| (∀ p∈ 𝔹) where 0<aj<bj<b<1. We show that if bjm<aj for some integer m and all j>0 then every bounded plurisubharmonic function on the non-autonomous basin is constant. Moreover, if in addition we have ∏i=1j−1bi≤ bjm then the basin is biholomorphic to ℂ2. For the quadratic Hénon family Fn(z,w)=(z2+cqnw,cqnz), 0<c<1, we study how the asymptotic behavior of the sequence qn→∞ influences the complex structure of the basin. In particular, we identify broad regimes in which the non-autonomous basin is either biholomorphic to ℂ2 or is a Short ℂ2, thereby isolating possible sequences for which other complex structures may occur.
Related
BibTeX
@misc{thaler2026complex,
title = {{Complex structures of non-autonomous basins}},
author = {Luka Boc Thaler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00627},
eprint = {2610.00627},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00627},
}
RIS
TY - UNPB
TI - Complex structures of non-autonomous basins
AU - Luka Boc Thaler
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00627
UR - https://arxiv.org/abs/2610.00627
AB - We study non-autonomous basins of F_j automorphisms of ℂ^2 satisfying a_j|p|≤|F_j(p)|≤ b_j|p| (∀ p∈ 𝔹) where 0<a_j<b_j<b<1. We show that if b_j^m<a_j for some integer m and all j>0 then every bounded plurisubharmonic function on the non-autonomous basin is constant. Moreover, if in addition we have ∏_i=1^j−1b_i≤ b_j^m then the basin is biholomorphic to ℂ^2. For the quadratic Hénon family F_n(z,w)=(z^2+c^q_nw,c^q_nz), 0<c<1, we study how the asymptotic behavior of the sequence q_n→∞ influences the complex structure of the basin. In particular, we identify broad regimes in which the non-autonomous basin is either biholomorphic to ℂ^2 or is a Short ℂ^2, thereby isolating possible sequences for which other complex structures may occur.
ER -