Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.00627

Complex structures of non-autonomous basins

Luka Boc Thaler
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study non-autonomous basins of $F_j$ automorphisms of $\mathbb C^2$ satisfying $$a_j\|p\|\leq\|F_j(p)\|\leq b_j\|p\|\qquad (\forall p\in \mathbb B)$$ where $0 0$ then every bounded plurisubharmonic function on the non-autonomous basin is constant. Moreover, if in addition we have $\prod_{i=1}^{j-1}b_i\leq b_j^m$ then the basin is biholomorphic to $\mathbb C^2$. For the quadratic Hénon family $$F_n(z,w)=\bigl(z^2+c^{q_n}w,c^{q_n}z\bigr), \qquad 0<c<1,$$ we study how the asymptotic behavior of the sequence $q_n\to\infty$ influences the complex structure of the basin. In particular, we identify broad regimes in which the non-autonomous basin is either biholomorphic to $\mathbb C^2$ or is a Short $\mathbb C^2$, thereby isolating possible sequences for which other complex structures may occur.

Explore related subjects

math.CVmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Luka Boc Thaler. 2026-09-30. Complex structures of non-autonomous basins. https://arxiv.org/abs/2610.00627

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Complex Dirichlet problem with unbounded boundary values

We study the Dirichlet problems with unbounded boundary data. Our main theorem proves interior continuity from a local biholomorphic deformation satisfying a one-sided compatibility condition near the infinite locus and barriers on the associated boundary cluster set. The deformed domains need not be nested, and the proof uses neither a Runge hypothesis nor a pseudoconvex ambient neighbourhood. We also prove nonmonotone Perron--Bremermann convergence outside an intrinsic neighbourhood pluripolar hull, a disk-fibre hull representation, stability under simultaneous nonmonotone variations of the domain and obstacle, and uniqueness in a fixed obstacle class.

math.CV↗
arXiv2026

Baernstein's quasi-norm monotonicity conjecture for polynomials with unimodular zero

Let $m$ denote the normalized Haar measure on the unit circle $\mathbb{T}$. For $0<r<\infty$, define $\lVert f\rVert_{r}:=\left(\int_{\mathbb{T}}|f|^{r}\,\mathrm{d} m\right)^{1/r}$, with $\lVert f\rVert_{0}$ and $\lVert f\rVert_{\infty}$ interpreted as the geometric mean and the supremum norm, respectively. Set $Q_n(z)=1+z^n$. We prove that, for every nonzero polynomial $p$ of degree $n$ whose zeros all lie on $\mathbb{T}$, $$ \frac{\lVert p\rVert_s}{\lVert Q_n\rVert_s} \le \frac{\lVert p\rVert_t}{\lVert Q_n\rVert_t}, \qquad 0\le s\le t\le\infty. $$ This settles Baernstein's quasi-norm monotonicity conjecture. As corollaries, we obtain an $L^r$ extension of Visser's coefficient inequality, the sharp O'Hara--Rodriguez inequality and its higher-power analogues, the Erdős--Szekeres product bound $ \left\lVert\prod_{j=1}^N(1-z^{s_j})\right\rVert_\infty\ge2\sqrt N $ for all positive integers $s_1,\ldots,s_N$ and Agler--McCarthy's entropy conjecture. We also provide a Lean 4 formalization of the main results.

math.CV↗