arXiv math/03092532003-09-16math.CVmath.DS

Invariant Nonrecurrent Fatou Components of Automorphisms of $C^2$

Daniel Jupiter, Krastio Lilov

英文摘要：

We examine invariant nonrecurrent Fatou components of automorphisms of $\mathbb{C}^2$ in the case where all limit maps are constant. We show that except in special cases there cannot be more than one such limit map. We also briefly examine such Fatou components where the limit maps may be nonconstant. Lastly we present a few examples of such Fatou components.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Fatou-Bieberbach domains as basins of attraction of automorphisms tangent to the identity

We prove that there exists an automorphism of C^2 tangent to the identity with a domain of attraction to the origin, biholomorphic to the origin, along a degenerate characteristic direction.

2009-07-12math.CVmath.DS

Invariant escaping Fatou components with two rank 1 limit functions for automorphisms of $\mathbb{C}^2$

We construct automorphisms of $\mathbb{C}^2$, and more precisely transcendental Hénon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank 1. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form $F(z,w)=(g(z,w),z)$ with $g(z,w):\mathbb{C}^2\rightarrow\mathbb{C}$ holomorphic.

2020-11-05math.CVmath.DS

Periodic cycles of attracting Fatou components of type $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ in automorphisms of $\mathbb{C}^{d}$

We generalise a recent example by F. Bracci, J. Raissy and B. Stensønes to construct automorphisms of $\mathbb{C}^{d}$ admitting an arbitrary finite number of non-recurrent Fatou components, each biholomorphic to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ and all attracting to a common boundary fixed point. These automorphisms can be chosen such that each Fatou component is invariant or such that the components are grouped into periodic cycles of any common period. We further show that no orbit in these attracting Fatou components can converge tangent to a complex submanifold, and that every sta…

2020-02-07math.CVmath.DS

Automorphisms of $\mathbb C^k$ with an invariant non-recurrent attracting Fatou component biholomorphic to $\mathbb C\times (\mathbb C^\ast)^{k-1}$

We prove the existence of automorphisms of $\mathbb C^k$, $k\ge 2$, having an invariant, non-recurrent Fatou component biholomorphic to $\mathbb C \times (\mathbb C^\ast)^{k-1}$ which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. Such a Fatou component also avoids $k$ analytic discs intersecting transversally at the fixed point. As a corollary, we obtain a Runge copy of $\mathbb C \times (\mathbb C^\ast)^{k-1}$ in $\mathbb C^k$.

2019-01-03math.CVmath.DS

Automorphisms of $\mathbb{C}^m$ with bounded wandering domains

We prove that the Euclidean ball can be realized as a Fatou component of a holomorphic automorphism of $\mathbb{C}^m$, in particular as the escaping and the oscillating wandering domain. Moreover, the same is true for a large class of bounded domains, namely for all bounded regular open sets $Ω\subset \mathbb{C}^m$ whose closure is polynomially convex. Our result gives in particular the first example of a bounded Fatou component with a smooth boundary in the category of holomorphic automorphisms.

2020-11-16math.CVmath.DS

Dynamics of Rational Surface Automorphisms: Rotation Domains

We consider rational surface automorphisms with positive entropy. A Fatou component is said to be a rotation domain if the automorphism induces a torus action on it. Here we construct a rational surface automorphism with positive entropy with the following property: it has a rotation domain which contains both a curve of fixed points and isolated fixed points. This Fatou component cannot be imbedded into complex euclidean space, so we introduce a global linear model space and show that it can be globally linearized in this model.

2009-07-20math.AGmath.CVmath.DS

更多相关论文收起更多论文

Punctured parabolic cylinders in automorphisms of $\mathbb{C}^{2}$

We show the existence of automorphisms $F$ of $\mathbb{C}^{2}$ with a non-recurrent Fatou component $Ω$ biholomorphic to $\mathbb{C}\times\mathbb{C}^{*}$ that is the basin of attraction to an invariant entire curve on which $F$ acts as an irrational rotation. We further show that the biholomorphism $Ω\to\mathbb{C}\times\mathbb{C}^{*}$ can be chosen such that it conjugates $F$ to a translation $(z,w)\mapsto(z+1,w)$, making $Ω$ a parabolic cylinder as recently defined by L.~Boc Thaler, F.~Bracci and H.~Peters. $F$ and $Ω$ are obtained by blowing up a fixed point of an automorphism of $\mathbb{C}…

2020-07-18math.CVmath.DS

射影K3曲面上的动力学：Herman环

Dynamics on projective K3 surfaces: Herman rings

本文构造了射影K3曲面上具有正拓扑熵和非空Fatou集的自同构，其不变域为Herman环的二维类比，解答了McMullen和Cantat的开放问题。

2026-09-30math.AGmath.CVmath.DS

Automorphisms of $\mathbb C^2$ with parabolic cylinders

A {\sl parabolic cylinder} is an invariant, non-recurrent Fatou component $Ω$ of an automorphism $F$ of $\mathbb C^2$ satisfying: (1) The closure of the $ω$-limit set of $F$ on $Ω$ contains an isolated fixed point, (2) there exists a univalent map $Φ$ from $Ω$ into $\mathbb C^2$ conjugating $F$ to the translation $(z,w) \mapsto (z+1, w)$, and (3) every limit map of $\{F^{\circ n}\}$ on $Ω$ has one-dimensional image. In this paper we prove the existence of parabolic cylinders for an explicit class of maps, and show that examples in this class can be constructed as compositions of shears and ove…

2020-02-19math.CVmath.DS

特征簇自同构群的动力学与 Painlevé 6 的 Fatou/Julia 分解

Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé 6

本文研究仿射三次曲面自同构群的动力学，定义并分析其Julia与Fatou集，证明Julia集中存在稠密轨道，并揭示Fatou/Julia二分法与局部离散性二分法的关联，为Painlevé 6方程及特征簇研究提供新视角。

2024-02-04math.CAmath.CVmath.DS

Polynomial shift--like maps in $\mathbb{C}^k$

The purpose of this article is to explore a few properties of polynomial shift-like automorphisms of $\mathbb{C}^k.$ We first prove that a $ν-$shift-like polynomial map (say $S_a$) degenerates essentially to a polynomial map in $ν-$dimensions as $a \to 0.$ Secondly, we show that a shift-like map obtained by perturbing a hyperbolic polynomial (i.e., $S_a$, where $|a|$ is sufficiently small) has finitely many Fatou components, consisting of basins of attraction of periodic points and the component at infinity.

2018-09-30math.CVmath.DS

Emergence of wandering stable components

We prove the existence of a locally dense set of real polynomial automorphisms of C 2 displaying a wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These Fatou components have non-empty real trace and their statistical behavior is historical with high emergence. The proof is based on a geometric model for parameter families of surface real mappings. At a dense set of parameters, we show that the dynamics of the model displays a historical, high emergent, stable domain. We show that this model can be embedded into fami…

2022-03-18math.CVmath.DS

关于$\boldsymbol{\text{C}^2}$中常雅可比剪切群映射的秩一极限集

On the rank-one limit sets for maps in the constant-Jacobian shear group of $\mathbb{C}^2$

Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）

该研究针对$\text{C}^2$中常雅可比剪切群映射，确定无限阶诱导动力学下Hénon映射有限复合的秩一极限集共形类型，排除三类类型并证明其不动点集无整曲线。

2026-10-14math.CVmath.DS