Basins of attraction in Loewner equations

一致非自治吸引盆

Uniform non-autonomous basins of attraction

本文证明一致吸引全纯映射族的吸引盆双全纯等价于复欧几里得空间，从而肯定地解决了双曲全纯自同构稳定流形双全纯型的猜想。

Examples of non-autonomous basins of attraction

The purpose of this paper is to present several examples of non--autonomous basins of attraction that arise from sequences of automorphisms of $\mathbb C^k$. In the first part, we prove that the non-autonomous basin of attraction arising from a pair of automorphisms of $\mathbb C^2$ of a prescribed form is biholomorphic to $\mathbb C^2$. This, in particular, provides a partial answer to a question raised in connection with Bedford's Conjecture about uniformizing stable manifolds. In the second part, we describe three examples of Short $\mathbb C^k$'s with specified properties. First, we show t…

Caratheodory convergence of immediate basins of attraction to a Siegel disk

Let f_n be a sequence of analytic functions in a domain U with a common attracting fixed point z_0. Suppose that f_n converges to f_0 uniformly on each compact subset of U and that z_0 is a Siegel point of f_0. We establish a sufficient condition for the immediate basins of attraction $\mathcal A^*(z_0,f_n,U)$ to form a sequence that converges to the Siegel disk of f_0 as to the kernel with respect to z_0. The same condition is shown to imply convergence of the Koenings functions associated with f_n to that of f_0. Our method allows us also to obtain a kind of quantitative result for analytic …

Non-Autonomous Basins of Attraction With 4-Dimensional Boundaries

We study whether the basin of attraction of a sequence of automorphisms of $\mathbb{C}^k$ is biholomorphic to $\mathbb{C}^k$. In particular we show that given any sequence of automorphisms with the same attracting fixed point, the basin is biholomorphic to $\mathbb{C}^k$ if the maps are repeated often enough. We also construct Fatou-Bieberbach domains whose boundaries are 4-dimensional.

Remarks on Automorphisms of $\mathbb{C}^* \times \mathbb{C}^*$ and their basins

We study basins of attraction of automorphisms of $\CC^2$ tangent to the identity that fix both axes. Our main result is that, if a well known conjecture about automorphisms of $\CC^*\times\CC^*$ holds, then there are no basins of attraction associated to the non-degenerate characteristic directions (in the sense of Hakim), and therefore we cannot find a Fatou-Bieberbach domain that does not intersect both axis with this method.

Partial Classification of Polynomials and an Orthonormal Basis Construction on the Associated Basin of Attraction

In the paper "Infinite product representations for kernels and iterations of functions", the authors associate certain Fatou subsets with reproducing kernel Hilbert spaces. They also present a method for constructing an orthonormal basis for said Hilbert space, but the method depends on the polynomial of the given Fatou set. We provide a partial classification of those polynomials the method applies to.

Tautness and Fatou Components in P^2

We investigate the tautness of invariant Fatou components for holomorphic endomorphisms of P^2. Previously, only basins of attraction were known to be taut. We show that two other kinds of recurrent Fatou components are taut. In the first of these cases, as well as for basins of attraction, we show that the Fatou components are in fact Kobayashi complete, which implies tautness.

A transcendental Hénon map with an oscillating wandering Short $\mathbb{C}^2$

Short $\mathbb{C}^2$'s were constructed in [F] as attracting basins of a sequence of holomorphic automorphisms whose rate of attraction increases superexponentially. The goal of this paper is to show that such domains also arise naturally as autonomous attracting basins: we construct a transcendental Hénon map with an oscillating wandering Fatou component that is a Short $\mathbb{C}^2$. The superexponential rate of attraction is not obtained at single iterations, but along consecutive oscillations.

牛顿法吸引盆地再审视

Newton's method's basins of attraction revisited

本文重新审视牛顿法收敛到根所需的混沌迭代次数，通过参数化的简单修改牛顿法展示收敛区域的复杂性。