转义集是蜘蛛网的整个函数

法图氏网

Fatou's web

设$f$为Fatou函数，即$f(z)= z+1+e^{-z}$。证明了$f$的逃逸集具有“蜘蛛网”的结构，并证明了这一结果意味着$f$的朱利亚集的非逃逸端点与无穷大一起构成一个全不连通的集.我们还给出了一个著名的超越整函数，由于Bergweiler，其中逃逸集是一个蜘蛛网，我们指出，同样的属性适用于家庭的功能。

Eremenko-Lyubich 类中的蜘蛛网

Spiders' webs in the Eremenko-Lyubich class

本文证明整函数 cosh(z) 的逃逸集乃至快速逃逸集均呈“蜘蛛网”结构，反驳并解答了相关猜想与问题。

蜘蛛网与超越整函数的locally connected朱利亚集

Spiders' webs and locally connected Julia sets of transcendental entire functions

我们表明，如果一个超越整函数的朱利亚集是局部连通的，那么它的形式的蜘蛛网的意义上定义的Rippon和Stallard。在相反的方向上，我们证明了蜘蛛网朱利亚集总是局部连通的一个稠密子集的掩埋点。我们还表明，一组埋点（剩余朱利亚集）可以是一个蜘蛛网。

Dynamical sets whose union with infinity is connected

Suppose that $f$ is a transcendental entire function. In 2014, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected, and an example a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class. It is known that for many transcendental entire functions the escaping set…

带蛛网的Poincaré函数

Poincaré functions with spiders' webs

对于具有排斥不动点w的多项式p，我们考虑p在w处的庞加莱函数，即对于复平面中的所有z，满足L（0）=w和p（L（z））=L（p '（w）*z）的整函数L。我们证明，如果p的包含w的朱利亚集的分量等于{w}，则L的（快速）逃逸集是一个蜘蛛网;特别是它是连通的。更确切地说，我们分类的所有线性化的多项式关于蜘蛛网结构的所有点的集合逃脱速度比迭代的最大模函数在一个足够大的点。

Non-escaping endpoints do not explode

The family of exponential maps $f_a(z)= e^z+a$ is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set $J(f_a)$. When $a\in (-\infty,-1)$, and more generally when $a$ belongs to the Fatou set of $f_a$, it is known that $J(f_a)$ can be written as a union of "hairs" and "endpoints" of these hairs. In 1990, Mayer proved for $a\in (-\infty,-1)$ that, while the set of endpoints is totally separated, its union with infinity is a connected set. Recently, Alhabib and the second author extended this result to t…

关于基本循环和快速逃逸集

On fundamental loops and the fast escaping set

超越整函数f的快速逃逸集A（f）已开始在超越动力学中起关键作用。在许多情况下，A（f）具有蜘蛛网的结构，其中包含一系列基本循环。我们研究了具有多连通Fatou分量的函数的基本环的结构，证明了存在超越整函数，其中一些基本环是解析曲线和近似圆，而另一些基本环是几何高度扭曲的.我们这样做，通过引入一个实值函数的措施逃逸率的点在A（f），并表明，这个功能有一些有趣的性质。

Eremenko points and the structure of the escaping set

Much recent work on the iterates of a transcendental entire function $f$ has been motivated by Eremenko's conjecture that all the components of the escaping set $I(f)$ are unbounded. Here we show that if $I(f)$ is disconnected, then the set $I(f)\setminus D$ has uncountably many unbounded components for any open disc $D$ that meets the Julia set of $f$. For the set $A_R(f)$, which is the `core' of the fast escaping set, we prove the much stronger result that for some $R>0$ either $A_R(f)$ is connected and has the structure of an infinite spider's web or it has uncountably many components each …

整函数的快速转义点

Fast escaping points of entire functions

设$f$是一个超越整函数，并设$A(f)$表示在迭代下“尽可能快地”逃逸到无穷大的点的集合。通过将$A(f)$写成闭集的可数并集，称为$A(f)$的“层”，我们对这个集合的结构有了新的理解。例如，我们表明，如果$U$是一个Fatou组件在$A(f)$，然后$\partial U\subset A(f)$，这将导致显着的新结果和相当大的改进，现有的结果约$A(f)$。特别是，我们研究的功能，其中$A(f)$，和它的每一个级别，有一个“无限的蜘蛛网”的结构。我们发现，有许多这样的功能，他们有一些强大的动力学性质。这个新的结构提供了一个意想不到的连接之间的猜想贝克关于组件的法图集和猜想Eremenko关于组件的逃逸集。

The topology of the set of non-escaping endpoints

There are several classes of transcendental entire functions for which the Julia set consists of an uncountable union of disjoint curves each of which joins a finite endpoint to infinity. Many authors have studied the topological properties of this set of finite endpoints. It was recently shown that, for certain functions in the exponential family, there is a strong dichotomy between the topological properties of the set of endpoints which escape and those of the set of endpoints which do not escape. In this paper, we show that this result holds for large families of functions in the Eremenko-…

Eremenko's conjecture for functions with real zeros: the role of the minimum modulus

We show that for many families of transcendental entire functions $f$ the property that $m^n(r)\to\infty$ as $n\to \infty$, for some $r>0$, where $m(r)=\min\{|f(z)|:|z|=r\}$, implies that the escaping set $I(f)$ of $f$ has the structure of a spider's web. In particular, in this situation $I(f)$ is connected, so Eremenko's conjecture holds. We also give new examples of families of functions for which this iterated minimum modulus condition holds and new families for which it does not hold.