Rigidity of escaping dynamics for transcendental entire functions
Lasse Rempe
备注：
28 pages; 2 figures. Final version (October 2008). Various modificiations were made, including the introduction of Proposition 3.6, which was not formally stated previously, and the inclusion of a new figure. No major changes otherwise
We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are *quasiconformally equivalent* in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane.
We also prove that this conjugacy is essentially unique. In particular, we show that an Eremenko-Lyubich class function f has no invariant line fields on its escaping set.
Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f and g which belong to the same parameter space are conjugate on their sets of escaping points.
In this paper, we have investigated the Bungee set of composition of two transcendental entire functions. We have provided a class of permutable entire functions for which their Bungee sets are equal. Moreover, we have obtained a result on permutable entire functions for which the escaping set of the composite entire function equals the union of the escaping sets of the two functions. In addition, we establish important relations between the Bungee set, Escaping set and Filled Julia set of composite entire functions with that of its individual functions.
A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this property, and this class has recently attracted much attention in complex dynamics. In the presence of escaping critical values, these curves break or split at (preimages of) critical points. In this paper, we develop combinatorial tools that allow us to provide a complete description of the escaping set of any criniferous function without asymptotic values on its…
A Study of Fatou Set, Julia set and Escaping Set in Conjugate Transcendental Semigroup
定义了超越半群的交换子，并在此基础上定义了共轭半群。证明了共轭半群是近阿贝尔的当且仅当给定的半群是近阿贝尔的。证明了换位子（仿射复共轭映射）下的逃逸集、朱利亚集和法图集的象分别等于共轭半群的逃逸集、朱利亚集和法图集。最后，证明了几乎交换半群$ S $的每个元素都可以表示为一个元素的合成，该元素来自于由生成子集$ Φ(S) $及其生成元的某些幂的合成所生成的集合.
We construct a transcendental entire $f:\mathbb{C}\rightarrow\mathbb{C}$ such that (1) $f$ has bounded singular set, (2) $f$ has a wandering domain, and (3) each singular value of $f$ escapes to infinity under iteration by $f$.
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.
Let $f$ be a transcendental entire function. The escaping set $I(f)$ consists of those points that tend to infinity under iteration of $f$. We show that $I(f)$ is not $σ$-compact, resolving a question of Rippon from 2009.
We introduce a new approximation technique into the context of complex dynamics that allows us to construct examples of transcendental entire functions with unbounded wandering domains. We provide examples of entire functions with an orbit of unbounded fast escaping wandering domains, answering a long-standing question of Rippon and Stallard. Moreover, these examples cover all possible types of simply connected wandering domains in terms of convergence to the boundary. In relation to a conjecture of Baker, it was unknown whether functions of order less than one could have unbounded wandering d…
