Entire functions with prescribed singular values

Recurrence of entire transcendental functions with simple post singular sets

We study how the orbits of the singularities of the inverse of a meromorphic function prescribe the dynamics on its Julia set, at least up to a set of (Lebesgue) measure zero. We concentrate on a family of entire transcendental functions with only finitely many singularities of the inverse, counting multiplicity, all of which either escape exponentially fast or are pre-periodic. For these functions we are able to decide, whether the function is recurrent of not. In the case that the Julia set is not the entire plane we also obtain estimates for the measure of the Julia set.

具有任意慢受控逃逸速率的Baker域

Baker domains with arbitrarily slow controlled rate of escape

本文构造了一个具有Baker域的超越整函数，其迭代以任意慢但规则的方式受控趋向无穷，证明基于共形映射畸变定理和逼近定理。

Direct singularities and completely invariant domains of entire functions

Let f be a transcendental entire function that omits a complex value a. We show that for every simply connected region D that does not contain a the full preimage of D is disconnected. We conjecture that the same holds if one only assumes that a is omitted locally. We were able to prove this conjecture under the additional assumption that f is of finite order. We include some auxilliary results on the singularities of the inverses of entire functions which are of independent interest.

$L^{1}$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds

Given $(X,ω)$ compact Kähler manifold and $ψ\in\mathcal{M}^{+}\subset PSH(X,ω)$ a model type envelope with non-zero mass, i.e. a fixed potential determing some singularities such that $\int_{X}(ω+dd^{c}ψ)^{n}>0$, we prove that the $ψ-$relative finite energy class $\mathcal{E}^{1}(X,ω,ψ)$ becomes a complete metric space if endowed with a distance $d$ which generalizes the well-known $d_{1}$ distance on the space of Kähler potentials. Moreover, for $\mathcal{A}\subset \mathcal{M}^{+}$ total ordered, we equip the set $X_{\mathcal{A}}:=\bigsqcup_{ψ\in\overline{\mathcal{A}}}\mathcal{E}^{1}(X,ω,ψ)$ …

单连通的小阶整函数的游荡域

Simply Connected Wandering Domains of Small Order Entire Functions

该研究展示了如何构造具有特定阶数的整函数，使其拥有有界单连通游荡域，并提供了首个此类函数的例子。

On metrics of constant positive curvature with four conic singularities on the sphere

We show that for given four points on the sphere and prescribed angles at these points, which are not multiples of $2π$, the number of metrics of curvature 1 having conic singularities with these angles at these points is finite.