Escaping points of commuting meromorphic functions with finitely many poles

亚纯函数逃逸集的豪斯多夫维数 II

Hausdorff dimension of escaping sets of meromorphic functions II

本文研究亚纯函数逃逸集的豪斯多夫维数，证明只需不超过4个奇异值即可实现区间[0,2]内的所有可能维数。

Dynamics of meromorphic functions outside a countable set of essential singularities

We consider a class of functions, denoted by K in this paper, which are meromorphic outside a compact and countable set B(f), investigated by A. Bolsch in his thesis in 1997. The set B(f) is the closure of isolated essential singularities. We review main definitions and properties of the Fatou and Julia sets of functions in class K. It is studied the role of B(f) in this context. Following Eremenko it is defined escaping sets and we prove some results related to them. For instance, the dynamics of a function is extended to its singularities using escaping hairs. We give an example of an escapi…

The exact value of Hausdorff dimension of escaping sets of class B meromorphic functions

We consider the subclass of class ${\mathcal B} $ consisting of meromorphic functions $f:{\mathbb C}\to\hat{\mathbb C}$ for which infinity is not an asymptotic value and whose all poles have orders uniformly bounded from above. This class was introduced in \cite{BwKo2012} and the Hausdorff dimension ${\rm HD}({{\mathcal I} (f)})$ of the set ${{\mathcal I} (f)}$ of all points escaping to infinity under forward iteration of $f$ was estimated therein. In this paper we provide a closed formula for the exact value of ${\rm HD}({{\mathcal I} (f)})$ identifying it with the critical exponent of the na…

On the Hausdorff dimension of the escaping set of certain meromorphic functions

The escaping set I(f) of a transcendental meromorphic function f consists of all points which tend to infinity under iteration. The Eremenko-Lyubich class B consists of all transcendental meromorphic functions for which the set of finite critical and asymptotic values of f is bounded. It is shown that if f is in B and of finite order of growth, if infinity is not an asymptotic value of f and if the multiplicities of the poles of f are uniformly bounded, then the Hausdorff dimension of I(f) is strictly smaller than 2. In fact, we give a sharp bound for the Hausdorff dimension of I(f) in terms o…

Hausdorff dimension of escaping sets of Nevanlinna functions

We determine the exact values of Hausdorff dimensions of escaping sets of meromorphic functions with polynomial Schwarzian derivatives. This will follow from the relation between these functions and the second order differential equations in the complex plane.

On the iterations and the argument distribution of meromorphic functions

This paper consists of tow parts. One is to study the existence of a point $a$ in the intersection of Julia set and escaping set such that $\arg z=θ$ is a singular direction if $θ$ is a limit point of $\{\arg f^n(a)\}$ under some growth condition of a meromorphic function. The other is to study the connection between the Fatou set and singular direction. We prove that the absent of singular direction deduces the non-existence of annuli in the Fatou set.

Dynamics of meromorphic functions with direct or logarithmic singularities

We show that if a meromorphic function has a direct singularity over infinity, then the escaping set has an unbounded component and the intersection of the escaping set with the Julia set contains continua. This intersection has an unbounded component if and only if the function has no Baker wandering domains. We also give estimates of the Hausdorff dimension and the upper box dimension of the Julia set of a meromorphic function with a logarithmic singularity over infinity. The above theorems are deduced from more general results concerning functions which have "direct or logarithmic tracts", …

On slow escaping and non-escaping points of quasimeromorphic mappings

We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.

Dimensions of non-autonomous meromorphic functions of finite order

In this paper we study two classes of meromorphic functions previously studied by Mayer, Kotus, and Urbański. In particular we estimate a lower bound for the Julia set and the set of escaping points for non-autonomous additive and affine perturbations of functions from these classes. For particular classes we are able to calculate these dimensions exactly. In these cases, we are able to reinterpret our results to show that the Hausdorff dimension of the set of points in the Julia set which escape to infinity is stable under sufficiently small additive perturbations. We accomplish this by const…

Escaping points in the boundaries of Baker domains

We study the dynamical behaviour of points in the boundaries of simply connected invariant Baker domains $U$ of meromorphic maps $f$ with a finite degree on $U$. We prove that if $f|_U$ is of hyperbolic or simply parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure escapes to infinity under iteration. On the contrary, if $f|_U$ is of doubly parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure has dense forward trajectory in the boundary of $U$, in particular the set of escaping points in the boundary of $U$ h…

两个可交换超越整函数的动力学集合相等

Equality of the dynamical sets of two commuting transcendental entire functions

本文研究了两个可交换的超越整函数f和g的动力学性质，其中g的形式为af^p + b，其中a,b∈C，p∈N，且a≠0,1。研究发现f和g的逃逸集、填充Julia集和蹦床集都相同。作为直接结果，我们得出f和g的Julia集相同。该定理扩展了Poon和Yang1998年的结果。此外，遵循Wang和Yang的研究，我们考虑了一个非常数多项式Q和可交换的整个函数f和g，满足关系Q(g)=aQ(f)+b，其中a≠0,1，b∈C。在更一般的情况下，我们证明了f和g的逃逸集、填充Julia集和蹦床集相等。

关于蹦床集的动力学研究：复合整个函数在Eremenko-Lyubich类中的结果

Results on Dynamics of Bungee set of Composite Entire Functions in the Eremenko-Lyubich Class

本文研究了复合整个函数在Eremenko-Lyubich类中的动力学特性，探讨了蹦床集、逃逸集和填充Julia集之间的关系，并通过实例验证了相关结论。

Permutable Quasiregular Maps

Let $f$ and $g$ be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$. We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form $f$ and $g=ϕ\circ f$, where $ϕ$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular …