Meromorphic functions with three singular values

On permutable meromorphic functions

We study the class $\mathcal{M}$ of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in $\mathcal{M}$, with at least one essential singularity, permutes with a non-constant rational map $g$, then $g$ is a Möbius map that is not conjugate to an irrational rotation. For a given function $ f \in\mathcal{M}$ which is not a Möbius map, we show that the set of functions in $\mathcal{M}$ that permute with $ f $ is countably infinite. Finally, we show that there exist transcendental meromorphic functions $f: \mathbb{C} \to \mathbb{C}$ such tha…

某些光滑情形下局部 zeta 函数的亚纯延拓与非极点奇点

Meromorphic continuation and non-polar singularities of local zeta functions in some smooth cases

本文研究了特定非实解析光滑函数的局部 zeta 函数，证明了其在某些奇点处具有非极点奇点，并验证了特定光滑函数类下亚纯延拓下界估计的最优性。

On meromorphic mappings admitting an Algebraic Addition Theorem

A proper or singular abelian mapping from $C^n$ to $\bar\C^n$ is parametrized by $n$ meromorphic functions with at most $2n$ periods. We develop the existence and structure theorems of the classical theory of an abelian mapping purely on the basis of its defining functional equation, the so-called algebraic addition theorem (AAT), with no appeal to any representation as quotients of theta functions. We offer two new proofs of the periodicity of a nonrational mapping admitting an AAT. We also prove by new arguments the existence of a rational group law on an associated algebraic variety, and th…

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。

On the vanishing cycles of a meromorphic function on the complement of its poles

We study vanishing cycles naturally attached to a meromorphic function with isolated singularities, in both local and global settings.

Hayman T Directions of Meromorphic Functions in Some Angular Domains

This paper is devoted to investigate the singular directions of mero- morphic functions in some angular domains. We will confirm the existence of Hayman T directions in some angular domains. This is a continuous work of Yang [Yang L., Borel directions of meromorphic functions in an angular domain, Science in China, Math. Series(I)(1979), 149-163.] and Zheng [Zheng, J.H., Value Distribution of Meromorphic Functions, preprint.].

Covering properties of meromorphic functions, negative curvature and spherical geometry

Every nonconstant meromorphic function in the plane univalently covers spherical discs of radii arbitrarily close to arctan(sqrt 8) ~ 70^\circ 32'. If in addition all critical points of the function are multiple, then a similar statement holds with pi/2. These constants are the best possible. The proof is based on the consideration of negatively curved singular surfaces associated with meromorphic functions.

The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class

We show that for each $d\in (0,2]$ there exists a meromorphic function $f$ such that the inverse function of $f$ has three singularities and the Julia set of $f$ has Hausdorff dimension $d$.

Expansion of real valued meromorphic functions into Fourier trigonometric series

In the main part of the paper, on the basis of contour integration of complex meromorphic functions whose singularities lie onto an integration contour, in the first step, a concept of improper integrals absolute existence of meromorphic functions, as more general one with respect to the concept of improper integrals convergence (existence), is introduced into analysis. In the second step, in the case when a modulus of complex parameter tends to infinity, an interval of improper integrals convergence of parametric meromorphic functions is defined. In accordance with this, it is shown that the …

An extension theorem for separately meromorphic functions with pluripolar singularities

Let $D_j\subset\mathbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N. $$ Let $M\subset X$ be relatively closed. For any $j\in\{1,...,N\}$ let $Σ_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1})\times(A_{j+1}\times...\times A_N)$ such that the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb C^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $Σ_1,...,Σ_N$ are pluripolar. Put ${multline*} X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1…

Characterization of polyanalytic functions by meromorphic extensions into chains of circles

One-parameter smooth families of circles in the complex plane with the following property are described: a function is polyanalytic if and only if it has meromorphic extension inside any circle from the family, with the only singularity-a pole at the center.

Meromorphic functions share three values with their difference operators

In the work, we focus on a conjecture due to Z.X. Chen and H.X. Yi[1] which is concerning the uniqueness problem of meromorphic functions share three distinct values with their difference operators. We prove that the conjecture is right for meromorphic function of finite order. Meanwhile, a result of J. Zhang and L.W. Liao[10] is generalized from entire functions to meromorphic functions.

Large scaled geometry of Julia sets of entire and meromorphic functions

In this paper, we study the large scaled geometric structure of Julia sets of entire and meromorphic functions. Roughly speaking, the structure gives us some asymptotic information about the Julia set near the essential singularity. We will show that part of this structure are determined by the transcendental directions coming from function theoretic point of view.

Equicontinuous families of meromorphic mappings with values in compact complex surfaces

We prove that a family of meromorphic mappings from a bidisc to a compact complex surface, which are equicontinuous in a neighborhood of the boundary of the bidisc, has the volumes of its graphs locally uniformly bounded.

关于加权值分担与亚纯函数唯一性的进一步研究

Further Investigations on Weighted Value Sharing and Uniqueness of Meromorphic Functions

本文探讨了两个非常数亚纯函数以指定权重二分担一个值时的不同结果，深化了对加权值分担与亚纯函数唯一性问题的研究。

Shared values of meromorphic functions on a compact Riemann surface

We prove a new bound on the number of shared values of distinct meromorphic functions on a compact Riemann surface, explain a mistake in a previous paper on this topic, and give a survey of related questions.

Value distribution and uniqueness for q-difference of meromorphic functions Sharing Two Sets

In this paper, we investigate the value distribution for linear q-difference polynomials of transcendental meromorphic functions of zero order which improves the results of Xu, Liu and Cao (\cite{Xu & Liu & Cao & 2015}). We also investigate the uniqueness of zero order meromorphic function with its q-difference operator sharing two sets with finite weight. Some examples have been exhibited which are relevant to the content of the paper.

Higher-order interlacing for matrix-valued meromorphic Herglotz functions

Scalar-valued meromorphic Herglotz-Nevanlinna functions are characterized by the interlacing property of their poles and zeros together with some growth properties. We give a characterization of matrix-valued Herglotz-Nevanlinna functions by means of a higher-order interlacing property. As an application we deduce a matrix version of the classical Hermite-Biehler Theorem for entire functions.

Bounded Geometry and Families of Meromorphic Functions with Two Asymptotic Values

In this paper we consider the topological class, denoted by AV2, of universal covering maps from the plane to the sphere with two removed points. We prove that an element f in AV2 with finite post-singular set is combinatorially equivalent to a meromorphic transcendental map g with constant Schwarzian derivative if any only if f satisfies a Bounded Geometry condition.

Slices of Parameter Space for Meromorphic Maps with Two Asymptotic Values

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work in [GK] on degree 2 rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further, is that, de…

Further results on normal families of meromorphic functions concerning shared values

In this paper, we prove two normality criteria for families of some functions concerning shared values, the results generalize those given by Hu and Meng. Some examples are given to show the sharpness of our results.

Meromorphic functions partially share three values with their difference operators

In this paper, we give a simple proof and strengthening of a uniqueness theorem of meromorphic functions which partially share 0, $\infty$ CM and 1 IM with their difference operators. Meanwhile, we partial solve a conjecture given by Chen-Yi in \cite{CY} and generalize some previous theorems in \cite{C, CX}.

Meromorphic functions with three radially distributed values

We consider transcendental meromorphic functions for which the zeros, 1-points and poles are distributed on three distinct rays. We show that such functions exist if and only if the rays are equally spaced. We also obtain a normal family analogue of this result.

Hausdorff dimension of escaping sets of meromorphic functions

We give a complete description of the possible Hausdorff dimensions of escaping sets for meromorphic functions with a finite number of singular values. More precisely, for any given $d\in [0,2]$ we show that there exists such a meromorphic function for which the Hausdorff dimension of the escaping set is equal to $d$. The main ingredient is to glue together suitable meromorphic functions by using quasiconformal mappings. Moreover, we show that there are uncountably many quasiconformally equivalent meromorphic functions for which the escaping sets have different Hausdorff dimensions.

Multiple values and uniqueness problem of meromorphic mappings sharing hypersurfaces

The purpose of this article is to deal with the multiple values and uniqueness problem of meromorphic mappings from $\mathbb{C}^{m}$ into the complex projective space $\mathbb{P}^{n}(\mathbb{C})$ sharing fixed and moving hypersurfaces. We obtain several uniqueness theorems which improve and extend some known results.

Multiple values and finiteness problem of meromorphic mappings sharing different families of moving hyperplanes

In this article, we show some uniqueness theorems for meromorphic mappings of $\C^n$ into the complex projective space $\pnc$ sharing different families of moving hyperplanes regardless of multiplicites, where all intersecting points between these mappings and moving hyperplanes with multiplicities more than a certain number do not need to be counted.

Value sharing of meromorphic functions and Fang's problem

In this paper, we shall study the uniqueness problems on meromorphic functions sharing a polynomial. We give a complete answer to a problem posed by Fang Mingliang. Our results improve or generalize those given by Fang and Hua, Yang and Hua, Fang, Fang and Qiu, Lin and Yi, Zhang, Xu, et al.

An index formula in connection with meromorphic approximation

Let $Φ$ be a continuous $n\times n$ matrix-valued function on the unit circle $\T$ such that the $(k-1)$th singular value of the Hankel operator with symbol $Φ$ is greater than the $k$th singular value. In this case, it is well-known that $Φ$ has a unique superoptimal meromorphic approximant $Q$ in $H^{\infty}_{(k)}$; that is, $Q$ has at most $k$ poles in the unit disc $\mathbb{D}$ (i.e. the McMillan degree of $Q$ in $\mathbb{D}$ is at most $k$) and $Q$ minimizes the essential suprema of singular values $s_{j}((Φ-Q)(ζ))$, $j\geq0$, with respect to the lexicographic ordering. For each $j\geq 0$…

Wedge extendability of CR-meromorphic functions: the minimal case

In this article, we consider metrically thin singularities E of the solutions of the tangential Cauchy-Riemann operators on a C^{2,a}-smooth embedded Cauchy-Riemann generic manifold M (CR functions on M - E) and more generally, we consider holomorphic functions defined in wedgelike domains attached to M - E. Our main result establishes the wedge- and the L^1-removability of E under the hypothesis that the (\dim M-2)-dimensional Hausdorff volume of E is zero and that M and M\backslash E are globally minimal. As an application, we deduce that there exists a wedgelike domain attached to an everyw…

解析函数与其一阶导数部分共享值的研究

Meromorphic functions that partially share values with their first derivative

研究解析函数与其导数部分共享值的唯一性，提出四个部分共享值的结论。

Value-sharing of meromorphic functions on a Riemann surface

We present some results on two meromorphic functions from S to the Riemann sphere sharing a number of values where S is a Riemann surface of one of the following types: compact, compact minus finitely many points, the unit disk, a torus, the complex plane.

共享值对的代数曲线与亚纯函数

Algebraic curves and meromorphic functions Sharing pairs of values

针对共享五对值（一对计重数）的非 Moebius 变换亚纯函数 4IM+1CM 问题，证明其解对应仅含五参数的零亏格低次代数曲线，为计算机代数求解奠定基础。

黎曼曲面上亚纯联络的平坦结构

Flat structure of meromorphic connections on Riemann surfaces

本文研究紧黎曼曲面上亚纯联络的无穷自相交测地线的 omega 极限集，建立其与亚纯 k-微分和奇异平坦度量的关系，并证明在特定单值群条件下的一类 Poincaré-Bendixson 定理。

Singularities of inverse functions

A survey of general results on the singularities of inverses to meromorphic functions is given, with applications to holomorphic dynamics. This is a lecture delivered at the workshop "The role of complex analysis in complex dynamics" in Edinburgh on May 22 2013, with corrected mistakes and updated references.

亚纯函数的幂与其在 C^m 中的 k 阶方向导数共享值的性质

Power of a meromorphic function sharing value with its k-th order directional derivative in C^m

本文研究了多复变 C^m 中亚纯函数的幂与其 k 阶方向导数共享值的唯一性问题，将单变量唯一性定理推广至高维，并通过实例证明结果的最佳性。

Value distribution of q-differences of meromorphic functions in several complex variables

In this paper, we study $q$-difference analogues of several central results in value distribution theory of several complex variables such as $q$-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura-Clunie theorem concerning partial $q$-difference polynomials is also obtained. Finally, we apply this theory to investigate the growth of meromorphic solutions of linear partial $q$-difference equations.

Formal meromorphic functions on manifolds of finite type

It is shown that a real-valued formal meromorphic function on a formal generic submanifold of finite Kohn-Bloom-Graham type is necessarily constant.

Linear Delay-Differential operator of a meromorphic function sharing two sets or small function together with values with its $c$-shift or $q$-shift

The paper is devoted to study the uniqueness problem of linear delay-differential operator of a meromorphic function sharing two sets or small function together with values with its $c$-shift and $q$-shift operator. Results of this paper drastically improve two recent results of Meng-Liu [J. Appl. Math. Inform. 37(1-2)(2019), 133-148] and Qi-Li-Yang [Comput. Methods Funct. Theo., 18(2018), 567-582]. In addition to this, one of our result improve and extend that of Qi-Yang [Comput. Methods Funct. Theo., 20(2020), 159-178].

Class of meromorphic functions partially shared values with their differences or shifts

Two meromorphic functions $ f $ and $ g $ are said to share a value $ s\in\mathbb{C}\cup\{\infty\} $ $ CM $ $ (IM) $ provided that $ f(z)-s $ and $ g(z)-s $ have the same set of zeros counting multiplicities (ignoring multiplicities). We say that a meromorphic function $ f $ share $ s\in\mathbb{C} $ partially $ CM $ with a meromorphic function $ g $ if $ E(s,f)\subseteq E(s,g)$. It is easy to see that the condition ``partially shared values $ CM $" is more general than the condition ``shared value $ CM $". With the idea of partially shared values, in this paper, we prove some uniqueness result…

偏微分方程中Borel和Nevanlinna例外值的亚纯解

Meromorphic Solutions of Difference Equations Involving Borel and Nevanlinna Exceptional Values

本文研究了差分方程中亚纯函数与其线性差分算子共享值的问题，证明了差分方程L_c^n(f)≡Af的亚纯解存在并给出了显式解，同时探讨了二阶差分方程的有理和超越亚纯解的存在性。

Remark on meromorphic functions that share five pairs

We determine all pairs $(f,g)$ of meromorphic functions that share four pairs of values $(a_ν,b_ν)$, $1\leν\le 4$, and a fifth pair $(a_5,b_5)$ under some mild additional condition.

Normality from one family of meromorphic functions to another through sharing of values

Let F and G be two families of meromorphic functions on a domain D, and let a, b and c be three distinct points in the extended complex plane. Let G be a normal family in D such that all limit functions of G are non-constant. If for each f in F, there exists g in G such that f and g share a, b and c partially, then F is normal in D. This gives a sharp improvement of a result due to X. J. Liu, S. H. Li and X. C. Pang. We also prove some interesting related sharp results.

Degenerate abelian function fields

Originally, an abelian function field is the field of meromorphic functions on the Jacobi variety J(X) of a compact Riemann surface X. It is generated by the fundamental abelian functions belonging to the meromorphic function field on X. We study this relation for singular curves.

Meromorphic functions with linearly distributed values and Julia sets of rational functions

If the preimage of a four-point set under a meromorphic function belongs to the real line, then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if the Julia set of a rational function is contained in a smooth curve then it is contained in a circle.

Two meromorphic mappings sharing 2n + 2 hyperplanes regardless of multiplicity

Nevanlinna showed that two non-constant meromorphic functions on $\mathbb C$ must be linked by a Möbius transformation if they have the same inverse images counted with multiplicities for four distinct values. After that this results is generalized by Gundersen to the case where two meromorphic functions share two values ignoring multiplicity and share other two values with multiplicities trucated by 2. Previously, the first author proved that for $n\ge 2,$ there are at most two linearly nondegenerate meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ sharing $2n+2$ hyperplane…

Extension Properties of Meromorphic Mappings with Values in Non-Kahler Manifolds

We prove an analogue of E. Levi's Continuity Principle for meromorphic mappings with values in arbitrary compact complex manifolds in place of the Riemann sphere $\cc\pp^1$. The result is achieved by introducing a new extension method for meromorphic mappings. One of the corollaries reads as follows: If a compact complex surface $X$ is not "among the known ones" then for every domain $Ω$ in a Stein surface every meromorphic mapping $f:Ω\to X$ is in fact holomorphic and extends as a holomorphic mapping $\hat f:\hat D\to X$ of the envelope of holomorphy $\hat D$ of $D$ into $X$. In this last ver…

Uniqueness of Meromorphic functions concerning k-th derivatives and difference operators

In this paper, we continue to study the sharing value problems for higher order derivatives of meromorphic functions with its linear difference and $q$-difference operators. Some of our results generalize and improve the results of Meng--Liu (J. Appl. Math. and Informatics, 37(2019), 133--148) to a large extent.

双曲面积方法在亚纯动力学与椭圆多项式斜积中的应用

Hyperbolic Area Methods in Meromorphic Dynamics and Elliptic Polynomial Skew Products

本文用双曲面积与覆盖几何统一证明并推广了无游荡定理，覆盖亚纯映射、整函数及椭圆多项式斜积，并给出高维局部与全局无游荡准则。

Walter Bergweiler 在亚纯函数值分布方面的工作

The work of Walter Bergweiler in value distribution of meromorphic functions

本文介绍Walter Bergweiler退休学术报告，概述其在亚纯函数值分布中关于Picard定理推广和Bloch原理应用的贡献，并简要引入Nevanlinna理论。

Meromorphic functions of one complex variable. A survey

This is an appendix to the English translation of the book by A. A. Goldberg and I. V. Ostrovskii, Distribution of values of meromorphic functions, Moscow, Nauka, 1970. An English translation of this book is to be published soon by the AMS. In this appendix we survey the results obtained on the topics of the book after 1970.

Omitted values and dynamics of meromorphic functions

Let $M$ be the class of all transcendental meromorphic functions $f: \mathbb{C} \to \mathbb{C} \bigcup\{\infty\}$ with at least two poles or one pole that is not an omitted value, and $M_o =\{f \in M:f{has at least one omitted value}\}$. Some dynamical issues of the functions in $M_o$ are addressed in this article. A complete classification in terms of forward orbits of all the multiply connected Fatou components is made. As a corollary, it follows that the Julia set is not totally disconnected unless all the omitted values are contained in a single Fatou component. Non-existence of both Baker…

关于Gary G. Gundersen所提出的问题：关于共分享三个不同值IM和一个第四值CM的亚纯函数

On a question of Gary G. Gundersen concerning meromorphic functions sharing three distinct values IM and a fourth value CM

本文证明了两个亚纯函数共享三个值IM和一个值CM时，必然共享所有四个值CM，并解决了Gundersen提出的关于亚纯函数不存在共享三个不同值DM和一个值CM的问题。

An extension of Picard's theorem for meromorphic functions of small hyper-order

A version of the second main theorem of Nevanlinna theory is proved, where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper-order. As a corollary of this result it is shown that if three distinct values of a meromorphic function f of hyper-order less than 1/n^2 have forward invariant pre-images with respect to a fixed branch of the algebraic function t(z)=z+a_{n-1} z^{1-1/n}+...+a_1 z^{1/n}+a_0 with constant coefficients, then f(t(z)) = f(z) for all z. This is a generalization of Picard's theorem for meromorphic fu…

On the uniqueness of power of a meromorphic function sharing a set with its k-th derivative

In the existing literature, many researchers consider the uniqueness of the power of a meromorphic function with its derivative counterpart share certain values or small functions. Here we consider the same problem under the aegis of a more general settings namely set sharing