Tien-Yu Peter Chern
It is proved that for any positive number $λ$, $1<λ<2$; there exists a meromorphic function $f$ with logarithmic order $λ$= $\displaystyle\limsup_{r\to+\infty}\frac{\log T(r,f)}{\log\log r}$ such that $f$ has no Julia directions, where $T(r,f)$ is the Nevanlinna characteristic function of $f$. (Note that A. Ostrowski has proved a {\it similar} result for $λ=2$ in 1926.)
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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 $d$.
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.
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.
Bounded Fatou and Julia components of meromorphic functions
本文完整刻画亚纯函数中有界 Fatou 分支与 Julia 分支的拓扑特征，并借助逼近理论构造含游荡连续统的亚纯函数给出证明。
A definition of hyperbolic meromorphic functions is given and then we discuss the dynamical behavior and the thermodynamic formalism of hyperbolic functions on the Julia set. We prove the important expanding properties for hyperbolic functions on the complex plane and with respect to the euclidean metric. We establish the Bowen formula for hyperbolic functions on the complex plane, that is, the Poincare exponent equals to the Hausdorff dimension of the radial Julia set and furthermore, we prove that all the results in the Walters' theory hold for hyperbolic functions on the Riemann sphere.
We study transcendental meromorphic functions having two prepole asymptotic values and no critical values. We prove that these functions acting on their Julia sets are non-ergodic, which illustrates the antithesis of the Keen-Kotus result in [KK2] on the ergodicity of another subfamily of functions with two asymptotic values and no critical values.
The Fatou-Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares many properties with those for transcendental meromorphic functions and for quasiregular mappings of punctured space.
We survey the definition of the radial Julia set of a meromorphic function (in fact, more generally, any "Ahlfors islands map"), and give a simple proof that the Hausdorff dimension of the reduced Julia set always coincides with the hyperbolic dimension.
We show that if a meromorphic function has two completely invariant Fatou components and only finitely many critical and asymptotic values, then its Julia set is a Jordan curve. However, even if both domains are attracting basins, the Julia set need not be a quasicircle. We also show that all critical and asymptotic values are contained in the two completely invariant components. This need not be the case for functions with infinitely many critical and asymptotic values.
We show that for any transcendental meromorphic function $f$ there is a point $z$ in the Julia set of $f$ such that the iterates $f^n(z)$ escape, that is, tend to $\infty$, arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which $f^n(z)$ tends to $\infty$ at a bounded rate, and establish the connections between these sets and the Julia set of $f$. To do this, we show that the iterates of $f$ satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a plane-filli…
In this note, we give an elementary proof of the absence of invariant line fields on the conical Julia set of an analytic function of one variable. This proof applies not only to rational as well as transcendental meromorphic functions (where it was previously known), but even to the extremely general setting of Ahlfors islands maps as defined by Adam Epstein. In fact, we prove a more general result on the absence of invariant_differentials_, measurable with respect to a conformal measure that is supported on the (unbranched) conical Julia set. This includes the study of cohomological equation…
The characterization and properties of Julia sets of one parameter family of transcendental meromorphic functions $ζ_λ(z)=λ\frac{z}{z+1} e^{-z}$, $λ>0$, $z\in \mathbb{C}$ is investigated in the present paper. It is found that bifurcations in the dynamics of $ζ_λ(x)$, $x\in {\mathbb{R}}\setminus \{-1\}$, occur at several parameter values and the dynamics of the family becomes chaotic when the parameter $λ$ crosses certain values. The Lyapunov exponent of $ζ_λ(x)$ for certain values of the parameter $λ$ is computed for quantifying the chaos in its dynamics. The characterization of the Julia set …
We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman-Valiron theory.
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", …
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…
The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. The Julia-Carathéodory theorem on the bidisk states that if the Julia quotient is bounded along some sequence of nontangential approach to some point in the torus, the function must have directional derivatives in all directions pointing into the bidisk. The directional derivative, however, need not be a linear function of the direction in that case. In this note, we show that if the Julia quotient is uniformly bounded along every sequence of nontangential approach, th…
Julia directions for holomorphic curves
该研究证明了映入复射影簇的整全纯映射的Picard型定理，其具备局部特性可证Julia方向存在，同时举例说明相关Borel定理无局部对应结论，推进了复分析领域相关研究。
The Hausdorff dimension of escaping sets of meromorphic functions in the Speiser class
本研究证明Bergweiler和Kotus给出的Eremenko-Lyubich类半纯函数逃逸集Hausdorff维数上界在Speiser类中也精确，还构造了具指定Julia集与逃逸集维数的Speiser类半纯函数。
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.].
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。
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.
In this paper we study, for the first time, Julia limiting directions of quasiregular mappings in $\mathbb{R}^n$ of transcendental-type. First, we give conditions under which every direction is a Julia limiting direction. Along the way, our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a contribution to the inverse problem in $\mathbb{R}^3$ of determining which compact subsets of $S^2$ can give rise to Julia limiting directions. The methods here will require showing that certain sectorial …
Frequently hypercyclic meromorphic curves with slow growth
研究通过Nevanlinna理论和Oka原理，构造了在射影空间中具有缓慢增长的频繁超循环整函数，并揭示了在增长约束下频繁超循环性质的二元对立。
Not much is known about the dynamics outside the support of the maximal entropy measure $μ$ for holomorphic endomorphisms of $\mathbb{CP}^k$. In this article we study the structure of the dynamics on the Julia set, which is typically larger than $Supp(μ)$. The Julia set is the support of the so-called Green current $T$, so it admits a natural filtration by the supports of the exterior powers of $T$. For $1\leq q \leq k$, let $J_q= Supp(T^q)$. We show that for a generic point of $J_q\setminus J_{q+1}$ there are at least $(k-q)$ "Fatou directions" in the tangent space. We also give estimates for…
Phragmén-Lindelöf Principles and Julia Limiting Directions of Quasiregular Mappings
本文研究超越型拟正则映射的 Julia 极限方向集结构，利用允许 $O(\log |x|)$ 边界增长的新型 Phragmén-Lindelöf 原理，建立其与维数、伸缩率和增长阶的关系。
In this paper, we have investigated the uniqueness problems of entire and meromorphic functions concerning differential polynomials sharing a small function. Our results radically extended and improved the results of Bhoosnurmath-Pujari and Harina - Anand not only by sharing small function instead of fixed point but also reducing the lower bound of $ n $. The authors Harina-Anand made plenty of mistakes in their paper. We have corrected all of them in a more convenient way. At last some open questions have been posed for further study in this direction.
A General Characterization on the Uniqueness Problem of L-Functions and General Meromorphic Functions
本文提出L-函数和一般整函数唯一性问题的通用标准，扩展了现有结果并提供了广泛应用的证明。
We show that, if the Julia set of a transcendental entire function is locally connected, then it takes the form of a spider's web in the sense defined by Rippon and Stallard. In the opposite direction, we prove that a spider's web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider's web.
We consider the family of all meromorphic functions $f$ of the form $$ f(z)=\frac{1}{z}+b_0+b_1z+b_2z^2+\cdots $$ analytic and locally univalent in the puncture disk $\mathbb{D}_0:=\{z\in\mathbb{C}:\,0<|z|<1\}$. Our first objective in this paper is to find a sufficient condition for $f$ to be meromorphically convex of order $α$, $0\le α<1$, in terms of the fact that the absolute value of the well-known Schwarzian derivative $S_f (z)$ of $f$ is bounded above by a smallest positive root of a non-linear equation. Secondly, we consider a family of functions $g$ of the form $g(z)=z+a_2z^2+a_3z^3+\c…
Uniqueness of first derivatives and differences in meromorphic functions and the characterization of entire function periodicity
本文研究亚纯函数一阶导数与差分分担值的唯一性，解决Qi等人开放问题，并给出超越整函数周期性的充分条件。
We study the asymptotic behavior of Pick functions, analytic functions which take the upper half plane to itself. We show that if a two variable Pick function $f$ has real residues to order $2N-1$ at infinity and the imaginary part of the remainder between $f$ and this expansion is of order $2N+1,$ then $f$ has real residues to order $2N$ and directional residues to order $2N+1.$ Furthermore, $f$ has real residues to order $2N+1$ if and only if the $2N+1$-th derivative is given by a polynomial, thus obtaining a two variable analogue of a higher order Julia-Carathéodory type theorem.
We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \in G$. In general, the Julia set of such a semigroup $G$ may be disconnected, and each Fatou component of such $G$ is either simply connected or doubly connected (\cite{Su01,Su9}). In this paper, we show that for any two distinct Fatou components of ce…
Ergodicity in some families of Nevanlinna Functions
本文研究具有 N 个渐近值且无临界值的 Nevanlinna 函数，证明当部分渐近值满足轨道聚点集上的排斥条件、其余为预极点时，函数仍在 Julia 集上遍历。
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。
Exceptional Sets of Positive Capacity in Gross's Star Theorem
本文证明Gross定理中的例外集不必有对数容度零，构造了超越整函数及其正则局部逆，使其例外集含正容度紧集，且对任意s<1，该紧集有正s维豪斯多夫测度。
The theory of 'zeta functions of fractal strings' has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called `distance zeta functions', which since then, has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta f…
Which Functions Admit a Positive Geometry? From Branch Cuts to String Amplitudes
本文探讨正几何中允许的函数，通过分支割线和弦幅度扩展了正几何的框架，引入伪亏格分类meromorphic函数，并展示d log形式与弦幅的正几何关系。
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