Fethi Neji
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.
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The article is devoted to the study of mappings with finite distortion in metric spaces. Analogues of results relating to equicontinuity and normality of families of quasiregular mappings are obtained. It is proved that the indicated families are equicontinuous if the characteristic of the mappings has a finite mean oscillation at each inner point, and the maps omit a certain fixed continuum. An equicontinuity of generalized quasiisometries on Riemannian manifolds is also obtained.
In this paper, we extend the uniqueness theorem for meromorphic mappings to the case where the family of hyperplanes depends on the meromorphic mapping and where the meromorphic mappings may be degenerate.
In the present paper, we introduced the extended bicomplex plane $\bar{\mathbb{T}}$, its geometric model: the bicomplex Riemann sphere, and the bicomplex chordal metric that enables us to talk about the convergence of the sequences of bicomplex meromorphic functions. Hence the concept of the normality of a family of bicomplex meromorphic functions on bicomplex domains emerges. Besides obtaining a normality criterion for such families, the bicomplex analog of the Montel theorem for meromorphic functions and the Fundamental Normality Tests for families of bicomplex holomorphic functions and bico…
Meromorphically normal families and a meromorphic Montel-Carathéodory theorem
本文研究了取值于 $\mathbb{P}^n$ 的亚纯映射族的亚纯正规性，受 Fujimoto 工作启发给出了一般充分条件，并据此建立了近期全纯映射正规族相关结果的亚纯类似物及 Montel-Carathéodory 定理的亚纯版本。
In this paper we prove some normality criteria for a family of meromorphic functions concerning shared analytic functions, which extend or generalized some result obtained by Y. F. Wang, M. L. Fang~\cite{WF} and J. Qui, T. Zhu ~\cite{QZ}.
In this paper we prove some normality criteria for a family of meromorphic functions, which involves the zeros of certain differential polynomials generated by the members of the family.
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.
By using Nevanlinna theory, we prove some normality criteria for a family of meromorphic functions under a condition on differential polynomials generated by the members of the family.
In this article we prove a sufficient condition of quasi-normality in higher dimension for a family of meromorphic mappings in which each pair of functions of family shares some moving hypersurfaces. We also prove a normality criterion concerning shared moving hypersurfaces.
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.
In this paper, we prove a general second main theorem for meromorphic mappings into a subvariety $V$ of $\mathbb P^N(\mathbb C)$ with an arbitrary family of moving hypersurfaces. Our second main theorem generalizes and improves all previous results for meromorphic mappings with moving hypersurfaces, in particular for meromorphic mappings and families of moving hypersurfaces in subgeneral position. The method of our proof is different from that of previous authors used for the case of moving hypersurfaces.
The main aim of this article is to give some sufficient conditions for a family of meromorphic mappings on a domain D in C^n into P^N(C) to be meromorphically normal if they satisfy only some very weak conditions with respect to moving hypersurfaces in P^N(C), namely that their intersections with these moving hypersurfaces, which may moreover depend on the meromorphic maps, are in some sense uniform. Our results generalise and complete previous results in this area, especially the works of Fujimoto, Tu, Tu-Li, Mai-Thai-Trang and the recent work of Quang-Tan.
Minor corrections, to appear in Annali SNS.
We show that a family of meromorphic functions in the unit disk $\dk$ whose spherical derivatives are uniformly bounded away from zero is normal. Furthermore, we show that for each $f$ meromorphic in $\dk$ we have $\inf_{z\in\dk} f^#(z)\le \frac{1}{2}$$ where $f^#$ denotes the spherical derivative of $f$.
In this paper, we prove normality criteria for families of meromorphic functions involving sharing of a holomorphic function by a certain class of differential polynomials. Results in this paper extends the works of different authors carried out in recent years.
Meromorphic mappings into projective varieties intersecting arbitrary families of moving hypersurfaces
本文针对从C^m到P^n(C)的子簇V的亚纯映射，建立了关于任意缓慢移动超曲面族Q={Q_1,...,Q_q}的一般第二基本定理，不要求映射在K_Q域上代数非退化，并显式估计了计数函数的截断水平，给出了与映射f无关的总缺陷界Δ_{Q,V}(3dim V-1)。
We prove that along any marked point the Green function of a meromorphic family of polynomials parameterized by the punctured unit disk explodes exponentially fast near the origin with a continuous error term.
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.
We study branching mappings that satisfy some condition of distortion of the modulus of families of paths. In a situation where the definition domain of mappings is locally connected on its boundary, the mapped domain is regular, and the majorant responsible for distortion of the modulus of families of paths is integrable, it is proved that the families of all specified mappings with one normalization condition are equicontinuous in the closure of the given domain.
A remark concerning normal families and shared values
本文改进了关于整函数正常族和共享值的结果，证明了在特定条件下整函数的正常性。
We study mappings defined in the domain of a metric space that distort the modulus of families of paths by the type of the inverse Poletskii inequality. Under certain conditions, it is proved that such mappings have a continuous extension to the boundary of the domain in terms of prime ends. Under some additional conditions, the families of such mappings are equicontinuous in the closure of the domain with respect to the space of prime ends.
The behavior of a class of mappings of a domain of Euclidean space is studied. It is established that the indicated class is equicontinuous both at the inner and at the boundary points of the domain if the mappings contained in it satisfy the general normalization condition, and the corresponding characteristic of quasiconformality has a weak growth.
It is an expanded form of Drasin's work on normality of family of meromorphic functions given in his seminal paper titled "Normal Families and the Nevanlinna Theory".
In this paper, by introducing the notion of "\textit{distributive constant}" of a family of hypersurfaces with respect to a projective variety, we prove a second main theorem in Nevanlinna theory for meromorphic mappings with arbitrary families of hypersurfaces in projective varieties. Our second main theorem generalizes and improves previous results for meromorphic mappings with hypersurfaces, in particular for algebraically degenerate mappings and for the families of hypersurfaces in subgeneral position. The analogous results for the holomorphic curves with finite growth index from a complex…
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.
On Montel theorem for mappings with inverse moduli inequalities
本文研究了有限扭曲映射，特别是满足逆Poletskii不等式的映射，探讨了在给定领域内此类映射族的等连续性问题，证明了在满足一定积分条件时，此类映射族是等连续的，从而推广了经典的Montel定理。
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.
We give several unequivalent notions of convergency of meromorphic functions and more generally meromorphic mappings (strong, weak, $Γ$-convergency and some others). Relations between them are investigated. A version of Rouche theorem for the strongly converging sequences of meromorphic mappings is given. Further we investigate the sets of normality of families of meromorphic mappings, connections with extension properties of those mappings, some apriori estimates for the volume. Finally we apply the technique developped to the study of Fatou sets of meromorphic endomorphisms of Kähler surface…
Let B be the open unit ball in C^2 and let a, b, c be three points in C^2 which do not lie in a complex line, such that the complex line through a and b meets B and such that <a|b> is different from 1 if one of the points a, b is in B and the other in the complement of B and such that at least one of the numbers <a|c>, <b|c> is different from 1. We prove that if a continuous function f on the sphere bB extends holomorphically into B along each complex line which passes through one of the points a, b, c then f extends holomorphically through B. This generalizes recent work of L.Baracco who prov…
A Note on Normal Families Concerning Nonexceptional Functions and Differential Polynomials
本文研究微分多项式满足特定条件的亚纯函数族的正规性，改进并推广了Chen的近期结果，并通过例子证明结论的精确性。
In this paper, we prove two normality criteria for a family of meromorphic functions. The first criterion extends a result of Fang and Zalcman[Normal families and shared values of meromorphic functions II, Comput. Methods Funct. Theory, 1(2001), 289 - 299] to a bigger class of differential polynomials whereas the second one leads to some counterexamples to the converse of the Bloch's principle.
In this paper we continue our earlier investigations on normal families of meromorphic functions\cite{CS2}. Here, we prove some value distribution results which lead to some normality criteria for a family of meromorphic functions involving the sharing of a holomorphic function by more general differential polynomials generated by members of the family and get some recently known results extended and improved. In particular, the main result of this paper leads to a counterexample to the converse of Bloch's principle.
In this paper we prove a normality criterion for the families of meromorphic functions involving sharing of functions. Our result generalizes some of the earlier results on Gu's normality criterion.
We show that a family ${\cal F}$ of meromorphic functions in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f^{(j)}|^α}(z)\ge C \qquad \mbox{for all} z\in D \mbox{and all} f\in {\cal F}$$ (where $k$ and $j$ are integers with $k>j\ge 0$ and $C>0$, $α>1$ are real numbers) is quasi-normal. Furthermore, if all functions in ${\cal F}$ are holomorphic, the order of quasi-normality of ${\cal F}$ is at most $j-1$. The proof relies on the Zalcman rescaling method and previous results on differential inequalities constituting normality.
We continue our study of the dynamics of meromorphic mappings with small topological degree on a compact Kähler surface $X$. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and admits a natural geometric description. Our hypotheses are always satisfied when $X$ has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of $\mathbf{C}^2$. They are new even in the birational case. We also exhibit families of mappings where our assumptions are generically satisfied and show that if counte…
On the global behavior of mappings and the correspondence of boundaries
本文研究了具有模不等式的映射族的全局行为，通过不同定义域的分析，证明了在附加条件下映射族的均匀等连续性，并探讨了四种主要情况，包括映射为同胚、有分支点、复杂几何域的同胚以及有分支点的映射，进一步分析了极限映射的性质及核边界点与边界点、内点与内点的对应关系。
General form of second main theorem on generalized $p$-Parabolic manifolds for arbitrary closed subschemes
本文提出了一种针对任意闭子方案的广义 $p$-抛物 manifold 上全纯映射的第二主定理的推广形式，并给出了与超曲面相交时的计数函数截断水平的显式结果。
We establish a criterion for local boundedness and hence normality of a family $\F$ of analytic functions on a domain $D$ in the complex plane whose corresponding family of derivatives is locally bounded. Furthermore we investigate the relation between domains of normality of a family $\F$ of meromorphic functions and its corresponding Schwarzian derivative family. We also establish some criterion for the Schwarzian derivative family of a family $\F$ of analytic functions on a domain $D$ in the complex plane to be a normal family.
We discuss families of meromorphic functions $f_h$ obtained from single functions $f$ by the re-scaling process $f_h(z)=h^{-α}f(h+h^{-β}z)$ generalising Yosida's process $f_h(z)=f(h+z)$. The main objective is to obtain information on the value distribution of the generating functions $f$ themselves. Among the most prominent generalised Yosida functions are first, second and fourth Painlevé transcendents. The Yosida class contains all limit functions of generalised Yosida functions--the Yosida class is universal.
Beardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform Hölder condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others, and y…
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。
A universal model for the bifurcations of asymptotic values
本文研究了渐近值分支的通用模型，通过定义Tandelbrot集，证明了Tangent-like映射的直化定理及其参数版本，揭示了多个解析映射族中拓扑Tandelbrot集的存在。
Finiteness of Fuchsian equations with unitarizable monodromy on compact Riemann surfaces
该研究证明紧黎曼曲面上特定亚纯射影联络的有限性，建立和乐的正则性与增长估计，推导球面锥形度量及稳定扩张分歧除子的有限性结果。
We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk.We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holdsin the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms.
This is a course of lectures given for students of the Regional Mathematical Center of the Novosibirsk State University from October 20 to November 3, 2017. The course is devoted to some geometric problems of ramified coverings of the Riemann sphere. A special attention is payed to compact surfaces of genus one (complex tori). In the first section we give a short introduction to the theory of elliptic functions. Section 2 is devoted to one-parametric families of holomorphic and meromorphic functions. We recall the role of such families on Loewner's equation in solving some problems of the theo…
Cycles of finite type
本书提出有限型循环的f-解析族，引入强拟固有映射并推广几何平坦化定理，证明了既约复空间的亚纯商存在性定理及经典Stein分解的推广。
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