Zero sets of some classes of entire functions
Alexander Russakovskii
给出了一种构造具有给定零点的整函数的方法和增长性估计。它给出了一种可能性来描述零集的某些类的整个功能的一个和几个变量的增长方面的体积，这些集在某些polycylinders。
A method of constructing an entire function with given zeros and estimates of growth is suggested. It gives a possibility to describe zero sets of certain classes of entire functions of one and several variables in terms of growth of volume of these sets in certain polycylinders.
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Distribution of Normalized Zero-Sets of Random Entire Functions with Small Random Perturbation
在本文中，我们推广了我们的早期结果（见[Y-2008]）的随机整函数的正规化零集的分布随机整函数的小随机扰动。
In this paper we investigate the primeness of a class of entire functions and discuss the dynamics of a periodic member f of this class with respect to a transcendental entire function g that permutes with f. In particular we show that the Julia sets of f and g are identical.
Zero sets of entire functions of exponential type with some conditions on real axis
给出了指数增长性整函数（有界于实轴，Cartwright类）的几个著名子类的零点集的完整描述
In 1895, Cantor showed that between every two countable dense real sets, there is an order isomorphism. In fact, there is always such an order isomorphism, which is the restriction of a universal entire function.
Uniqueness sets with angular density for spaces of entire functions, II: $ρ=1$
本文研究指数型整函数空间中具有角密度的唯一性集合，在放弃Lindelöf条件限制的情况下展开分析，并保持内容自足可独立阅读。
Uniqueness sets with angular density for spaces of entire functions, III: how to minimize the type
本文研究了整个函数空间中具有角密度的唯一性集，通过分析满足一定正则性条件的离散集Λ，证明存在一个类型最小化的测度Δ₀，其质量少于2ρ。对于ρ=2的情况，确定了临界唯一性类型的几何表达。
Entire curves avoiding given sets in C^n
设$F\subset\Bbb C^n$是$\Bbb C^n$的真闭子集，且$A\subset\Bbb C^n\setminus F$至多可数（$n\geq 2$）.给出了$F$和$A$的条件，在此条件下，存在一个全纯浸入（或一个真全纯嵌入）$ϕ:\Bbb C\to\Bbb C^n$与$A\subsetϕ(\Bbb C)\subset\Bbb C^n\setminus F$ .
Uniqueness sets with angular density for spaces of entire functions, I. Basics
本文研究了具有角密度的整个函数空间的唯一性集，确定了临界零集类型并给出了其与临界唯一性集类型相吻合的条件，同时探讨了其在福克型空间中的应用。
We give criteria for the escaping set and the Julia set of an entire function to have positive measure. The results are applied to Poincaré functions of semihyperbolic polynomials and to the Weierstraß $σ$-function.
Entire curves, Integral sets and Principal bundles
我们比较整个曲线和积分集的行为，特别是在局部平凡纤维丛，代数群和有限分歧覆盖半阿贝尔簇。
Distribution of Normalized Zero-Sets of Random Entire Functions
本文研究了随机整函数正规化零点集的分布。零集的归一化与Nevanlinna理论中整函数的计数函数的归一化相同。所得结果将Shiffman和Zelditch关于正Hermitian全纯线丛幂的随机全纯截面零点分布的理论从多项式函数推广到整函数.我们的结果也可以看作是随机整函数零点集分布理论中Nevanlinna第一大定理的类比。
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.
Let $f$ and $g$ be transcendental entire functions, each with a bounded set of singular values, and suppose that $f$ and $g$ are affinely equivalent (that is, $g \circ ϕ= ψ\circ f$, where $ϕ,ψ:\C\to\C$ are affine). We show that the escaping sets of $f$ and $g$ have the same Hausdorff dimension. Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equal to one.
We investigate decay near boundary of the volume of sublevel sets in Cegrell classes of $m-$ subharmonic function on bounded domains in $\mathbb C^n.$ On the reverse direction, some sufficient conditions for membership in certain Cegrell's classes, in terms of the decay of the sublevel sets, are also discussed.
On approximation of continuous functions by entire functions on subsets of the real line
本文推广了经典的Bernstein定理，该定理是关于实直线上一致连续函数类的构造性描述。研究了指数型整函数在实直线的无界闭真子集上逼近连续有界函数的问题。
Local connectivity of Julia sets of some transcendental entire functions with Siegel disks
本文利用拟Blaschke乘积长迭代的弱扩张性质，证明了一类具有有界型Siegel盘的超越整函数Julia集局部连通，并给出正弦函数实例及含渐近值情形的存在性结果。
We give conditions ensuring that the Julia set and the escaping set of an entire function of completely regular growth have positive Lebesgue measure. The essential hypotheses are that the indicator is positive except perhaps at isolated points and that most zeros are located in neighborhoods of finitely many rays. We apply the result to solutions of linear differential equations.
Let $f$ be a transcendental entire function. For $n \in \mathbb{N},$ let $ f^{n}$ denote the $n^{th}$ iterate of $f$. Let $ I(f) = \{z \in \mathbb{C} : f^n \rightarrow \infty $ as $ n \rightarrow \infty \} $ and $ K(f) = \{z: \textrm{ there exists } R > 0 \textrm{ such that } | f^n(z) | \leq R \textrm{ for } n \geq 0 \}. $ Then the set $ \mathbb{C}\ \setminus (I(f) \cup K(f)) $ denoted by $ BU(f) $ is called Bungee set of $f$. In this paper we give an alternate definition for $ BU(f)$ which is very easy to work with, and we illustrate it by proving some properties of Bungee sets of composite t…
Steiner-Minkowski Polynomials of Convex Sets in High Dimension, and Limit Entire Functions
对于n维欧几里得空间的凸集（K），Steiner-Minkowski多项式（M_K（t））定义为半径（t）邻域的n维欧氏体积.定义为正的（t），Steiner-Minkowski多项式被认为是所有复杂的（t）。提出了Steiner多项式的重整化方法。对应于所有可能的全维立体凸集的重整化Steiner-Minkowski多项式在整个复平面上形成一个正规族。对于四个凸集族中的每一个：欧氏球，立方体，正则交叉多面体和维数（n）的正则辛，极限整函数，因为（n）趋于无穷大，明确计算。
In the spirit of very recent articles by J. Bonet, W. Lusky and J. Taskinen we are studying the so-called solid hulls and cores of spaces of weighted entire functions when the weights are given in terms of associated weight functions coming from weight sequences. These sequences are required to satisfy certain (standard) growth and regularity properties which are frequently arising and used in the theory of ultradifferentiable and ultraholomorphic function classes (where also the associated weight function plays a prominent role). Thanks to this additional information we are able to see which …
Recurrence of entire transcendental functions with simple post singular sets
研究了亚纯函数的逆函数的奇点轨道如何在其朱利亚集上定义动力学，至少直到一个Lebesgue测度为零的集合.我们集中在一个家庭的整个超越函数只有1000个奇异的逆，计数的多重性，所有这些要么逃逸指数快速或前周期。对于这些函数，我们能够决定函数是否是递归的。在朱利亚集不是整个平面的情况下，我们还得到了朱利亚集的测度的估计.
We find some sufficient conditions for a system of partial derivatives of an entire function to be complete in the space $H(\mathbb{C}^d)$ of all entire functions of $d$ variables. As an appliation of this result we describe new classes of frequently hypercyclic operators on $H(\mathbb{C}^d)$.
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.
In this paper we study an algebraic and topological structure inside the following sets of special functions: Bloch functions defined on the open unit disk that are unbounded and analytic functions of bounded type defined a Banach algebra E into E, which are not Lorch-analytic.
In this note we consider a certain class of Gaussian entire functions, characterized by some asymptotic properties of their covariance kernels, which we call admissible (as defined by Hayman). A notable example is the Gaussian Entire Function, whose zero set is well-known to be invariant with respect to the isometries of the complex plane. We explore the rigidity of the zero set of Gaussian Taylor series, a phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire Function. In particular, we find that for a function of infinite order of growth, and having an admissible kern…
The present paper studies structure of the ring of integer-valued entire functions. We characterize certain classes of prime and maximal ideals and investigate some of their properties.
In this paper we solve the Dirichlet problems for different classes of plurisubharmonic functions on compact sets in $\mathbb C^n$ including continuous, pluriharmonic and maximal functions.
We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed by Lindsey and Younsi and analyze the rate of approximation. We use slightly modified fundamental Lagrange interpolation polynomials and show that taking certain classes of nodes with subexponential growth of Lebesgue constants improves the approximation rate. To this end we investigate properties of some arrays of points in $\mathbb{C}$. In particular we prove subexponential growth of Lebesgue constants for pseudo Leja sequences with bounded Edrei growth on finite unions of quasiconfo…
Let f be a real entire function whose set S(f) of singular values is real and bounded. We show that, if f satisfies a certain function-theoretic condition (the "sector condition"), then $f$ has no wandering domains. Our result includes all maps of the form f(z)=λsinh(z)/z + a, where a is a real constant and λ is positive. We also show the absence of wandering domains for certain non-real entire functions for which S(f) is bounded and the iterates of f tend to infinity uniformly on S(f). As a special case of our theorem, we give a short, elementary and non-technical proof that the Julia set of …
In this paper, we have investigated the Bungee set of composition of two transcendental entire functions. We have provided a class of permutable entire functions for which their Bungee sets are equal. Moreover, we have obtained a result on permutable entire functions for which the escaping set of the composite entire function equals the union of the escaping sets of the two functions. In addition, we establish important relations between the Bungee set, Escaping set and Filled Julia set of composite entire functions with that of its individual functions.
We prove density of hyperbolicity in spaces of (i) real transcendental entire functions, bounded on the real line, whose singular set is finite and real and (ii) transcendental self-maps of the punctured plane which preserve the circle and whose singular set (apart from zero and infinity) is contained in the circle. In particular, we prove density of hyperbolicity in the famous Arnol'd family of circle maps and its generalizations, and solve a number of other open problems for these functions, including three conjectures by de Melo, Salomão and Vargas. We also prove density of (real) hyperboli…
Entire functions with Julia sets of positive measure
设f是一个超越整函数，其临界值和渐近值的集合是有界的。Denjoy-Carleman-Ahlfors定理意味着，如果所有z的集合，|f（z）|>R有N个分量，对于某个R>0，则f的阶至少为N/2。更精确地说，我们有log log M（r，f）>（N/2）log r - O（1），其中M（r，f）表示f的最大模。证明了如果f的增长速度不超过这个值，则f的逃逸集和朱利亚集都有正的勒贝格测度.然而，只要f的阶数超过N/2，这就不一定是真的。证明需要一个尖锐的形式的估计有关的Tsuji Denjoy-Carleman-Ahlfors定理。
We construct several new classes of transcendental entire functions, f, such that both the escaping set, I(f), and the fast escaping set, A(f), have a structure known as a spider's web. We show that some of these classes have a degree of stability under changes in the function. We show that new examples of functions for which I(f) and A(f) are spiders' webs can be constructed by composition, by differentiation, and by integration of existing examples. We use a property of spiders' webs to give new results concerning functions with no unbounded Fatou components.
Suppose that $f$ is a transcendental entire function. In 2014, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected, and an example a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class. It is known that for many transcendental entire functions the escaping set…
Sets non-thin at \infty in \Bbb C ^m, and the growth of sequences of entire functions of genus zero
在本文中，我们定义的概念，非薄在$\infty$如下：让$E$是一个子集的$\Bbb C^m$。对于任何$R>0$，定义$E_R=E\cap \{z\in \Bbb C ^m :|z|\leq R\}$。如果对于所有$z\in \Bbb C^m$，\lim_{R\to\infty} V_{E_R}（z）=0，则我们说$E$在$\infty$处是非瘦的，其中$V_E$是$E$的复格林函数。在$\infty$上的非瘦定义具有良好的性质：如果$E\subset \Bbb C^m$在$\infty$上是非瘦的，并且$A$是多极的，则$E\backslash A$在$\infty$上是非瘦的，如果$E\subset \Bbb C^m$和$F\subset \Bbb C^n$是在$\infty$上非瘦的闭集，则$E\times F\subset \Bbb C^m\times \Bbb C^n$在$\infty$上是非瘦的（参见引理\ref{Lem1}）。然后我们研究了$\infty$集上的非瘦集的性质，并将其推广到\cite{mul-yav}和\cite{trong-tuyen}中。
We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.
In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex non-parametric version of Moser's Bernstein Theorem. More precisely, we prove that any entire complex analytic set in $\mathbb{C}^n$ which i…
Entire functions with Cantor bouquet Julia sets
本文研究超越整函数的Cantor bouquet Julia集，证明了criniferous性质与发丝结构的等价性，给出基于吸收集的新刻画，并在几何假设下证明head-start条件既是充分的也是必要的。
We partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider interest. It was shown by Rippon and Stallard that the fast escaping set has no bounded components. In contrast, by studying a function considered by Hardy, we give an example of a transcendental entire functi…
We show that for large classes of entire functions the Julia set and the escaping set have packing dimension two. For example, this is the case for entire functions which are bounded on a curve tending to infinity. More generally, we show that the result holds under suitable growth conditions involving the minimum and maximum modulus.
Let $f$ be a transcendental entire function. The quite fast escaping set, $Q(f)$, and the set $Q_2(f),$ which was defined recently, are equal to the fast escaping set, $A(f),$ under certain conditions. In this paper we generalise these sets by introducing a family of sets $Q_m(f)$, $m \in \mathbb{N}.$ We also give one regularity and one growth condition which imply that $Q_m(f)$ is equal to $A(f)$ and we show that all functions of finite order and positive lower order satisfy $Q_m(f)=A(f)$ for any $m$. Finally, we relate the new regularity condition to a sufficient condition for $Q_2(f)=A(f)$ …
Decrease of bounded holomorphic functions along discrete sets
我们提供的唯一性的全纯函数的Nevanlinna类桥接海曼和Lyubarskiii-Seip以前获得的结果。也就是说，我们提出了某些类的双曲分离序列的磁盘，在非切向积累的速度到边界（该类谱的端点分别是具有正测度的非切向聚值集的序列和违反Blaschke条件的序列）;并对每一类给出了使Nevanlinna类函数同零的模径向减小的临界条件。
On inclusion relations between weighted spaces of entire functions
本文研究整函数加权空间的包含关系，通过关联权重函数与关联权重序列在权重函数和权重序列情形间建立刻画，并应用于逐点乘法封闭性。
Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture
本文研究了分离型整个函数的Julia集拓扑性质，证明了其Julia连续体为弧状连续体，并构造了具有特定性质的函数，与Eremenko猜想相关。
We study the subsets of metric spaces that are negligible for the infimal length of connecting curves; such sets are called metrically removable. In particular, we show that every totally disconnected set with finite Hausdorff measure of codimension 1 is metrically removable, which answers a question raised by Hakobyan and Herron. The metrically removable sets are shown to be related to other classes of "thin" sets that appeared in the literature. They are also related to the removability problems for classes of holomorphic functions with restrictions on the derivative.
A survey on permutable transcendental entire functions and their dynamics
综述1958年至2025年可交换超越整函数动力学成果，探讨其函数形式、动力学性质，分析基于点轨道动力学分类的子集关系及逃逸集分类，还讨论了相关集合的动力学关系与蹦极集和游荡域的情况。
The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these sets with respect to certain gauge functions.
The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2$. We are particularly interested in studying these sets for semigroups generated by various classes of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2.$ We prove that if the Julia set of a semigroup $G$ which is generated by endomorphisms of maximal generic rank $k$ in $\mathbb{C}^k$ contains an isolated point, then $G$ must contain an element that is conjugate to an upper triangular automorphism of $\mathbb{C}^k.$ This generalizes a t…
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