Entire periodic functions with plane zeros

Primeness and dynamics of some classes of entire functions

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.

Subharmonic Almost Periodic Functions of Slow Growth

We obtain a complete description of the Riesz measures of almost periodic subharmonic functions with at most of linear growth on the complex plane; as a consequence we get a complete description of zero sets for the class of entire functions of exponential type with almost periodic modulus.

Triply Periodic Minimal Surfaces Bounded by Vertical Symmetry Planes

We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz-Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces.

On exponential type entire functions without zeros in the open lower half-plane

We obtain sufficient conditions for an exponential type entire function not to have zeros in the open lower half-plane. An exact inequality containing the real and imaginary parts of such functions and their derivatives restricted to the real axis is deduced. A connection is established to the positive definite functions.

关于C. C. Yang猜想与亚纯函数的周期性

On the Conjecture of C. C. Yang and periodicity of meromorphic functions

本文研究杨重骏提出的两个关于整函数周期性的猜想，通过建立新的周期性定理，显著改进并推广了前人的结果到更广泛的亚纯函数类。

On the difference equations with periodic coefficients

In this paper, we study entire solutions of the difference equation $ψ(z+h)=M(z)ψ(z)$, $z\in{\mathbb C}$, $ψ(z)\in {\mathbb C}^2$. In this equation, $h$ is a fixed positive parameter and $M: {\mathbb C}\to SL(2,{\mathbb C})$ is a given matrix function. We assume that $M(z)$ is a $2π$-periodic trigonometric polynomial. We construct the minimal entire solutions, i.e. entire solutions with the minimal possible growth simultaneously as for im$z\to+\infty$ so for im$z\to-\infty$. We show that the monodromy matrices corresponding to the minimal entire solutions are trigonometric polynomials of the s…

Zero (sub-)sequences of entire functions

We announce conditions under which a given sequence of points on the complex plane is a subsequence of zeros of an entire function with weight restrictions on growth.

Multipliers of periodic orbits of quadratic polynomials and the parameter plane

We prove an extension results for the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove that the size of p/q-limb of a hyperbolic component of the Mandelbrot set of period n is O(4^n/p), and give an explicit condition on internal arguments under which the Julia set of corresponding (unique) infinitely renormalizable quadratic polynomial is not locally connected

Minimal surfaces with the area growth of two planes; the case of infinite symmetry

We prove that a connected properly immersed minimal surface in Euclidean 3-space with infinite symmetry group whose intersection with a ball of radius R is less than 2\piR^2 is a plane, a catenoid or a Scherk singly-periodic minimal surface. In particular, we prove that the only periodic minimal desingularization of a pair of intersecting planes is Scherk's singly-periodic minimal surface.

Almost periodic functions in finite-dimensional space with the spectrum in a cone

An almost periodic function in finite-dimensional space extends to a holomorphic bounded function in a tube domain with a cone in the base if and only if the spectrum belongs to the conjugate cone. Also, an almost periodic function in finite-dimensional space has the bounded spectrum if and only if it extends to an entire function of exponential type.

Representation of mean-periodic functions in series of exponential polynomials

Let $θ$ be a Young function and consider the space $\mathcal{F}_θ(\C)$ of all entire functions with $θ$-exponential growth. In this paper, we are interested in the solutions $f\in \mathcal{F}_θ(\C)$ of the convolution equation $T\star f=0$, called mean-periodic functions, where $T$ is in the topological dual of $\mathcal{F}_θ(\C)$. We show that each mean-periodic function can be represented in an explicit way as a convergent series of exponential polynomials.

Entire holomorphic curves into projective plane intersecting few generic algebraic curves

For $q\leq 3$ smooth plane algebraic curves $\mathcal{C}_i$ having simple normal crossings, if the invariant logarithmic $2$-jet differential bundle associated to $(\mathbb{P}^2(\mathbb{C}), \sum_{i=1}^q \mathcal{C}_i)$ has a nonzero section vanishing on some ample divisor, then, for every algebraically nondegenerate entire holomorphic curve $f\colon\mathbb{C}\rightarrow\mathbb{P}^2(\mathbb{C})$, we have a Second Main Theorem type estimate: \[ T_f(r) \leq c\sum_{i=1}^q\,N_f^{[1]}(r,\mathcal{C}_i) + o\big(T_f(r) \big)\parallel, \] where $T_f(r)$ and $N_f^{[1]}(r,C_i)$ stand for the order functi…

关于常规平面多项式仿射变换族的公共周期点的统一界

Uniform bound on common periodic points for families of regular plane polynomial automorphisms

研究了常规平面多项式仿射变换族中公共周期点的统一界问题，证明了在特定条件下公共周期点数量被统一常数限制。

加权复合算子在全纯函数空间上的动力学刚性

Dynamical rigidity for weighted composition operators on holomorphic function spaces

本文研究了加权复合算子在准巴纳赫全纯函数空间上的动力学刚性，通过诱导作用于周期轨道上的 jets，揭示了符号的局部全纯动力学限制，并在不同维度下证明了多项式自同构的加权刚性。