Transcendental Entire Function Research Articles

Amenability of bounded automata groups on infinite alphabets

AbstractWe study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first‐level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire transcendental functions in holomorphic dynamics.

On a version of the Problem B of Mahler involving derivatives

In 1902, Paul St\"ackel constructed an analytic function $f(z)$ in a neighborhood of the origin, which was transcendental, and with the property that both $f(z)$ and its inverse, as well as its derivatives, assumed algebraic values at all algebraic points in this neighborhood. Inspired by this result, Mahler in 1976 questioned the existence of an transcendental entire function $f(z)$ such that $f(\overline{\mathbb{Q}})$ and $f^{-1}(\overline{\mathbb{Q}})$ are subsets of $\overline{\mathbb{Q}}.$ This problem was solved by Marques and Moreira in 2017. As St\"acklel's result involved derivatives, it is natural to question whether we have an analogous result for transcendental entire functions involving derivatives. In this article, we show that there are an uncountable amount of such functions.

The link on extraneous non-repelling cycles of Schröder's methods of the first and second kind

Let Sf,n and Kf,n be the functions defined in Schröder's method of the first and second kind for an entire function f with given order n(n≥2), respectively. Based on unrefined algebra characterizations of Sf,n and Kf,n, we obtain some sufficient conditions on f such that both Sf,n and Kf,n possess given finite pairs of extraneous non-repelling cycles. Here, these conditions are a pair of equations, which have infinitely many polynomials or transcendental entire functions as its solutions. For obtaining some solutions f of such equations, we provide a step-by-step method. We start from any solution g in corresponding equations so that the function Sg,2 possesses the above finite pairs of extraneous cycles but all are super-attracting, and then f can be obtained by a series of formulas concerning the function g, points and multipliers of those cycles. Note that Sf,2=Kf,2 is identical with Newton's method for f. In a sense, this fact reveals that some extraneous super-attracting cycles of Newton's method imply certain extraneous non-repelling cycles of any method from the two families of methods. More generally, for any given orders n and m, some extraneous non-repelling cycles of Sf,n or Kf,n imply that of SF,m or KF,m for some entire functions f and F. These give a partial answer for the problem of finding possible link between the two families of methods, which was posed by Steven Smale in 1994.

Uniqueness of Meromorphic Functions With Nonlinear Differential Polynomials Sharing a Small Function IM

In the paper, we discuss the distribution of uniqueness and its elements over the extended complex plane from different polynomials of view. We obtain some new results regarding the structure and position of uniqueness. These new results have immense applications like classifying different expressions to be or not to be unique. The principal objective of the paper is to study the uniqueness of meromorphic functions when sharing a small function $a(z)$ IM with restricted finite order and its nonlinear differential polynomials. The lemma on the logarithmic derivative by Halburb and Korhonen (Journal of Mathematical Analysis and Applications, \textbf{314} (2006), 477--87) is the starting point of this kind of research. In this direction, the current focus in this field involves exploring unique results for the differential-difference polynomials of meromorphic functions, covering both derivatives and differences. Liu et al. (Applied Mathematics A Journal of Chinese Universities, \textbf{27} (2012), 94--104) have notably contributed to this research. Their research establishes that when $n \leq k + 2$ for a finite-order transcendental entire function $f$ the differential-difference polynomial$[f^{n}f(z+c)]^{(k)} - \alpha(z)$ has infinitely many zeros. Here, $\alpha(z)$ is characterized by its smallness relatively to $f$. Additionally, for two distinct meromorphic functions $f$ and $g$, both of finite order, if the differential-difference polynomials $[f^{n}f(z+c)]^{(k)}$\ and\ $[g^{n}g(z+c)]^{(k)}$ share the value $1$ in the same set, then $f(z)=c_1e^{dz},$ $g(z)=c_2e^{-dz}.$ We prove two results, which significantly generalize the results of Dyavanal and Mathai (Ukrainian Math. J., \textbf{71} (2019), 1032--1042), and Zhang and Xu (Comput. Math. Appl., \textbf{61} (2011), 722-730) and citing a proper example we have shown that the result is true only for a particular case. Finally, we present the compact version of the same result as an improvement.

ON THE EXCEPTIONAL SET OF TRANSCENDENTAL ENTIRE FUNCTIONS IN SEVERAL VARIABLES

Abstract We prove that any subset of $\overline {\mathbb {Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountably many transcendental entire functions over $\mathbb {C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer-Verlag, Berlin, 1976)].

Inequalities concerning maximum modulus and zeros of random entire functions

AbstractLet $f(z)=\sum\limits_{j=0}^{\infty} a_j z^j$ be a transcendental entire function and let $f_\omega(z)=\sum\limits_{j=0}^{\infty}\chi_j(\omega) a_j z^j$ be a random entire function, where $\chi_j(\omega)$ are independent and identically distributed random variables defined on a probability space $(\Omega, \mathcal{F}, \mu)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher and Steinhaus entire functions. We prove that, for almost all functions in the family and for any constant C > 1, there exist a constant $r_0=r_0(\omega)$ and a set $E\subset [e, \infty)$ of finite logarithmic measure such that, for $r \gt r_0$ and $r\notin E$, \begin{equation*} |\log M(r, f)- N(r,0, f_\omega)|\le (C/A)^{\frac1{B}}\,\log^{\frac1{B}}\,\log M(r,f) +\log\,\log M(r, f), \qquad a.s. \end{equation*}where $A, B$ are constants, $M(r, f)$ is the maximum modulus and $N(r, 0, f)$ is the integrated zero-counting function of f. As a by-product of our main results, we prove Nevanlinna’s second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by an integrated counting function, rather than by two integrated counting functions as in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions fω and for any ϵ > 0, there is r0 such that, for $r \gt r_0$, \begin{equation*} T(r, f) \le N(r,0, f_\omega)+\left(\tfrac12+\epsilon\right) \log T(r, f). \end{equation*}

REFINEMENT OF MACINTYRE - EVGRAFOV TYPE THEOREMS

The study of the asymptotic behavior of an entire transcendental function of the form f(z) = Pn anzpn, pn ∈ N, on curves γ going to infinity arbitrarily, is a classical problem, goes back to the works of Hadamard, Littlewood and Polia. Polia posed the following problem: under what conditions on pn does there unbounded sequence {ξn} ⊂ γ exist such that lnMf(|ξn|) ∼ ln|f(ξn)| for ξn → ∞ (Polya’s problem). Here Mf(r) is the maximum of the modulus f on a circle of radius r. He showed that if the sequence {pn} has zero density and f is of finite order, then the indicated relation between lnMf(|ξn|) and ln|f(ξn)| is always present. This assertion is also true in the case when f has a finite lower order: the final results for this case were obtained by A.M.Gaisin, I.D.Latypov and N.N.Yusupova-Aitkuzhina. We consider the situation when the lower order is equal to infinity. A.M. Gaisin received an answer to Polia’s problem in 2003. This is the criterion. If the conditions of this criterion are satisfied not by the sequence {pn} itself, but only by a subsequence — a sequence of central exponents, then the logarithms of the maximum modulus and modulus of the sum of the series will also be equivalent in the indicated sense on any curve γ going to infinity.

An approximate solution to Erdős' maximum modulus points problem

In this note we investigate the asymptotic behavior of the number of maximum modulus points, of an entire function, sitting in a disc of radius r. In 1964, Erdős asked whether there exists a non-monomial function so that this quantity is unbounded? tends to infinity? In 1968 Herzog and Piranian constructed an entire map for which it is unbounded. Nevertheless, it is still unknown today whether it is possible that it tends to infinity or not. In this paper, we construct a transcendental entire function that is arbitrarily close to satisfying this property, thereby giving the strongest evidence supporting a positive answer to this question.

Second order linear differential equations with a basis of solutions having only real zeros

AbstractLet A be a transcendental entire function of finite order. We show that, if the differential equation w″ + Aw = 0 has two linearly independent solutions with only real zeros, then the order of A must be an odd integer or one half of an odd integer. Moreover, A has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.

On meromorphic functions which are Brody curves

We discuss meromorphic functions on the complex plane which are Brody curves regarded as holomorphic maps to P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {P}}}_1$$\\end{document}, i.e., which have bounded spherical derivative. For some special classes we gave explicit criteria which functions are Brody. We also discuss which divisors of very slow growth may occur as zero divisor of a Brody function and show that there are transcendental entire functions of arbitrarily slow growth which are not Brody.

A model for boundary dynamics of Baker domains

We consider the transcendental entire function f(z)=z+e^{-z} , which has a doubly parabolic Baker domain U of degree two, i.e. an invariant stable component for which all iterates converge locally uniformly to infinity, and for which the hyperbolic distance between successive iterates converges to zero. It is known from general results that the dynamics on the boundary is ergodic and recurrent and that the set of points in partial U whose orbit escapes to infinity has zero harmonic measure. For this model we show that stronger results hold, namely that this escaping set is non-empty, and it is organized in curves encoded by some symbolic dynamics, whose closure is precisely partial U . We also prove that nevertheless, all escaping points in partial U are non-accessible from U, as opposed to points in partial U having a bounded orbit, which are all accessible. Moreover, repelling periodic points are shown to be dense in partial U , answering a question posted in (Barański et al. in J Anal Math 137:679–706, 2019). None of these features are known to occur for a general doubly parabolic Baker domain.

Unbounded fast escaping wandering domains

We introduce a new approximation technique into the context of complex dynamics that allows us to construct examples of transcendental entire functions with unbounded wandering domains. We provide examples of entire functions with an orbit of unbounded fast escaping wandering domains, answering a question of Rippon and Stallard. Moreover, these examples cover all possible types of simply-connected wandering domains in terms of convergence to the boundary. In relation to a conjecture of Baker, it was unknown whether functions of order less than one could have unbounded wandering domains. For any given order greater than 1/2 and smaller than 1, we provide an entire function of such order with an unbounded wandering domain.

Some transcendental entire functions with irrationally indifferent fixed points

Let $S$ be the set of all transcendental entire functions of the form $P(z) \exp (Q(z))$, where $P$ and $Q$ are polynomials. In this paper, by using the theory of polynomial-like mappings, we construct various kinds of functions in $S$ with irrationally indifferent fixed points as follows: (1) We construct functions in $S$ with bounded type Siegel disks centered at points other than the origin bounded by quasicircles containing critical points. This is an extension of Zakeri's result in [24] for $f \in S$. (2) We construct functions in $S$ with Cremer points whose multipliers satisfy some Cremer's condition in [6] only for rational functions. Our method shows that this condition can be applicable even in some transcendental cases. (3) For any integer $d \geq 2$ and some $c \in {\mathbf C} \setminus \{0\}$, we show that the function of the form $e^{2\pi i \theta}z(1 + cz)^{d-1}e^z\,(\theta \in {\mathbf R} \setminus {\mathbf Q})$ has a Siegel point at the origin if and only if $\theta$ is a Brjuno number. This is an extension of Geyer's result in [11]. (4) For the function of the form $(e^{2\pi i\theta}z+\alpha z^2)e^z \,(\theta \in {\mathbf R} \setminus {\mathbf Q}, \alpha \in {\mathbf C} \setminus \{0\})$, we show that if $\alpha$ and $\theta$ satisfy some condition, then the Siegel disk centered at the origin is bounded by a Jordan curve containing a critical point, which is not a quasicircle. Moreover, we can choose $\alpha$ and $\theta$ so that the Lebesgue measure of the Julia set is positive and can also choose them so that it is zero. This is an extension of Keen and Zhang's result in [13].

Best possibility of the Fatou-Shishikura inequality for transcendental entire functions in the Speiser class

The Speiser class S S is the set of all entire functions with finitely many singular values. Let S q ⊂ S S_q\subset S be the set of all transcendental entire functions with exactly q q distinct singular values. The Fatou-Shishikura inequality for f ∈ S q f\in S_q gives an upper bound q q of the sum of the numbers of its Cremer cycles and its cycles of immediate attractive basins, parabolic basins, and Siegel disks. In this paper, we show that the inequality for f ∈ S q f\in S_q is best possible in the following sense: For any combination of the numbers of these cycles which satisfies the inequality, some T ∈ S q T\in S_q realizes it. In our construction, T T is a structurally finite transcendental entire function.

NON-EXISTENCE OF ENTIRE SOLUTIONS OF NON-LINEAR GENERAL DIFFERENCE EQUATIONS

The main objective of this article is to investigate the solvability of a non-linear difference equation generated by general difference polynomial of a transcendental entire function of finite order.

Periodic Components of the Fatou Set in Angular Region

Here we discuss, for a given integer, the existence of transcendental entire function such that its number of periodic Fatou components lie in angular regions and their periodicity are related to the integer.

Splitting Hairs with Transcendental Entire Functions

Abstract In recent years, there has been significant progress in the understanding of the dynamics of transcendental entire functions with bounded postsingular set. In particular, for certain classes of such functions, a complete description of their topological dynamics in terms of a simpler model has been given inspired by methods from polynomial dynamics. In this paper, and for the 1st time, we give analogous results in cases when the postsingular set is unbounded. More specifically, we show that if $f$ is of finite order, has bounded criticality on its Julia set $J(f)$, and its singular set consists of finitely many critical values that escape to infinity and satisfy a certain separation condition, then $J(f)$ is a collection of dynamic rays or hairs, which split at critical points, together with their corresponding landing points. In fact, our result holds for a much larger class of functions with bounded singular set. Moreover, this result is a consequence of a significantly more general one: we provide a topological model for the action of $f$ on its Julia set.

A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme

In this research, we look at the Julia set patterns that are linked to the entire transcendental function f(z)=aezn+bz+c, where a,b,c∈C and n≥2, using the Mann iterative scheme, and discuss their dynamical behavior. The sophisticated orbit structure of this function, whose Julia set encompasses the entire complex plane, is described using symbolic dynamics. We also present bifurcation diagrams of Julia sets generated using the proposed iteration and function, which altogether contain four parameters, and discuss the graphical analysis of bifurcation occurring in the family of this function.

Baker domains and non-convergent deformations

For an entire transcendental function $f$ with a non-completely invariant Baker domain $U$, we study the pinching process of paths in $U$ with certain restrictions, that we call Baker laminations. We show that if some curve in the Baker lamination of $f$ joins a point in the boundary of $U$ with infinity, then the deformation does not converge. Thus, in this particular case, the boundary of the space of deformations of $f$ is incomplete.

Geometrically finite transcendental entire functions

For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree $d\geq 2$ is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle $d$-tupling on the circle. For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of "docile" functions: a transcendental entire function with bounded postsingular set is docile if it is the quotient of a suitable disjoint-type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This can be seen as an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author and of Mihaljevi\'c for more restrictive classes of entire functions.