Transcendental Entire Function Research Articles

Models for the Eremenko–Lyubich class

If f is in the Eremenko–Lyubich class B (transcendental entire functions with bounded singular set), then Ω = { z : | f ( z ) | > R } and f | Ω must satisfy certain simple topological conditions when R is sufficiently large. A model ( Ω , F ) is an open set Ω and a holomorphic function F on Ω that satisfy these same conditions. We show that any model can be approximated by an Eremenko–Lyubich function in a precise sense. In many cases, this allows the construction of functions in B with a desired property to be reduced to the construction of a model with that property, and this is often much easier to do.

The radial oscillation of entire solutions of complex differential equations

This paper is devoted to studying the radial oscillation of solutions of linear differential equations f″+A(z)f′+B(z)f=F(z), where A(z), B(z) and F(z) are transcendental entire functions. With some added conditions on coefficients, we show that there exists some close relation between the Borel directions of solutions and that of F(z), and also obtain some estimates of growth of solutions in the angular domains.

On the Best Polynomial Approximations of Entire Transcendental Functions of Many Complex Variables in Some Banach Spaces

For the entire transcendental functions f of many complex variables m (m > 2) of finite generalized order of growth ⇢m(f; ↵, β), we obtain the limiting relations between the indicated characteristic of growth and the sequences of best polynomial approximations of f in the Hardy Banach spaces Hq(U m ) and in the Banach spaces Bm(p,q, λ) studied by Gvaradze. The presented results are extensions of the corresponding assertions made by Varga, Batyrev, Shah, Reddy, Ibragimov, and Shikhaliev to the multidimensional case. 1. For functions of one complex variable, the history of the problem of relationship between the rate of convergence of the sequences of best polynomial approximations of the entire transcendental functions to zero and some generalized characteristics of their growth is fairly comprehensively presented in [1]. The number of available results for the entire transcendental functions of many complex variables is much smaller (see, e.g., [2–4]). In the present paper, we continue our investigation of this problem in the multidimensional case. To formulate the required notions and definitions, we introduce the following notation: C is a space of complex

Density of hyperbolicity for classes of real transcendental entire functions and circle maps

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, Salom\~ao and Vargas. We also prove density of (real) hyperbolicity for certain families as in (i) but without the boundedness condition. Our results apply, in particular, when the functions in question have only finitely many critical points and asymptotic singularities, or when there are no asymptotic values and the degree of critical points is uniformly bounded.

Dimensions of slowly escaping sets and annular itineraries for exponential functions

We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity ‘slowly’, and which have Hausdorff dimension equal to$1$. We prove these results by using the idea of anannular itinerary. In the case of a general transcendental entire function we show that one of these sets, theuniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.

Meromorphic Functions Sharing Three Values with Their Linear Differential Polynomials in Some Angular Domains

We study the uniqueness question of transcendental meromorphic functions sharing three distinct finite values with their linear differential polynomials in some angular domains instead of the whole complex plane, and study the uniqueness question of transcendental entire functions sharing two distinct finite values with their linear differential polynomials in some angular domains instead of the whole complex plane. The results in this paper improve the corresponding results from Frank and Weissenborn (Complex Var 7:33–43, 1986), Frank and Schick (Results Math 22:679–684, 1992), Bernstein et al. (Forum Math 8:379–396, 1996) and improve Theorem 3 in Zheng (Can Math Bull 47:152–160, 2004).

Bounded Fatou components of transcendental entire functions with order less than 1/2

Let f be a transcendental entire function with order ρ 1 such that, for all r > r0 and any small ɛ > 0, $$M(r^\sigma ,f) \geqslant M(r,f)^{\sigma + \varepsilon } ,$$ then every component of the Fatou set F(f) is bounded.

Functions of genus zero for which the fast escaping set has Hausdorff dimension two

We study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a half-plane. In general these functions lie outside the Eremenko-Lyubich class. We show that for functions in this family the fast escaping set has Hausdorff dimension equal to two.

Poincaré linearizers in higher dimensions

It is well-known that a holomorphic function near a repelling fixed point may be conjugated to a linear function. The function which conjugates is called a Poincar\'e linearizer and may be extended to a transcendental entire function in the plane. In this paper, we study the dynamics of a higher dimensional generalization of Poincar\'e linearizers. These arise by conjugating a uniformly quasiregular mapping in $\R^m$ near a repelling fixed point to the mapping $x\mapsto 2x$. In particular, we show that the fast escaping set of such a linearizer has a spider's web structure.

The dynamics of semigroups of transcendental entire functions I

We consider the dynamics associated with an arbitrary semigroup of transcendental entire functions. Fatou-Julia theory is used to investigate the dynamics of these semigroups. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroup case. We also investigate the dynamics of conjugate semigroups, abelian transcendental semigroups and wandering and Baker domains of transcendental semigroups.

Maximally and non-maximally fast escaping points of transcendental entire functions

AbstractWe 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 function for which the maximally and non-maximally fast escaping sets each have uncountably many singleton components.

A Generalization of Brück’s Conjecture for a Class of Entire Functions

In this paper, we study the uniqueness problem of entire functions sharing polynomials IM with their first derivative. As an application, we generalize Bruck’s conjecture from sharing value CM to sharing polynomial IM for a class of functions. In fact, we prove a result as follows: Let $${a({\not\equiv} 0)}$$ be a polynomial and $${n \geq 2}$$ be an integer, let f be a transcendental entire function, and let $${F = f^n}$$ . If F and F′ share a IM, then $${f(z) = Ae^{z/n},}$$ where A is a nonzero constant. It extends some previous related theorems.

Nonreal zero decreasing operators related to orthogonal polynomials

Laguerre's theorem regarding the number of non-real zeros of a polynomial and its image under certain linear operators is generalized. This generalization is then used to (1) exhibit a number of previously undiscovered complex zero decreasing sequences for the Chebyshev, Legendre, and generalized Laguerre polynomial bases and (2) simultaneously generate a basis $B$ and a corresponding $B$-CZDS. Some extensions to transcendental entire functions in the Laguerre-Polya class are given which, in turn, give a new and short proof of a previously known result due to one of the authors. The paper concludes with several open questions.

Dynamical convergence of a certain polynomial family to f_a(z) = z + e^z + a

A transcendental entire function fa(z) = z +e z +a may have a Baker domain or a wandering domain, which never appear in the dynamics of polynomials. We consider a sequence of polynomials Pa,d(z) = (1+a/d)z +(1+z/d) d+1 +a, which converges uniformly on compact sets to fa as d → ∞. We show its dynamical convergence under a certain assumption, even though fa has a Baker domain or a wandering domain. We also investigate the parameter spaces of fa and Pa,d.

Bounded Fatou Components of Composite Transcendental Entire Functions with Gaps

We prove that composite transcendental entire functions with certain gaps have no unbounded Fatou component.

Constructing entire functions by quasiconformal folding

We give a method for constructing transcendental entire functions with good control of both the singular values of f and the geometry of f. Among other applications, we construct a function f with bounded singular set, whose Fatou set contains a wandering domain.

Julia and Escaping Set Spiders’ Webs of Positive Area

We study the dynamics of a collection of families of transcendental entire functions which generalises the well-known exponential and cosine families. We show that for functions in many of these families the Julia set, the escaping set and the fast escaping set are all spiders' webs of positive area. This result is unusual in that most of these functions lie outside the Eremenko-Lyubich class Β. This is also the first result on the area of a spider's web.

A separation theorem for entire transcendental maps

We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix $p\in\N$ and assume that all dynamic rays which are invariant under $f^p$ land. An interior $p$-periodic point is a fixed point of $f^p$ which is not the landing point of any periodic ray invariant under $f^p$. Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above we show that rays which are invariant under $f^p$, together with their landing points, separate the plane into finitely many regions, each containing exactly one interior $p-$periodic point or one parabolic immediate basin invariant under $f^p$. This result generalizes the Goldberg-Milnor Separation Theorem for polynomials, and has several corollaries. It follows, for example, that two periodic Fatou components can always be separated by a pair of periodic rays landing together; that there cannot be Cremer points on the boundary of Siegel discs; that "hidden components" of a bounded Siegel disc have to be either wandering domains or preperiodic to the Siegel disc itself; or that there are only finitely many non-repelling cycles of any given period, regardless of the number of singular values.

Non-linear differential equations and Hayman’s theorem on differential polynomials

In this paper, we first prove that if the following differential equationadmits a meromorphic function with finitely many poles, where and is a differential polynomial in with degree and rational functions as its coefficients, is a non-zero rational function and is a non-constant polynomial, then has the form and where is a rational function and is a polynomial with With this in hand, we prove if is a transcendental entire function, is a polynomial of degree , then assumes every complex number infinitely many times, except a possible value . On the other hand, if assumes the complex value finitely many times, then and .

ON THE TRANSCENDENTAL ENTIRE SOLUTIONS OF A CLASS OF DIFFERENTIAL EQUATIONS

In this paper, we consider the differential equation <TEX>$$F^{\prime}-Q_1=Re^{\alpha}(F-Q_2)$$</TEX>, where <TEX>$Q_1$</TEX> and <TEX>$Q_2$</TEX> are polynomials with <TEX>$Q_1Q_2{\neq}0$</TEX>, R is a rational function and <TEX>${\alpha}$</TEX> is an entire function. We consider solutions of the form <TEX>$F=f^n$</TEX>, where f is an entire function and <TEX>$n{\geq}2$</TEX> is an integer, and we prove that if f is a transcendental entire function, then <TEX>$\frac{Q_1}{Q_2}$</TEX> is a polynomial and <TEX>$f^{\prime}=\frac{Q_1}{nQ_2}f$</TEX>. This theorem improves some known results and answers an open question raised in [16].