Transcendental Entire Function Research Articles

A note on transcendental entire functions mapping uncountable many Liouville numbers into the set of Liouville numbers

A note on transcendental entire functions mapping uncountable many Liouville numbers into the set of Liouville numbers

On topological properties of Fatou sets and Julia sets of transcendental entire functions

On topological properties of Fatou sets and Julia sets of transcendental entire functions

On the Dynamics of Composition of Transcendental Entire Functions in Angular Region-II

In [14] we showed that for transcendental entire functions f and g, there exist finitely many domains in an angular region, which lie in wandering component of f, wandering component of g and also in wandering component of f ° g and in wandering component of g ° f. Several other related results were discussed in that paper. In this paper we show the existence of such infinite components in angular region, using approximation theory.

A Note on Transcendental Power Series Mapping the Set of Rational Numbers into Itself

Abstract In this note, we prove that there is no transcendental entire function f(z) ∈ ℚ[[z]] such that f(ℚ) ⊆ ℚ and den f(p/q) = F(q), for all sufficiently large q, where F(z) ∈ ℤ[z].

Some results on differential-difference analogues of Brück conjecture

Abstract In this paper, we will consider value sharing problems on a transcendental entire function f with its differential-difference polynomials. We establish some results that can be viewed as the differential-difference analogues of Brück conjecture.

Regularity and growth conditions for fast escaping points of entire functions

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 $Q_2(f)=A(f)$ introduced in recent work.

Julia set of λ exp(z)/z with real parameters λ

In this paper, we investigate the Julia set of the family λ exp(z)/z with real parameters λ. We look for what values of real parameters λ such that the Julia set of λ exp(z)/z does not coincide with the whole plane, and thus gives a complete classification for real parameters, which is similar to Jang’s result of a family of transcendental entire functions. Moreover, We also discuss the shape and size of Fatou sets and Julia sets of λ exp(z)/z with real parameters λ when the Julia sets are not the whole plane.

Singular values and bounded Siegel disks

AbstractLet f be an entire transcendental function of finite order and Δ be a forward invariant bounded Siegel disk for f with rotation number in Herman's class $\mathcal{H}$. We show that if f has two singular values with bounded orbit, then the boundary of Δ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow.

Entire functions whose Julia sets include any finitely many copies of quadratic Julia sets

We prove that for any finite collection of quadratic Julia sets, a polynomial and a transcendental entire function exist whose Julia sets include copies of the given quadratic Julia sets. In order to prove the result, we construct quasiregular maps with required dynamics and employ the quasiconformal surgery to obtain the desired functions.

Periodic domains of quasiregular maps

We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^{d}$ to $\mathbb{R}^{d}$. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is the best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function. We construct a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of bi-Lipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ which is equal to the identity map in a half-space.

Unitarity and microscopic acausality in a nonlocal theory

We consider unitarity and causality in a higher-derivative theory of infinite order, where propagators fall off more quickly in the ultraviolet due to the presence of a transcendental entire function of the momentum. Like Lee-Wick theories, these field theories might provide new avenues for addressing the hierarchy problem; unlike Lee-Wick theories, tree-level propagators do not have additional poles corresponding to unobserved particles with unusual properties. We consider microscopic acausality in these nonlocal theories. The acausal ordering of production and decay vertices for ordinary resonant particles may provide a phenomenologically distinct signature for these models.

Models for the Speiser class

The Eremenko–Lyubich class B consists of transcendental entire functions with bounded singular set and the Speiser class S ⊂ B is made up of functions with a finite singular set. In an earlier work (J. Lond. Math. Soc. 92 (2015) 202–221), I gave a method for constructing Eremenko–Lyubich functions that approximate certain simpler functions called models. In this paper, I show that all models can be approximated in a weaker sense by Speiser class functions, and that the stronger approximation of the earlier work (J. Lond. Math. Soc. 92 (2015) 202–221) can fail for the Speiser class. In particular, I give geometric restrictions on the geometry of a Speiser class function that need not be satisfied by general Eremenko–Lyubich functions.

ON THE SHAPE OF MAXIMUM CURVE OF eaz2+bz+c

In this paper, we investigate the proper shape and location of the maximum curve of transcendental entire functions <TEX>$e^{az^2+bz+c}$</TEX>. We show that the alpha curve of <TEX>$e^{az^2+bz+c}$</TEX> is a subset of a rectangular hyperbola, and the maximum curve is the connected set originating from the origin as a subset of the alpha curve.

The dynamics of quasiregular maps of punctured space

The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties are stronger than those known to hold for the Julia set of a general quasiregular map of space. We define the quasi-Fatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of Marti-Pete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set.

Approximation of entire transcendental functions of several complex variables in some Banach spaces for slow growth

AbstractIn the present paper, the coefficients characterizations of generalized type

A Review on the Structure and Properties of the Escaping Set of Transcendental Entire Functions

For a transcendental entire function f, we study the structure and properties of the escaping set I(f) which consists of points whose iterates under f escape to infinity. We concentrate on Eremenko’s conjecture and we review some attempts of its proofs. A significant amount of progress in Eremenko’s conjecture has been made possible via fast escaping set A(f) which consists points that escape to infinity as fast as possible. This set can be written as union of closed sets, called levels of A(f). We review classes of functions for which A(f) and each of its levels has the structure of infinite spider’s web. In general, we study classes of entire functions for which the escaping set I(f) is a spider’s web. Spider’s web is a recently investigated structure of I(f) that gives new results in the direction of Eremenko’s conjecture.

Hyperbolic entire functions and the Eremenko\u2013Lyubich class: Class mathcal {B} or not class mathcal {B}?

Hyperbolicity plays an important role in the study of dynamical systems, and is a key concept in the iteration of rational functions of one complex variable. Hyperbolic systems have also been considered in the study of transcendental entire functions. There does not appear to be an agreed definition of the concept in this context, due to complications arising from the non-compactness of the phase space. In this article, we consider a natural definition of hyperbolicity that requires expanding properties on the preimage of a punctured neighbourhood of the isolated singularity. We show that this definition is equivalent to another commonly used one: a transcendental entire function is hyperbolic if and only if its postsingular set is a compact subset of the Fatou set. This leads us to propose that this notion should be used as the general definition of hyperbolicity in the context of entire functions, and, in particular, that speaking about hyperbolicity makes sense only within the Eremenko–Lyubich classmathcal {B} of transcendental entire functions with a bounded set of singular values. We also considerably strengthen a recent characterisation of the class mathcal {B}, by showing that functions outside of this class cannot be expanding with respect to a metric whose density decays at most polynomially. In particular, this implies that no transcendental entire function can be expanding with respect to the spherical metric. Finally we give a characterisation of an analogous class of functions analytic in a hyperbolic domain.

Hollow quasi-Fatou components of quasiregular maps

AbstractWe define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ℝd is called hollow if it has a bounded complementary component. We show that for each d ⩾ 2 there exists a quasiregular map of transcendental type f: ℝd → ℝd with a quasi-Fatou component which is hollow.Suppose that U is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if U is bounded, then U has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if U is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if J(f) has an isolated point, or if J(f) is not equal to the boundary of the fast escaping set. Finally, we deduce that if J(f) has a bounded component, then all components of J(f) are bounded.

Semigroups of transcendental entire functions and their dynamics

We investigate the dynamics of semigroups of transcendental entire functions using Fatou–Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some condition for connectivity of the Julia set of the transcendental semigroups. We also study finitely generated transcendental semigroups, abelian transcendental semigroups and limit functions of transcendental semigroups on its invariant Fatou components.

The size and topology of quasi-Fatou components of quasiregular maps

We consider the iteration of quasiregular maps of transcendental type from $\mathbb {R}^d$ to $\mathbb {R}^d$. In particular we study quasi-Fatou components, which are defined as the connected components of the complement of the Julia set. Many authors have studied the components of the Fatou set of a transcendental entire function, and our goal in this paper is to generalise some of these results to quasi-Fatou components. First, we study the number of complementary components of quasi-Fatou components, generalising, and slightly strengthening, a result of Kisaka and Shishikura. Second, we study the size of quasi-Fatou components that are bounded and have a bounded complementary component. We obtain results analogous to those of Zheng, and of Bergweiler, Rippon and Stallard. These are obtained using techniques which may be of interest even in the case of transcendental entire functions.