Transcendental Julia sets with fractional packing dimension

We construct transcendental entire functions whose Julia sets have packing dimension in ( 1 , 2 ) (1,2) . These are the first examples where the computed packing dimension is not 1 1 or 2 2 . Our analysis will allow us further show that the set of packing dimensions attained is dense in the interval ( 1 , 2 ) (1,2) , and that the Hausdorff dimension of the Julia sets can be made arbitrarily close to the corresponding packing dimension.

Homotopy Hubbard trees for post-singularly finite exponential maps

We extend the concept of a Hubbard tree, well established and useful in the theory of polynomial dynamics, to the dynamics of transcendental entire functions. We show that Hubbard trees in the strict traditional sense, as invariant compact trees embedded in $\mathbb {C}$ , do not exist even for post-singularly finite exponential maps; the difficulty lies in the existence of asymptotic values. We therefore introduce the concept of a homotopy Hubbard tree that takes care of these difficulties. Specifically for the family of exponential maps, we show that every post-singularly finite map has a homotopy Hubbard tree that is unique up to homotopy, and that post-singularly finite exponential maps can be classified in terms of homotopy Hubbard trees, using a transcendental analogue of Thurston’s topological characterization theorem of rational maps.

Causal Gödel-type metrics in non-local gravity theories

It is well known that non-local theories of gravity have been a flourish arena of studies for many reasons, for instance, the UV incompleteness of General Relativity (GR). In this paper we check the consistency of ST-homogeneous Gödel-type metrics within the non-local gravity framework. The non-local models considered here are ghost-free but not necessarily renormalizable since we focus on the classical solutions of the field equations. Furthermore, the non-locality is displayed in the action through transcendental entire functions of the d’Alembert operator $$\Box $$ □ that are mathematically represented by a power series of the $$\Box $$ □ operator. We find two exact solutions for the field equations correspondent to the degenerate ($$\omega =0$$ ω = 0 ) and hyperbolic ($$m^{2}=4\omega ^2$$ m 2 = 4 ω 2 ) classes of ST-homogeneous Gödel-type metrics.

Classifying simply connected wandering domains

While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable.

On Subharmonic and Entire Functions of Small Order: After Kjellberg

We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves a novel technique to obtain an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, based on comparing the Harnack metric and hyperbolic metric, which gives a sharp estimate for the growth in many cases. Dedicated to the memory of Paddy Barry.

Orbifold expansion and entire functions with bounded Fatou components

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

Criniferous entire maps with absorbing Cantor bouquets

<p style='text-indent:20px;'>It is known that, for many transcendental entire functions in the Eremenko-Lyubich class <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are <i>criniferous</i>. In this paper, we extend this result to a new class of maps in <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>. Furthermore, we show that if a map belongs to this class, then its Julia set contains a <i>Cantor bouquet</i>; in other words, it is a subset of <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{C} $\end{document}</tex-math></inline-formula> ambiently homeomorphic to a straight brush.</p>

Periodic Components of the Fatou Set of Three Transcendental Entire Functions and Their Compositions

We prove that there exist three different transcendental entire functions that can have infinite number of domains which lie in the different periodic component of each of these functions and their compositions.

Uniqueness Problems about Entire Functions with Their Difference Operator Sharing Sets

In this paper, we study the uniqueness questions of finite order transcendental entire functions and their difference operators sharing a set consisting of two distinct entire functions of finite smaller order. Our results in this paper improve the corresponding results from Liu (2009) and Li (2012).