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.

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.

Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure

We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f both have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge \exp (|z|^\alpha )$ for some $\alpha>0$ and all z outside a set of finite Lebesgue measure.

Entire functions with prescribed singular values

We introduce a new class of entire functions [Formula: see text] which consists of all [Formula: see text] for which there exists a sequence [Formula: see text] and a sequence [Formula: see text] satisfying [Formula: see text] for all [Formula: see text]. This new class is closed under the composition and it is dense in the space of all nonvanishing entire functions. We prove that every closed set [Formula: see text] containing the origin and at least one more point is the set of singular values of some locally univalent function in [Formula: see text], hence, this new class has nontrivial intersection with both the Speiser class and the Eremenko–Lyubich class of entire functions. As a consequence, we provide a new proof of an old result by Heins which states that every closed set [Formula: see text] is the set of singular values of some locally univalent entire function. The novelty of our construction is that these functions are obtained as a uniform limit of a sequence of entire functions, the process under which the set of singular values is not stable. Finally, we show that the class [Formula: see text] contains functions with an empty Fatou set and also functions whose Fatou set is nonempty.

Dynamics on the Pre-periodic Components of the Fatou Set of Three Transcendental Entire Functions and Their Compositions

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

Transcendental Semigroup that has Simply Connected Fatou Components

We prove that there exists non-trivial transcendental semigroup S such that the periodic or pre-periodic or wandering components of Fatou set F(S) have simply connected domain D.

Structurally Similar Sets of Transcendental Entire Function

In complex dynamics, the complex plane is partitioned into invariant subsets. In classical sense, these subsets are of course Fatou set and Julia set. Rest of the abstract available with the full text