On slow escaping and non-escaping points of quasimeromorphic mappings

We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.

On the iteration of quasimeromorphic mappings

The Fatou–Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares many properties with those for transcendental meromorphic functions and for quasiregular mappings of punctured space.

A Lehto–Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space

We establish a Lehto–Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space, which are useful for establishing a big Picard-type theorem and a big Brody-type one for holomorphic curves.

A Numerical Test of Padé Approximation for Some Functions with Singularity

The aim of this study is to examine some numerical tests of Padé approximation for some typical functions with singularities such as simple pole, essential singularity, brunch cut, and natural boundary. As pointed out by Baker, it was shown that the simple pole and the essential singularity can be characterized by the poles of the Padé approximation. However, it was not fully clear how the Padé approximation works for the functions with the branch cut or the natural boundary. In the present paper, it is shown that the poles and zeros of the Padé approximated functions are alternately lined along the branch cut if the test function has branch cut, and poles are also distributed around the natural boundary for some lacunary power series and random power series which rigorously have a natural boundary on the unit circle. On the other hand, Froissart doublets due to numerical errors and/or external noise also appear around the unit circle in the Padé approximation. It is also shown that the residue calculus for the Padé approximated functions can be used to confirm the numerical accuracy of the Padé approximation and quasianalyticity of the random power series.

ASYMPTOTIC BEHAVIOR OF THE PROPER LENGTH AND VOLUME OF THE SCHWARZSCHILD SINGULARITY

Though popular presentations give the Schwarzschild singularity as a point, it is known that it is spacelike and not timelike. Thus, it has a "length" and is not a "point." In fact, its length is necessarily infinite. It has been proven that the proper length of the Qadir–Wheeler suture model goes to infinity,1 while its proper volume shrinks to zero, and the asymptotic behavior of the length and volume has been calculated. That model consists of two Friedmann sections connected by a Schwarzschild "suture." The question arises whether a similar analysis could provide the asymptotic behavior of the Schwarzschild black hole near the singularity. It is proven here that, unlike the behavior for the suture model, for the Schwarzschild essential singularity Δs ~ K1/3 ln K and V ~ K-1 ln K, where K is the mean extrinsic curvature, or the York time.