The rational maps F λ (z) = z m + λ/z d have no Herman rings

Abstract

The rational maps no Herman ringsAbstract. It is proved that the rational maps in the family {z → z m + λ/z d : λ ∈ C\{0}} for integers m, d ≥ 2 have no Herman rings.

Proof of a folklore Julia set connectedness theorem and connections with elliptic functions

“…There are also several published statements closely related to the next result (it follows from [6], Proposition (Dynamical Symmetry)) and is proved in [16], Lemma 2. Lemma 2.2 also follows from Proposition 2.1 proved below.…”

“…Then the cycle lies in C, and we have one of the following: (1) an attracting (or superattracting) periodic cycle; (2) a parabolic periodic cycle; or (3) a cycle of Siegel disks. The existence of a Herman ring was ruled out in [16], with a different proof for the case n = p in [8].…”

Singular Perturbations with Multiple Poles of the Simple Polynomials

“…For more rational maps whose Julia sets are Sierpiński curves, see [6]. For the study of non-existence of Herman rings for rational families, see [27,30] and the references therein.…”

“…In particular, the Julia set of F λ is a Cantor set of circles if 1/n + 1/d < 1 and λ = 0 is small. If the orbits of the free critical points of F λ are bounded, then F λ has no Herman rings [27] and actually, the corresponding Julia set is connected [8]. Since the McMullen family behaves extremely rich dynamics, this family has also been studied in [21,24].…”

Optical Absorption Property and Photo‐catalytic Activity of Tin Dioxide‐doped Titanium Dioxides

“…In fact, the separation and recombination of photo-induced charge carriers are in a competition process and the photo-catalytic reaction is effective only when the photo-induced electrons and holes are trapped on the surfaces, respectively. 8 How to decrease the recombination of the photo-induced carriers is crucial for improvement of photo-catalytic performance of TiO 2 . There have been a number of studies about doping and coupling Sn species for the purpose of improving TiO 2 photo-catalytic activity.…”