Abstract
In this work, it is shown that a set of holomorphic functions, normal in any set, remains normal in the same set under the action of another holomorphic function. And therefore, we verify that Julia's sets of two-function composition (no matter the order of composition) coincide, up to a rescaling.
Highlights
In 2005, Professor SHERETOV Vladimir Gueorguevich in (Grigorief, 2005), proposes to generalize the sets of Fatou and Julia on the case of a system of holomorphic functions on the sphere of Riemann C.By applying the classical results of the complex analysis (notably the convergence of a sequence of elements, notion of a normal family (cf. for example (Shabat, 1969) and (Goluzin, 1966)), the stability of the set of Julia or of Fatou by composition with a holomorphic function is verified.This article is dedicated to the study of some properties of such a system
We study the influence of a holomorphic function on a holomorphic family
Let U be the neighborhood of this point z0, such that the family {( fk ◦ fk−1 ◦ ... ◦ f1)◦n} be normal in U
Summary
Rufin EYELANGOLI OKANDZE1, Vital Delmas MABONZO1 & Dieudonne AMPINI2 1 Departement des Mathematiques, Ecole Normale Superieure, Universite Marien NGOUABI, Brazzaville, Congo 2 Departement des Mathematiques, Facultedes Sciences et Technique, Universite Marien NGOUABI, Brazzaville, Congo. Correspondence: Rufin EYELANGOLI OKANDZE, Universite Marien NGOUABI, BP: 69, Brazzaville, Congo. Received: August 3, 2018 Accepted: August 20, 2018 Online Published: September 11, 2018 doi:10.5539/jmr.v10n5p153
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