Iteration of Holomorphic Function Systems on the Riemann Sphere

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.


Introduction
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.Indeed, we study the influence of a holomorphic function on a holomorphic family.
Let f •n be the n-th iteration of the holomorphic function f (which is different to any constant) of a surface of Riemann S defined in itself, so we call the set of Fatou for this function, the set on which the iterations family { f •n } will be normal (a pre-compact family of continuous functions)(cf.(Grigorief, 2005)).We consider the system { f 1 , f 2 , • • • , f k } of holomorphic functions, which induces the following series of compositions: For a good understanding of this article, note the following definition: Definition 1 System (1) is normal at the point z 0 ∈ C, if there exists a neighborhood U of this point such that, from every infinite sequence of elements of (1), we can determine a subsequence, locally and uniformly convergent, either to a holomorphic function or to infinity.
We conclude that , system (1) is normal on a set, if it is normal in every point of this set.This set will be denoted Fatou In this paper we will limit to the study of holomorphic systems.
The following theorem plays a vital role in this present work: is normal to the neighborhood of z 0 , so for any holomorphic function h, the family {h Proof.Let U be the neighborhood of this point z 0 , such that the family According to the hypotheses of the theorem, g •n is locally and uniformly convergent in U, either to a holomorphic function, or to infinity.a) if h is bounded near the point f (z 0 ), therefore in a neighborhood of z 0 , the compound h • f (z) ∞ and continue.Let's analyze the expression h • f •n − h • f : h being holomorphic function and bounded near the point f (z 0 ), then asymptotically, one can write h(w + ∆w) = h ′ (w)∆w + o(|∆w|).h ′ is also holomorphic at the point g(z 0 ), then ∃M ∈ R such that, M = sup |h ′ | in the neighborhood of f (z 0 ).We will have the following increase: The right hand side of system (2) tends to zero when n → ∞ , because g •n locally and uniformly converge towards The family of functions p n is holomorphic and bounded in a neighborhood of zero.So for the function We have The right hand side of system (3) tends to zero locally and uniformly in U(z 0 ), when n tend to ∞.It is for this reason that h • g •n locally and uniformly in U(z 0 ) converge to ∞.
2nd case.Suppose the g •n locally and uniformly tends to ∞ in the neighborhood U(z 0 ) of the point z 0 .
Let's take It is clear that for n → ∞, the G •n locally and uniformly tends to zero.As before, we prove the local and uniform convergence of the sequence h ) , either to a holomorphic function or to the constant ∞.
The Theorem 1 is then proved.

Consequence.
Let f 1 and f 2 be two holomorphic functions, then Julia sets for the functions  Proof.
1. Let us first show that Fatou ( •n , functions of this family,we can extract a convergent subsequence, that is to say the family of iterations ( So any family sub-sequence (1) is constituted by the elements of the form f It is also clear that we can choose a subsequence is also normal.Thus, from the elements of the family (1), it is possible to extract a subset locally and uniformly convergent, that is (1) is normal.
The theorem 2 is proved.
Remark 2 Theorems 1 and 2 could be stated relatively to Julia sets.