INNER COMPOSITION OF ANALYTIC MAPPINGS ON THE UNIT DISK

A basic theorem of iteration theory (Henrici [6]) states that f analytic on the interior of the closed unit disk D and continuous on D with Int(D)f(D) carries any point z ϵ D to the unique fixed point α ϵ D of f. That is to say, fn(z)→α as n→∞. In [3] and [5] the author generalized this result in the following way: Let Fn(z):=f1∘…∘fn(z). Then fn→f uniformly on D implies Fn(z)λ, a constant, for all z ϵ D. This kind of compositional structure is a generalization of a limit periodic continued fraction. This paper focuses on the convergence behavior of more general inner compositional structures f1∘…∘fn(z) where the fj's are analytic on Int(D) and continuous on D with Int(D)fj(D), but essentially random. Applications include analytic functions defined by this process.

Let f(z) be a function that is analytic on the interior of the unit closed disk D (Izl I) and continuous on D. Suppose that f(D) lies in the interior of D. It is well-known that f must have exactly one fixed point in the set f(D) and the nth iterate fn(z) of any point z e D converges to as n =.
From Henrici (theorem 6.12a[6]) we have the following result with slightly more liberal hypotheses.
THEOREM I. Let f be analytic in a simply connected region S and continuous on the closure S' of S, and let f(S') be a bounded set contained in S. Then f has exactly one fixed point and the sequence {fn(z)} converges to the fixed point for arbitrary z g S'.
The proof of theorem is predicated on the proof of the theorem for the special case in which S is the open unit disk.A simple application of the Riemann mapping may accelerate the convergence o these expansions [2], [3].
The following theorem due to Hlllam and Thron (Lemma 4.38 [7]) demonstrates the preliminary ideas under discussion in the context of fairly general Mobius transformations.where Ibnl > I. ICII + r < <==> Ibnl > 2. Therefore, D fn It is interesting to compare the convergence behaviors of "outer" and "inner" compositions.
We shall see that the convergence of {Fn(Z)} is guaranteed if the f's map D into (lwl .6),e.g., whereas, it is trivial to find such functions n that will produce oscillatory divergence of {J (z)} no matter how small a disk n (lwl R < l) the f's map D into.n EXAMPLE I. Let f(z) be a Mobius transformation mapping D onto (Iz-R/21 R/8) and let g(z) be a similar function mapping D onto (Iz + R/21 R/8).If f2n(Z) f(z) and f2n_l(Z) g(z), then {Jn(Z)} diverges for each z e D.
We begin our expoloration of the convergence behavior of {F (z)} with the observation that some kind of condition is required in order to insure convergence to a constant.i0 Next, we introduce a very simple lemma involving a Lipschltz condition on the fn'S" Set Fn,n+m (z): fn+1 o fn+m(Z).
If If '(z) K for all z E U and K 0, then F (z) for all z E D. Ifn(Zl fn(Z2 )I KnlZl-z21 for z I, z2 U implies theorem then suffices to extend the result to a more general set S.
The author, in [5], extended theorem by considering limit periodic sequences of the form {Fn(Z)} where Fl(z) fl(z), Fn(Z) Fn_l(fn(z)), with fn f in a region S.
(A slightly weaker result not requiring the Riemann mapping theorem is found in [3]).
THEOREM 2. Let f be analytic in a simply connected region S and continuous on the closure S' of S, and let f(S') be a bounded set contained in S. Suppose f f n uniformly on S. Then F (z)/ , a constant, for each z e S'. n Limit periodic sequences occur naturally in the study of limit periodic continued fractions and quasl-geometrlc series, and may be generalized in complete metric spaces [2].
Such sequences when employed in the context of functional expansions are inherently more interesting and productively richer than simple iteration or what might be considered "outer" composition (J (z)   f of o...of (z)) for the following n n n-1 reason: en f and a simple Lipschitz condlt[on holds on the f's these latter n n two sequences converge to the attractive fixed point of the limit function f, whereas the limit periodic sequence converges, but to a limit that depends upon the structures of the individual f 's.n In the present paper the following question is posed, and, to some extent, answered: Suppose each member of the sequence {fn is analytic on Int(D) and continuous on D with D f (D) (it is not assumed that f f).
Under what n n condit[ons does F (z) f o...of (z)/ %, a constant, for all z D, as n ?Thus n n we are considering "inner" compositions of essentially random sequences of functions mapping the unit disk into itself.
Although our approach focuses on mappings of D into D, more general results are possible.Let S F Int(F) where F is a Jordan curve.Let be the Riemann mapping function giving (S) D.
Suppose that gn is analytic in Int(F) and continuus on S, -I with Gn(S) contained in S. Then OgnO :=fn maps D into D. It easily follows that the convergence of {F n} implies the convergence of {G n} where Gn(Z): glo...Ogn(Z).
We shall present several theorems describing conditions on the fn'S that imply F (D) . After proving each of these basic theorems we will exhibit an alternative n and extended version of the result describing a class of analytic functions that can be generated in the following way: for each n let f (z)= fn(,z)be analytlc for n both S, a compact region, and z D. Let F (,z): n Apart from elementary details concerning unform boundedness and uni form convergence, the proof of these alterntive theorems are pratically identical to the proofs that are given for the simpler versions, and are therefore omitted.This will minimize notational complexity.
Although the method of constructing ()seems unusual several common modes of functional expansion may be categorized in this way.In fact, a judicious choice of z lfn(,z)/z Kn for all z U and for all S, and Kn O, then F (,D) X() uniformly on S. n We then easily obtain a result concerning the case in which the f's map D into a n smaller circle whose center is the origin.THEOREM 4. (a) Suppose fn(Z)l R: (,5 I)/2 < .6181for all n for zl I.
Then F (,D) X() uniformly on S.
n The fact that fn(Z)l R < for all zl is not sufficient to guarantee the Lipschitz condition If '(z) < for Izl R. Thls can be easily seen in the example n n n n a+to n EXAMP LE 3. Set f }fn(,z)l R < .61for the Indicated values of and z.Therefore F (,z)/ X() analytic on (tt 1).EXAMPLE 4. We define a continued square fraction by setting 2 f (,z): a ()/(b n() + z for S and z g D. If we assume that Ibn()l 9 2 and lan()l R < (5-I)/2 for S, then Ifn(,z) R and F (,z) X(), analytic on S.
If the values of f (0)are fairly close to 0, the critical value of R can be a n bit larger.
We turn now to conditions on the fixed points of the fn'S that insure convergence of {Fn(,z)}.
Invest[gatlons of limit periodic phenomena suggest that these fixed points may play a strong role in the kind of generalized iteration ,low being explored [I], [3], [4], [8].
Our next theorem is, in a sense, a generalization of theorem I.
6(a).Suppose that ,If (z)l R < for all n for all z e D, and that THEOREM a.
n/m sufficiently large.
Therefore, for large n and m, JGn,n+m (z)j and (Gn,n+m (z)[ provided k is large.
We will now show, in three steps, that fn (%)I p < for all n and that this small.It will then be possible to use thls information to establish the convergence of {Gn,n+m(z)}as m * .
-1 I.For each N set c (z) (z a )/(I anZ) and hn(Z) t of ot (z) where n n n n n t () O. Thus h (0) 0.
n n n large.If this were not the case there would exist {z n} such that z a and n Ifn (Zn) ) I-I/n However

THEOREM 3 .
Suppose that f (z) (a z + b )/(c z + d )< I. Then F (z)+ for all z e INT(D).
constant for llzll I. PROOF.The center C and radius r of f (D) are C b

EXAMPLE 2 .
Let f (z) maps D into Int D and {Fn(ZO)} n n

F
uniformly on S, and k() is analytic on S.