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Infinite compositions of analytic functions

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Mathematical theory about infinitely iterated function composition

In mathematics,infinitecompositions ofanalytic functions (ICAF) offer alternative formulations ofanalytic continued fractions,series,products and other infinite expansions, and the theory evolving from such compositions may shed light on theconvergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions offixed point equations involving infinite expansions.Complex dynamics offers another venue foriteration of systems of functions rather than a single function. For infinite compositions of asingle function seeIterated function. For compositions of a finite number of functions, useful infractal theory, seeIterated function system.

Although the title of this article specifies analytic functions, there are results for more generalfunctions of a complex variable as well.

Notation

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There are several notations describing infinite compositions, including the following:

Forward compositions: $F_{k,n}(z)=f_{k}\circ f_{k+1}\circ \dots \circ f_{n-1}\circ f_{n}(z).$

Backward compositions: $G_{k,n}(z)=f_{n}\circ f_{n-1}\circ \dots \circ f_{k+1}\circ f_{k}(z).$

In each case convergence is interpreted as the existence of the following limits:

\lim _{n\to \infty }F_{1,n}(z),\qquad \lim _{n\to \infty }G_{1,n}(z).

For convenience, setF_n(z) =F_1,n(z) andG_n(z) =G_1,n(z).

One may also write $F_{n}(z)={\underset {k=1}{\overset {n}{\mathop {R} }}}\,f_{k}(z)=f_{1}\circ f_{2}\circ \cdots \circ f_{n}(z)$ and $G_{n}(z)={\underset {k=1}{\overset {n}{\mathop {L} }}}\,g_{k}(z)=g_{n}\circ g_{n-1}\circ \cdots \circ g_{1}(z)$

Comment: It is not clear when the first explorations of infinite compositions of analytic functions not restricted to sequences of functions of a specific kind occurred. Possibly in the 1980s.^[1]

Contraction theorem

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Many results can be considered extensions of the following result:

Contraction Theorem for Analytic Functions^[2]—Letf be analytic in a simply-connected regionS and continuous on the closureS ofS. Supposef(S) is a bounded set contained inS. Then for allz inS there exists anattractive fixed point α off inS such that: $F_{n}(z)=(f\circ f\circ \cdots \circ f)(z)\to \alpha .$

Infinite compositions of contractive functions

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Let {f_n} be a sequence of functions analytic on a simply-connected domainS. Suppose there exists a compact set Ω ⊂S such that for eachn,f_n(S) ⊂ Ω.

Forward (inner or right) Compositions Theorem—{F_n} converges uniformly on compact subsets ofS to a constant functionF(z) =λ.^[3]

Backward (outer or left) Compositions Theorem—{G_n} converges uniformly on compact subsets ofS toγ ∈ Ω if and only if the sequence of fixed points {γ_n} of the {f_n} converges toγ.^[4]

Additional theory resulting from investigations based on these two theorems, particularly Forward Compositions Theorem, include location analysis for the limits obtained in the following reference.^[5] For a different approach to Backward Compositions Theorem, see the following reference.^[6]

Regarding Backward Compositions Theorem, the examplef_2n(z) = 1/2 andf_2n−1(z) = −1/2 forS = {z : |z| < 1} demonstrates the inadequacy of simply requiring contraction into a compact subset, like Forward Compositions Theorem.

For functions not necessarily analytic theLipschitz condition suffices:

Theorem^[7]—Suppose $S {\displaystyle S}$ is a simply connected compact subset of $\mathbb {C}$ and let $t_{n}:S\to S$ be a family of functions that satisfies $\forall n,\forall z_{1},z_{2}\in S,\exists \rho :\quad \left|t_{n}(z_{1})-t_{n}(z_{2})\right|\leq \rho |z_{1}-z_{2}|,\quad \rho <1.$ Define: ${\begin{aligned}G_{n}(z)&=\left(t_{n}\circ t_{n-1}\circ \cdots \circ t_{1}\right)(z)\\F_{n}(z)&=\left(t_{1}\circ t_{2}\circ \cdots \circ t_{n}\right)(z)\end{aligned}}$ Then $F_{n}(z)\to \beta \in S$ uniformly on $S . {\displaystyle S.}$ If $\alpha _{n}$ is the unique fixed point of $t_{n}$ then $G_{n}(z)\to \alpha$ uniformly on $S {\displaystyle S}$ if and only if $|\alpha _{n}-\alpha |=\varepsilon _{n}\to 0$ .

Infinite compositions of other functions

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Non-contractive complex functions

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Results involvingentire functions include the following, as examples. Set

{\begin{aligned}f_{n}(z)&=a_{n}z+c_{n,2}z^{2}+c_{n,3}z^{3}+\cdots \\\rho _{n}&=\sup _{r}\left\{\left|c_{n,r}\right|^{\frac {1}{r-1}}\right\}\end{aligned}}

Then the following results hold:

Theorem E1^[8]—Ifa_n ≡ 1, $\sum _{n=1}^{\infty }\rho _{n}<\infty$ thenF_n →F is entire.

Theorem E2^[9]—Setε_n = |a_n−1| suppose there exists non-negativeδ_n,M₁,M₂,R such that the following holds: ${\begin{aligned}\sum _{n=1}^{\infty }\varepsilon _{n}&<\infty ,\\\sum _{n=1}^{\infty }\delta _{n}&<\infty ,\\\prod _{n=1}^{\infty }(1+\delta _{n})&<M_{1},\\\prod _{n=1}^{\infty }(1+\varepsilon _{n})&<M_{2},\\\rho _{n}&<{\frac {\delta _{n}}{RM_{1}M_{2}}}.\end{aligned}}$ ThenG_n(z) →G(z) is analytic for |z| <R. Convergence is uniform on compact subsets of {z : |z| <R}.

Additional elementary results include:

Theorem GF3^[7]—Suppose $f_{k}(z)=z+\rho _{k}\varphi _{k}(z)$ where there exist $R,M>0$ such that $|z|<R$ implies $|\varphi _{k}(z)|<M,\forall k,\$ Furthermore, suppose ${\textstyle \rho _{k}\geq 0,\sum _{k=1}^{\infty }\rho _{k}<\infty }$ and ${\textstyle R>M\sum _{k=1}^{\infty }\rho _{k}.}$ Then for ${\textstyle R*<R-M\sum _{k=1}^{\infty }\rho _{k}}$ $G_{n}(z)\equiv \left(f_{n}\circ f_{n-1}\circ \cdots \circ f_{1}\right)(z)\to G(z)\qquad {\text{ for }}\{z:|z|<R*\}$

Theorem GF4^[7]—Suppose $f_{k}(z)=z+\rho _{k}\varphi _{k}(z)$ where there exist $R,M>0$ such that $|z|<R$ and $|\zeta |<R$ implies $|\varphi _{k}(z)|<M$ and $|\varphi _{k}(z)-\varphi _{k}(\zeta )|\leq r|z-\zeta |,\forall k.\$ Furthermore, suppose ${\textstyle \rho _{k}\geq 0,\sum _{k=1}^{\infty }\rho _{k}<\infty }$ and ${\textstyle R>M\sum _{k=1}^{\infty }\rho _{k}.}$ Then for ${\textstyle R*<R-M\sum _{k=1}^{\infty }\rho _{k}}$ $F_{n}(z)\equiv \left(f_{1}\circ f_{2}\circ \cdots \circ f_{n}\right)(z)\to F(z)\qquad {\text{ for }}\{z:|z|<R*\}$

Linear fractional transformations

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Results^[9] for compositions oflinear fractional (Möbius) transformations include the following, as examples:

Theorem LFT1—On the set of convergence of a sequence {F_n} of non-singular LFTs, the limit function is either:

a non-singular LFT,
a function taking on two distinct values, or
a constant.

In (a), the sequence converges everywhere in the extended plane. In (b), the sequence converges either everywhere, and to the same value everywhere except at one point, or it converges at only two points. Case (c) can occur with every possible set of convergence.^[10]

Theorem LFT2^[11]—If {F_n} converges to an LFT, thenf_n converge to the identity functionf(z) =z.

Theorem LFT3^[12]—Iff_n →f and all functions arehyperbolic orloxodromic Möbius transformations, thenF_n(z) →λ, a constant, for all ${\textstyle z\neq \beta =\lim _{n\to \infty }\beta _{n}}$ , where {β_n} are the repulsive fixed points of the {f_n}.

Theorem LFT4^[13]—Iff_n →f wheref isparabolic with fixed pointγ. Let the fixed-points of the {f_n} be {γ_n} and {β_n}. If $\sum _{n=1}^{\infty }\left|\gamma _{n}-\beta _{n}\right|<\infty \quad {\text{and}}\quad \sum _{n=1}^{\infty }n\left|\beta _{n+1}-\beta _{n}\right|<\infty$ thenF_n(z) →λ, a constant in the extended complex plane, for allz.

Examples and applications

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Continued fractions

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The value of the infinite continued fraction

{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+\cdots }}}}

may be expressed as the limit of the sequence {F_n(0)} where

f_{n}(z)={\frac {a_{n}}{b_{n}+z}}.

As a simple example, a well-known result (Worpitsky's circle theorem^[14]) follows from an application of Theorem (A):

Consider the continued fraction

{\cfrac {a_{1}\zeta }{1+{\cfrac {a_{2}\zeta }{1+\cdots }}}}

with

f_{n}(z)={\frac {a_{n}\zeta }{1+z}}.

Stipulate that |ζ| < 1 and |z| <R < 1. Then for 0 <r < 1,

|a_{n}|<rR(1-R)\Rightarrow \left|f_{n}(z)\right|<rR<R\Rightarrow {\frac {a_{1}\zeta }{1+{\frac {a_{2}\zeta }{1+\cdots }}}}=F(\zeta )

, analytic for |z| < 1. SetR = 1/2.

Example. $F(z)={\frac {(i-1)z}{1+i+z{\text{ }}+}}{\text{ }}{\frac {(2-i)z}{1+2i+z{\text{ }}+}}{\text{ }}{\frac {(3-i)z}{1+3i+z{\text{ }}+}}\cdots ,$ $[-15,15]$

Example: Continued fraction1 – Topographical (moduli) image of a continued fraction (one for each point) in the complex plane. [−15,15]

Example.^[9] Afixed-point continued fraction form (a single variable).

f_{k,n}(z)={\frac {\alpha _{k,n}\beta _{k,n}}{\alpha _{k,n}+\beta _{k,n}-z}},\alpha _{k,n}=\alpha _{k,n}(z),\beta _{k,n}=\beta _{k,n}(z),F_{n}(z)=\left(f_{1,n}\circ \cdots \circ f_{n,n}\right)(z)

\alpha _{k,n}=x\cos(ty)+iy\sin(tx),\beta _{k,n}=\cos(ty)+i\sin(tx),t=k/n

Example: Infinite Brooch - Topographical (moduli) image of acontinued fraction form in the complex plane. (6<x<9.6),(4.8<y<8)

Direct functional expansion

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Examples illustrating the conversion of a function directly into a composition follow:

Example 1.^[8]^[15] Suppose $\phi$ is an entire function satisfying the following conditions:

{\begin{cases}\phi (tz)=t\left(\phi (z)+\phi (z)^{2}\right)&|t|>1\\\phi (0)=0\\\phi '(0)=1\end{cases}}

Then

f_{n}(z)=z+{\frac {z^{2}}{t^{n}}}\Longrightarrow F_{n}(z)\to \phi (z)

Example 2.^[8]

f_{n}(z)=z+{\frac {z^{2}}{2^{n}}}\Longrightarrow F_{n}(z)\to {\frac {1}{2}}\left(e^{2z}-1\right)

Example 3.^[7]

f_{n}(z)={\frac {z}{1-{\tfrac {z^{2}}{4^{n}}}}}\Longrightarrow F_{n}(z)\to \tan(z)

Example 4.^[7]

g_{n}(z)={\frac {2\cdot 4^{n}}{z}}\left({\sqrt {1+{\frac {z^{2}}{4^{n}}}}}-1\right)\Longrightarrow G_{n}(z)\to \arctan(z)

Calculation of fixed-points

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Theorem (B) can be applied to determine the fixed-points of functions defined by infinite expansions or certain integrals. The following examples illustrate the process:

Example FP1.^[4] For |ζ| ≤ 1 let

G(\zeta )={\frac {\tfrac {e^{\zeta }}{4}}{3+\zeta +{\cfrac {\tfrac {e^{\zeta }}{8}}{3+\zeta +{\cfrac {\tfrac {e^{\zeta }}{12}}{3+\zeta +\cdots }}}}}}

To find α =G(α), first we define:

{\begin{aligned}t_{n}(z)&={\cfrac {\tfrac {e^{\zeta }}{4n}}{3+\zeta +z}}\\f_{n}(\zeta )&=t_{1}\circ t_{2}\circ \cdots \circ t_{n}(0)\end{aligned}}

Then calculate $G_{n}(\zeta )=f_{n}\circ \cdots \circ f_{1}(\zeta )$ with ζ = 1, which gives: α = 0.087118118... to ten decimal places after ten iterations.

Theorem FP2^[9]—Letφ(ζ,t) be analytic inS = {z : |z| <R} for allt in [0, 1] and continuous int. Set $f_{n}(\zeta )={\frac {1}{n}}\sum _{k=1}^{n}\varphi \left(\zeta ,{\tfrac {k}{n}}\right).$ If |φ(ζ,t)| ≤r <R forζ ∈S andt ∈ [0, 1], then $\zeta =\int _{0}^{1}\varphi (\zeta ,t)\,dt$ has a unique solution,α inS, with $\lim _{n\to \infty }G_{n}(\zeta )=\alpha .$

Evolution functions

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Consider a time interval, normalized toI = [0, 1]. ICAFs can be constructed to describe continuous motion of a point,z, over the interval, but in such a way that at each "instant" the motion is virtually zero (seeZeno's Arrow): For the interval divided into n equal subintervals, 1 ≤k ≤n set $g_{k,n}(z)=z+\varphi _{k,n}(z)$ analytic or simply continuous – in a domainS, such that

\lim _{n\to \infty }\varphi _{k,n}(z)=0

for allk and allz inS,

and $g_{k,n}(z)\in S$ .

Principal example

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Source:^[9]

{\begin{aligned}g_{k,n}(z)&=z+{\frac {1}{n}}\phi \left(z,{\tfrac {k}{n}}\right)\\G_{k,n}(z)&=\left(g_{k,n}\circ g_{k-1,n}\circ \cdots \circ g_{1,n}\right)(z)\\G_{n}(z)&=G_{n,n}(z)\end{aligned}}

implies

\lambda _{n}(z)\doteq G_{n}(z)-z={\frac {1}{n}}\sum _{k=1}^{n}\phi \left(G_{k-1,n}(z){\tfrac {k}{n}}\right)\doteq {\frac {1}{n}}\sum _{k=1}^{n}\psi \left(z,{\tfrac {k}{n}}\right)\sim \int _{0}^{1}\psi (z,t)\,dt,

where the integral is well-defined if ${\tfrac {dz}{dt}}=\phi (z,t)$ has a closed-form solutionz(t). Then

\lambda _{n}(z_{0})\approx \int _{0}^{1}\phi (z(t),t)\,dt.

Otherwise, the integrand is poorly defined although the value of the integral is easily computed. In this case one might call the integral a "virtual" integral.

Example. $\phi (z,t)={\frac {2t-\cos y}{1-\sin x\cos y}}+i{\frac {1-2t\sin x}{1-\sin x\cos y}},\int _{0}^{1}\psi (z,t)\,dt$

Example 1: Virtual tunnels – Topographical (moduli) image of virtual integrals (one for each point) in the complex plane. [−10,10]

Two contours flowing towards an attractive fixed point (red on the left). The white contour (c = 2) terminates before reaching the fixed point. The second contour (c(n) = square root ofn) terminates at the fixed point. For both contours,n = 10,000

Example. Let:

g_{n}(z)=z+{\frac {c_{n}}{n}}\phi (z),\quad {\text{with}}\quad f(z)=z+\phi (z).

Next, set $T_{1,n}(z)=g_{n}(z),T_{k,n}(z)=g_{n}(T_{k-1,n}(z)),$ andT_n(z) =T_n,n(z). Let

T(z)=\lim _{n\to \infty }T_{n}(z)

when that limit exists. The sequence {T_n(z)} defines contours γ = γ(c_n,z) that follow the flow of the vector fieldf(z). If there exists an attractive fixed point α, meaning |f(z) − α| ≤ ρ|z − α| for 0 ≤ ρ < 1, thenT_n(z) →T(z) ≡ α along γ = γ(c_n,z), provided (for example) $c_{n}={\sqrt {n}}$ . Ifc_n ≡c > 0, thenT_n(z) →T(z), a point on the contour γ = γ(c,z). It is easily seen that

\oint _{\gamma }\phi (\zeta )\,d\zeta =\lim _{n\to \infty }{\frac {c}{n}}\sum _{k=1}^{n}\phi ^{2}\left(T_{k-1,n}(z)\right)

and

L(\gamma (z))=\lim _{n\to \infty }{\frac {c}{n}}\sum _{k=1}^{n}\left|\phi \left(T_{k-1,n}(z)\right)\right|,

when these limits exist.

These concepts are marginally related toactive contour theory in image processing, and are simple generalizations of theEuler method

Self-replicating expansions

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Series

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The series defined recursively byf_n(z) =z +g_n(z) have the property that the nth term is predicated on the sum of the firstn − 1 terms. In order to employ theorem (GF3) it is necessary to show boundedness in the following sense: If eachf_n is defined for |z| <M then |G_n(z)| <M must follow before |f_n(z) − z| = |g_n(z)| ≤ Cβ_n is defined for iterative purposes. This is because $g_{n}(G_{n-1}(z))$ occurs throughout the expansion. The restriction

|z|<R=M-C\sum _{k=1}^{\infty }\beta _{k}>0

serves this purpose. ThenG_n(z) →G(z) uniformly on the restricted domain.

Example (S1). Set

f_{n}(z)=z+{\frac {1}{\rho n^{2}}}{\sqrt {z}},\qquad \rho >{\sqrt {\frac {\pi }{6}}}

andM = ρ². ThenR = ρ² − (π/6) > 0. Then, if $S=\left\{z:|z|<R,\operatorname {Re} (z)>0\right\}$ ,z inS implies |G_n(z)| <M and theorem (GF3) applies, so that

{\begin{aligned}G_{n}(z)&=z+g_{1}(z)+g_{2}(G_{1}(z))+g_{3}(G_{2}(z))+\cdots +g_{n}(G_{n-1}(z))\\&=z+{\frac {1}{\rho \cdot 1^{2}}}{\sqrt {z}}+{\frac {1}{\rho \cdot 2^{2}}}{\sqrt {G_{1}(z)}}+{\frac {1}{\rho \cdot 3^{2}}}{\sqrt {G_{2}(z)}}+\cdots +{\frac {1}{\rho \cdot n^{2}}}{\sqrt {G_{n-1}(z)}}\end{aligned}}

converges absolutely, hence is convergent.

Example (S2): $f_{n}(z)=z+{\frac {1}{n^{2}}}\cdot \varphi (z),\varphi (z)=2\cos(x/y)+i2\sin(x/y),>G_{n}(z)=f_{n}\circ f_{n-1}\circ \cdots \circ f_{1}(z),\qquad [-10,10],n=50$

Example (S2)- A topographical (moduli) image of a self generating series.

Products

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The product defined recursively by

f_{n}(z)=z(1+g_{n}(z)),\qquad |z|\leqslant M,

has the appearance

G_{n}(z)=z\prod _{k=1}^{n}\left(1+g_{k}\left(G_{k-1}(z)\right)\right).

In order to apply Theorem GF3 it is required that:

\left|zg_{n}(z)\right|\leq C\beta _{n},\qquad \sum _{k=1}^{\infty }\beta _{k}<\infty .

Once again, a boundedness condition must support

\left|G_{n-1}(z)g_{n}(G_{n-1}(z))\right|\leq C\beta _{n}.

If one knowsCβ_n in advance, the following will suffice:

|z|\leqslant R={\frac {M}{P}}\qquad {\text{where}}\quad P=\prod _{n=1}^{\infty }\left(1+C\beta _{n}\right).

ThenG_n(z) →G(z) uniformly on the restricted domain.

Example (P1). Suppose $f_{n}(z)=z(1+g_{n}(z))$ with $g_{n}(z)={\tfrac {z^{2}}{n^{3}}},$ observing after a few preliminary computations, that |z| ≤ 1/4 implies |G_n(z)| < 0.27. Then

\left|G_{n}(z){\frac {G_{n}(z)^{2}}{n^{3}}}\right|<(0.02){\frac {1}{n^{3}}}=C\beta _{n}

and

G_{n}(z)=z\prod _{k=1}^{n-1}\left(1+{\frac {G_{k}(z)^{2}}{n^{3}}}\right)

converges uniformly.

Example (P2).

g_{k,n}(z)=z\left(1+{\frac {1}{n}}\varphi \left(z,{\tfrac {k}{n}}\right)\right),

G_{n,n}(z)=\left(g_{n,n}\circ g_{n-1,n}\circ \cdots \circ g_{1,n}\right)(z)=z\prod _{k=1}^{n}(1+P_{k,n}(z)),

P_{k,n}(z)={\frac {1}{n}}\varphi \left(G_{k-1,n}(z),{\tfrac {k}{n}}\right),

\prod _{k=1}^{n-1}\left(1+P_{k,n}(z)\right)=1+P_{1,n}(z)+P_{2,n}(z)+\cdots +P_{k-1,n}(z)+R_{n}(z)\sim \int _{0}^{1}\pi (z,t)\,dt+1+R_{n}(z),

\varphi (z)=x\cos(y)+iy\sin(x),\int _{0}^{1}(z\pi (z,t)-1)\,dt,\qquad [-15,15]:

Example (P2): Picasso's Universe – a derived virtual integral from a self-generating infinite product. Click on image for higher resolution.

Continued fractions

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Example (CF1): A self-generating continued fraction.^[9]

{\begin{aligned}F_{n}(z)&={\frac {\rho (z)}{\delta _{1}+}}{\frac {\rho (F_{1}(z))}{\delta _{2}+}}{\frac {\rho (F_{2}(z))}{\delta _{3}+}}\cdots {\frac {\rho (F_{n-1}(z))}{\delta _{n}}},\\\rho (z)&={\frac {\cos(y)}{\cos(y)+\sin(x)}}+i{\frac {\sin(x)}{\cos(y)+\sin(x)}},\qquad [0<x<20],[0<y<20],\qquad \delta _{k}\equiv 1\end{aligned}}

Example CF1: Diminishing returns – a topographical (moduli) image of a self-generating continued fraction.

Example (CF2): Best described as a self-generating reverseEuler continued fraction.^[9]

G_{n}(z)={\frac {\rho (G_{n-1}(z))}{1+\rho (G_{n-1}(z))-}}\ {\frac {\rho (G_{n-2}(z))}{1+\rho (G_{n-2}(z))-}}\cdots {\frac {\rho (G_{1}(z))}{1+\rho (G_{1}(z))-}}\ {\frac {\rho (z)}{1+\rho (z)-z}},

\rho (z)=\rho (x+iy)=x\cos(y)+iy\sin(x),\qquad [-15,15],n=30

Example CF2: Dream of Gold – a topographical (moduli) image of a self-generating reverse Euler continued fraction.

References

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