Infinite compositions of analytic functions

In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/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 of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.

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

NotationEdit

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

Forward compositions:  

Backward compositions:  

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

 

For convenience, set Fn(z) = F1,n(z) and Gn(z) = G1,n(z).

One may also write   and  

Contraction theoremEdit

Many results can be considered extensions of the following result:

Contraction Theorem for Analytic Functions[1] — Let f be analytic in a simply-connected region S and continuous on the closure S of S. Suppose f(S) is a bounded set contained in S. Then for all z in S there exists an attractive fixed point α of f in S such that:

 

Infinite compositions of contractive functionsEdit

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

Forward (inner or right) Compositions Theorem — {Fn} converges uniformly on compact subsets of S to a constant function F(z) = λ.[2]

Backward (outer or left) Compositions Theorem — {Gn} converges uniformly on compact subsets of S to γ ∈ Ω if and only if the sequence of fixed points {γn} of the {fn} converges to γ.[3]

Additional theory resulting from investigations based on these two theorems, particularly Forward Compositions Theorem, include location analysis for the limits obtained here [1]. For a different approach to Backward Compositions Theorem, see [2].

Regarding Backward Compositions Theorem, the example f2n(z) = 1/2 and f2n−1(z) = −1/2 for S = {z : |z| < 1} demonstrates the inadequacy of simply requiring contraction into a compact subset, like Forward Compositions Theorem.

For functions not necessarily analytic the Lipschitz condition suffices:

Theorem[4] — Suppose   is a simply connected compact subset of   and let   be a family of functions that satisfies

 
Define:
 
Then   uniformly on   If   is the unique fixed point of   then   uniformly on   if and only if  .

Infinite compositions of other functionsEdit

Non-contractive complex functionsEdit

Results involving entire functions include the following, as examples. Set

 

Then the following results hold:

Theorem E1[5] — If an ≡ 1,

 
then FnF is entire.

Theorem E2[6] — Set εn = |an−1| suppose there exists non-negative δn, M1, M2, R such that the following holds:

 
Then Gn(z) → G(z) is analytic for |z| < R. Convergence is uniform on compact subsets of {z : |z| < R}.

Additional elementary results include:

Theorem GF3[4] — Suppose   where there exist   such that   implies   Furthermore, suppose   and   Then for  

 

Theorem GF4[4] — Suppose   where there exist   such that   and   implies   and   Furthermore, suppose   and   Then for  

 


Example GF1:  [7]

 
Example GF1:Reproductive universe – A topographical (moduli) image of an infinite composition.

Example GF2:  

 
Example GF2:Metropolis at 30K – A topographical (moduli) image of an infinite composition.

Linear fractional transformationsEdit

Results[6] for compositions of linear fractional (Möbius) transformations include the following, as examples:

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

  1. a non-singular LFT,
  2. a function taking on two distinct values, or
  3. 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.[8]

Theorem LFT2[9] — If {Fn} converges to an LFT, then fn converge to the identity function f(z) = z.

Theorem LFT3[10] — If fnf and all functions are hyperbolic or loxodromic Möbius transformations, then Fn(z) → λ, a constant, for all  , where {βn} are the repulsive fixed points of the {fn}.

Theorem LFT4[11] — If fnf where f is parabolic with fixed point γ. Let the fixed-points of the {fn} be {γn} and {βn}. If

 
then Fn(z) → λ, a constant in the extended complex plane, for all z.

Examples and applicationsEdit

Continued fractionsEdit

The value of the infinite continued fraction

 

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

 

As a simple example, a well-known result (Worpitsky Circle*[12]) follows from an application of Theorem (A):

Consider the continued fraction

 

with

 

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

 , analytic for |z| < 1. Set R = 1/2.

Example.    

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

Example.[6] A fixed-point continued fraction form (a single variable).

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

Direct functional expansionEdit

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

Example 1.[5][13] Suppose   is an entire function satisfying the following conditions:

 

Then

 .

Example 2.[5]

 

Example 3.[4]

 

Example 4.[4]

 

Calculation of fixed-pointsEdit

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.[3] For |ζ| ≤ 1 let

 

To find α = G(α), first we define:

 

Then calculate   with ζ = 1, which gives: α = 0.087118118... to ten decimal places after ten iterations.

Theorem FP2[6] — Let φ(ζ, t) be analytic in S = {z : |z| < R} for all t in [0, 1] and continuous in t. Set

 
If |φ(ζ, t)| ≤ r < R for ζS and t ∈ [0, 1], then
 
has a unique solution, α in S, with  

Evolution functionsEdit

Consider a time interval, normalized to I = [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 (see Zeno's Arrow): For the interval divided into n equal subintervals, 1 ≤ kn set   analytic or simply continuous – in a domain S, such that

  for all k and all z in S,

and  .

Principal example[6]Edit

 

implies

 

where the integral is well-defined if   has a closed-form solution z(t). Then

 

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.  

 
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 of n) terminates at the fixed point. For both contours, n = 10,000

Example. Let:

 

Next, set   and Tn(z) = Tn,n(z). Let

 

when that limit exists. The sequence {Tn(z)} defines contours γ = γ(cn, z) that follow the flow of the vector field f(z). If there exists an attractive fixed point α, meaning |f(z) − α| ≤ ρ|z − α| for 0 ≤ ρ < 1, then Tn(z) → T(z) ≡ α along γ = γ(cn, z), provided (for example)  . If cnc > 0, then Tn(z) → T(z), a point on the contour γ = γ(c, z). It is easily seen that

 

and

 

when these limits exist.

These concepts are marginally related to active contour theory in image processing, and are simple generalizations of the Euler method

Self-replicating expansionsEdit

SeriesEdit

The series defined recursively by fn(z) = z + gn(z) have the property that the nth term is predicated on the sum of the first n − 1 terms. In order to employ theorem (GF3) it is necessary to show boundedness in the following sense: If each fn is defined for |z| < M then |Gn(z)| < M must follow before |fn(z) − z| = |gn(z)| ≤ n is defined for iterative purposes. This is because   occurs throughout the expansion. The restriction

 

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

Example (S1). Set

 

and M = ρ2. Then R = ρ2 − (π/6) > 0. Then, if  , z in S implies |Gn(z)| < M and theorem (GF3) applies, so that

 

converges absolutely, hence is convergent.

Example (S2):  

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

ProductsEdit

The product defined recursively by

 

has the appearance

 

In order to apply Theorem GF3 it is required that:

 

Once again, a boundedness condition must support

 

If one knows n in advance, the following will suffice:

 

Then Gn(z) → G(z) uniformly on the restricted domain.

Example (P1). Suppose   with   observing after a few preliminary computations, that |z| ≤ 1/4 implies |Gn(z)| < 0.27. Then

 

and

 

converges uniformly.

Example (P2).

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

Continued fractionsEdit

Example (CF1): A self-generating continued fraction.[6]


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

Example (CF2): Best described as a self-generating reverse Euler continued fraction.[6]

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

See alsoEdit

ReferencesEdit

  1. ^ Henrici, P. (1988) [1974]. Applied and Computational Complex Analysis. Vol. 1. Wiley. ISBN 978-0-471-60841-7.
  2. ^ Lorentzen, Lisa (November 1990). "Compositions of contractions". Journal of Computational and Applied Mathematics. 32 (1–2): 169–178. doi:10.1016/0377-0427(90)90428-3.
  3. ^ a b Gill, J. (1991). "The use of the sequence Fn(z)=fn∘⋯∘f1(z) in computing the fixed points of continued fractions, products, and series". Appl. Numer. Math. 8 (6): 469–476. doi:10.1016/0168-9274(91)90109-D.
  4. ^ a b c d e Gill 2017
  5. ^ a b c Kojima, Shota (May 2012). "On the convergence of infinite compositions of entire functions". Archiv der Mathematik. 98 (5): 453–465. doi:10.1007/s00013-012-0385-z. S2CID 121444171.
  6. ^ a b c d e f g Gill, J. (2012). "Convergence of Infinite Compositions of Complex Functions" (PDF). Communications in the Analytic Theory of Continued Fractions. XIX.
  7. ^ https://www.researchgate.net/publication/351764310_A_Short_Note_On_the_Dynamical_System_of_the_Reproductive_Universe
  8. ^ Piranian, G.; Thron, W. J. (1957). "Convergence properties of sequences of linear fractional transformations". Michigan Mathematical Journal. 4 (2). doi:10.1307/mmj/1028989001.
  9. ^ de Pree, J. D.; Thron, W. J. (December 1962). "On sequences of Moebius transformations". Mathematische Zeitschrift. 80 (1): 184–193. doi:10.1007/BF01162375. S2CID 120487262.
  10. ^ Mandell, Michael; Magnus, Arne (1970). "On convergence of sequences of linear fractional transformations". Mathematische Zeitschrift. 115 (1): 11–17. doi:10.1007/BF01109744. S2CID 119407993.
  11. ^ Gill, John (1973). "Infinite compositions of Möbius transformations". Transactions of the American Mathematical Society. 176: 479. doi:10.1090/S0002-9947-1973-0316690-6.
  12. ^ Lorentzen, L.; Waadeland, H. (1992). Continued Fractions with Applications. Elsevier Science. ISBN 978-0-444-89265-2.[page needed]
  13. ^ Steinmetz, N. (2011) [1993]. Rational Iteration. de Gruyter. ISBN 978-3-11-088931-4.

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