John Gill

Colorado State University - Pueblo (retired) · Mathematics

Convergence criteria are established for infinite compositions of functions defined on a Banach space.

A sequence of functions in C that converges to a limit function having an attractive fixed point forms one of two infinite compositional chains. The convergence behavior of these chains then depends upon the attracting nature of the limit function. The expression "limit-periodic" arose from the study of analytic continued fractions. There is some o...

A parabolic Mobius transformation (MT) has a single (attracting) fixed point in the complex plane, whereas other types of MTs have two fixed points, the most common having an attracting and a repelling fixed point. There is a smooth transformational process relating MTs; but how does one create a second fixed point as the parabolic case morphs into...

A vector field in the complex plane acts on a point with each iteration. If a Lipshitz constant (in the Banach theorem sense) varies with time, then a limiting process reveals the behavior of the point over a unit period. The image below demonstrates the unusual non-fractal graphics resulting from these processes.

A dynamical system involves a differential equation and/or iterative process with a single evolution function. This process with an infinite sequence of evolution functions no longer correlates with a single DE but with a sequence of coupled DEs. How can these two scenarios be reconciled?

There are sequences of continuously differentiable contours in the complex plane that converge uniformly to the line segment [0,1], even to z=0, but have lengths approaching any predetermined size, even infinite. Work (or potential energy) arising from these sequences is explored briefly.

A short exposition of a specific form of complex functions suitable for infinite compositions in C, both in the contexts of limit functions and definite integrals describing contours.

Möbius transformations can be considered geometric/algebraic entities or functions of a complex variable that move points around in C by iteration. This latter interpretation arises in the study of analytic continued fractions. However, iterative formats for hyperbolic/loxodromic/elliptic MTs differ markedly from those of parabolic MTs. This note p...

Reproductive Universe is an elaborate topographical image emerging unpredictably from the infinite composition of a certain sequence of non-holomorphic complex functions. An initial investigation of the dynamics of the components of the system requires elementary techniques.

SU(2) corresponds to a group of linear fractional transformations. This note briefly explores the dynamics of these functions in the complex plane.

Described here is an elementary dynamical system in the complex plane in which iteration (or more accurately, infinite composition) is intensified, generating tunnels from starting points to the centroid of a finite collection of attracting fixed points.

Series, continued fractions, products, and integrals are examples of infinite expansions in C that can be expressed as compositions of simple functional units. The attracting fixed points of many of these expansions can be obtained by using a derived second type of composition. This note extends the result for analytic compositional functions in [1...

A family of sequences of elementary contours is described, each sequence consisting of continuously differentiable functions, with each sequence converging uniformly to the unit interval, but with arc lengths in the limit ranging from one to infinite.

Rational and polynomial functions can at times be written in terms of their fixed points. Replacing an attracting fixed point by an attractor function or transform changes the original function. This derived function may iterate to new attracting fixed points. This note continues the informal investigations in [1] and [2].

Linear Fractional Transformations (LFTs) can be expressed in terms of their fixed points and multiplier. If these are themselves functions of the underlying variable, LFT forms (LFTfs) are produced, generalizations that may coincide with their parent LFTs at numerous fixed points upon iteration. Unusual image cells may be generated by using trig fu...

Initial convergence theory is developed for limit periodic iterations involving a sequence of indifferent fixed points of a class of rational functions in the complex plane, arising as a generalization of a simple iteration example in which any neighborhood of an indifferent fixed point z=p contains points that are proven to converge to p or diverg...

The Joukowski transform (JT) is generalized in a certain way, and is written in a fashion involving fixed points, similar to its cousin the Möbius transformation(MT), allowing some elementary convergence/divergence properties of the dynamical system to be easily described as they are for MTs. Beyond simple iteration, there are more sensitive result...

A brief discussion of iterative convergence/divergence behavior of Joukowskian Transforms (JTs) and Möbius Transformations (MTs) that are hyperbolic. Plus convergence results for infinite compositions of JTs (using the extended Banach FP theorem) and MTs (using Lorentzens' theorem) in which there are no repeated or convergent coefficients. This not...

Previous theory of infinite compositions of Möbius transformations (LFTs) that approach f(z)=z require rapid convergence of the three parameters of the functions , and computations are onerous [1]. In this note, a relatively simple argument shows that one parameter can be left simply bounded and convergence to a limit function is assured, although...

Vector fields having fixed points are embodied in certain general infinite compositional expansions about the origin or in neighborhoods of these points. The dynamic behavior in the two cases can differ.

Outer (or left) infinite compositions of complex functions occur in two formats. We examine a special trigonometry-based case under one of these formats, then derive a similar result for Inner (or right) compositions. In addition, images show implied convergence rates corresponding to the number of composition steps and ratios.

Convergence theory of infinite compositions is easily derived for expanding about the origin. However, if the defining function has a singularity one might wish to expand about some other point. A procedure for these expansions is demonstrated in two special cases: a pole and an essential singularity.

A brief collection of expansions of elementary complex functions, f(z), as infinite compositions on the variable z. Backward recursion of these expansions provides accuracy within a few iterations.

Three specific formats arising in the expansions of functions into infinite compositions in the complex plane are investigated, extending existing theory. Images showing moduli symmetries are shown.

The Cauchy Product of two series in C has terms that are, in a sense, symmetrically interlaced. Analogues of this property in infinite compositions in general are examined, including a method for solving numerically a pair of coupled differential equations. "Head and tail" continued fraction forms also demonstrate this property.

Tannery type compositions in the complex plane may contain ambiguous variables of the form k/n where k:1 -> n.

A brief discussion of infinite compositions of linear fractional transformation forms. There is no requirement the LFTs be periodic or limit-periodic: the first such results. Images are moduli contours.

Abstract: A pseudo-functional derivative is defined on sets of contours in the complex plane with associated measures and used to algorithmically solve a functional derivative equation, with imagery.

A complex time variable, T, that distorts normal time, can be used to generate contours. Examples of imaginary time, including a functional integral are given. Time distortion forces contours to attractors, demonstrated by hypothetical historical event planes and time/event contours. Perception of time intervals with age leads to a concept of imagi...

When does the integral of an infinite composition of complex functions exist? When does the infinite composition of integrals exist? This note provides some answers to these questions in elementary settings not requiring functions be analytic. The images are modulus contours with dark=small values.

Starting from a rudimentary definition of functional integral and basic complex variable theory several examples of functional integrals and their values are derived. In addition, images of the complex plane in which each point is associated with a functional integral are presented. This is not the functional integral used in physics, nor are these...

Certain infinite products in the complex plane can be viewed as compositions and as such give rise to expansions that are Riemann sums (poorly defined at times). These infinite products are " self-generating. "

In the Banach theorem simple iteration of a single function converges to a unique fixed point. A variation is described in which infinite sequences of functions are composed, uniformly converging to unique points in the metric space analogous to the Banach fixed point.

Linear fractional transformations (LFTs) that generate continued fractions can be written entirely in terms of their two fixed points, leading to fixed-point continued fractions. These continued fractions form the basis of expansions that resemble continued fractions but are much more general. This note describes rudimentary convergence theory and...

There is a simple expression for the velocity of the tip of a contour. Colored images describe this velocity. The rate of change of tip-velocity with regard to initial points is also described through imagery.

There are at least three distinct definitions of contraction that are useful in the theory of infinite compositions of complex functions.

Elementary means not requiring the complex functions be holomorphic. Theorem proofs are fairly simple and are outlined. This is a brief compilation of such basic results, with examples, prefaced by three fundamental theorems about holomorphic functions. Images are moduli contours: black(zero), red, green, blue (very large)

An informal presentation of several theorems on convergence of (outer) infinite compositions of complex functions, plus examples of functional expansions and images.

An elementary discrete constructive algorithm defines contours in C , generating a simple Banach space & images.

Informal discussions of unusual continued fraction examples & theory. The images – apart from vector fields-are simple topographical (contour) maps with dark=small moduli and light=large moduli, and do not necessarily correspond to theoretical hypotheses.

Further elementary explorations and the imagery of contour and virtual integrals of implicit complex functions.

Further comments on implicit functions and vector fields.

Elementary commentary on Implicit functions and graphs.

The approximants of continued fractions lie on natural curves in the complex plane.

Contours in the complex plane can be combined in various ways and also subject to several operations.

Outer of left compositions of complex functions are rarely if ever encountered. By expanding a function to the right (inner) it is possible to find the outer expansion of its inverse in many instances.

The value of a contour integral is described as a simple vector arising from an associated "hybrid" contour.

Siamese Contours – having the same initial points -arise in two or more time-dependent complex vector fields and can be combined into simple sums and products that incorporate the features of the vector fields. These elementary classroom notes illustrate the processes. Imagine the following scenario: We know the various paths a particle in the comp...

Direct expansion of certain complex functions into infinite compositions. The first such expansions as outer or left infinite compositions.

Contours in the complex plane are investigated, both from a calculus perspective and an iteration approach. Zeno contours arise from theoretical infinite iteration, and are connected to certain integrals that are "virtual" in that their values are easily computed, but their integrands are abstractions that frequently are not obtainable.

The images that follow are topographical graphs of various complex functions having isolated singularities including poles, essential singularities and branch points. There are also examples of singularities that are dense in certain regions of the complex plane.

Time-dependent vector fields are illustrated as vector clusters ranging from black to bright green as time progresses from t=0 to t=1. Zeno contours are pathlines that can be generated easily in these TDVFs. In a complex and changing fluid environment, a boat is set adrift and must arrive at a certain point at a certain time. Where must it be place...

Computer generated images in the complex plane of iterations of certain analytic continued fractions.

Tannery's Theorem can be generalized in several ways.

Inner compositions of analytic functions and outer compositions of analytic functions � ���are variations on simple iteration, and their convergence behaviors as n becomes infinite may reflect that of simple iterations of contraction mappings�. In addition, results by the author and others provide convergence information about such compositions tha...

Euler's Connection describes an exact equivalence between certain continued fractions and power series. If the partial numerators and denominators of the continued fractions are perturbed slightly, the continued fractions equal power series plus easily computed error terms. These continued fractions may be integrated by the series with another easi...

A continued fraction in the complex plane is a discrete expansion having approximants {Fn} for n=1,2,…. There is a naturally derived function F(t), continuous and piecewise differentiable for t>0, such that F(n)=Fn for each n.

This paper investigates convergence behavior of composition sequences f1∘f2∘…∘fn(z) and fn∘fn−1∘…∘f1(z) where the fn's are bilinear transformations and fn→z. Additional results are provided for the case when the fn's are more general functions.

Although it is difficult to differentiate analytic functions defined by continued fractions, it is relatively easy in some cases to determine uniform bounds on such derivatives by perceiving the continued fraction as an infinite composition of linear fractional transformations and applying an infinite chain rule for differentiation.

It is shown, using classical means, that the outer composition of hyperbolic or loxodromic linear fractional transformations {fn}, where fn→f, converges to α, the attracting fixed point of f, for all complex numbers z, with one possible exception, z0. I.e.,Fn(z):=fn∘fn−1∘…∘f1(z)→αWhen z0 exists, Fn(z0)→...

The basic convergence behavior of the sequence {Fn} where F1=f1 and Fn(z)=fn(Fn−1(z)) is described when each fn is analytic in a region S, S⊃fn(S), and the fixed points of {fn} converge. The sequence {Fn} is then used as a process to efficiently compute the attractive fixed points of functions defined by certain continued fractions, infinite produc...

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...

The sequence {Fn(z)} is one kind of generalization of limit periodic continued fractions. The convergence/divergence of {Fn(z)} relative to that of the iterative sequence {Fn(z)} is determined when ⨍n→⨍ uniformly on a domain S. When ⨍ maps the closure of S into S,Fn(z)→λ for all zϵS. If rapidly on S, the sequence {Fn} uniformly shadows, and the two...

Frequently, in applications, a function is iterated in order to determine its fixed point, which represents the solution of some problem. In the variation of iteration presented in this paper fixed points serve a different purpose. The sequence {Fn(z)} is studied, where F1(z) = f1(z) and Fn(z) = Fn−1(fn(z)), with fn → f. Many infinite arithmetic ex...

Compositional chains of the form f{hook}1 {ring operator} f{hook}2 {ring operator} f{hook}3 {ring operator} ⋯, where f{hook}n→f{hook} occur in several infinite proc esses, including continued fractions and power series. This generalization of simple iteration of a single function is investigated in the context of complete metric spaces. Simple fixe...

New and improved truncation error bounds are derived for continued fractions K(a n /1), where a n →0. The geometrical approach is somewhat unusual in that it involves both isometric circles and fixed points of bilinear transformations.

New and improved truncation error bounds are derived for continued fractions K(an1), where an→ 0. The geometrical approach is somewhat unusual in that it involves both isometric circles and fixed points of bilinear transformations.

Limit periodic continued fractions can be accelerated, and, in some instances, analytically extended by the use of certain modifying factors. This procedure is actually Aitken's Δ2-method when applied to equivalent continued fractions/power series. Both acceleration and continuation results are given.

Converging factors for continued fractions K(an/1) are used to enhance convergence either by accelerating the convergence process or by altering the region of convergence if the an’s are functions of a complex variable. The first results concerning the use of converging factors to accelerate convergence in the important case an → 0 are presented in...

Converging factors for continued fractions $K(a_n/1)$ are used to enhance convergence either by accelerating the convergence process or by altering the region of convergence if the $a_n$'s are functions of a complex variable. The first results concerning the use of converging factors to accelerate convergence in the important case $a_n \rightarrow...

A continued fraction can be interpreted as a composition of MObius transformations. Frequently these transformations have powerful attractive fixed points which, under certain circumstances, can be used as converging factors for the continued fraction. The limit of a sequence of such fixed points can be employed as a constant converging factor.

Typescript (photocopy) Dissertation (Ph.D.)--Colorado State University, 1971. Bibliography: leaf 45.

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