DOI

10.17077/etd.1uobfy7c

Document Type

Dissertation

Date of Degree

Fall 2016

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Palle E. Jorgensen

First Committee Member

Paul S. Muhly

Second Committee Member

Ionut Chifan

Third Committee Member

Gerhard O. Strohmer

Fourth Committee Member

Surjit S. Khurana

Abstract

Both complex dynamics and the theory of reproducing kernel Hilbert spaces have found widespread application over the last few decades. Although complex dynamics started over a century ago, the gravity of it's importance was only recently realized due to B.B. Mandelbrot's work in the 1980's. B.B. Mandelbrot demonstrated to the world that fractals, which are chaotic patterns containing a high degree of self-similarity, often times serve as better models to nature than conventional smooth models. The theory of reproducing kernel Hilbert spaces also having started over a century ago, didn't pick up until N. Aronszajn's classic was written in 1950. Since then, the theory has found widespread application to fields including machine learning, quantum mechanics, and harmonic analysis.

In the paper, Infinite Product Representations of Kernel Functions and Iterated Function Systems, the authors, D. Alpay, P. Jorgensen, I. Lewkowicz, and I. Martiziano, show how a kernel function can be constructed on an attracting set of an iterated function system. Furthermore, they show that when certain conditions are met, one can construct an orthonormal basis of the associated Hilbert space via certain pull-back and multiplier operators.

In this thesis we take for our iterated function system, the family of iterates of a given rational map. Thus we investigate for which rational maps their kernel construction holds as well as their orthornormal basis construction. We are able to show that the kernel construction applies to any rational map conjugate to a polynomial with an attracting fixed point at 0. Within such rational maps, we are able to find a family of polynomials for which the orthonormal basis construction holds. It is then natural to ask how the orthonormal basis changes as the polynomial within a given family varies. We are able to determine for certain families of polynomials, that the dynamics of the corresponding orthonormal basis is well behaved. Finally, we conclude with some possible avenues of future investigation.

Public Abstract

A fractal is an object which exhibits a property called self-similarity; no matter how far one zooms into the image, one will find slightly perturbed copies of the original image. Their are many natural phenomena which are currently best modeled by fractals. These include lightning, plants, animals, the coast line, and brownian motion, to name a few.

A Hilbert space is a collection of objects which satisfy two special properties. The first property is the existence of a two-variable function called an inner product. An inner product allows one to assign a notion of length and angle for the objects in the space. The second property is that the space is complete; intuitively this means there aren't any “naturally obtained" objects missing from the space. A reproducing kernel Hilbert space is a Hilbert space consisting of functions, which satisfy a third property. The third property, called the reproducing property, is the existence of a function, called the kernel function, which allows one to compute values of other functions in the space using the inner product. Hilbert spaces have the nice feature that any object in the space can be obtained as a sum involving a collection of orthogonal normal objects within the set. The term normal means each object has length one, and the term orthogonal means that each pair of objects meet at a 90° angle. The collection of these orthogonal normal objects is called an orthonormal basis.

In this thesis we consider fractals which are obtained from rational maps. A rational map is a fraction in which both the numerator and denominator are polynomials. We investigate a technique that associates to each fractal a reproducing kernel Hilbert space. We ask how does the orthonormal basis associated to the reproducing kernel Hilbert space change as we let the fractal vary? We are able to find a class of fractals for which the dynamics of the associated orthonormal bases is well-behaved in a certain sense.

Keywords

complex dynamics, fractal, functional analysis, kernel function, reproducing kernel

Pages

ix, 61 pages

Bibliography

Includes bibliographical references (pages 59-61).

Copyright

Copyright © 2016 James Edward Tipton

Included in

Mathematics Commons

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