Date of Award

1-1-2019

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Content Description

1 online resource (v, 84 pages) : illustrations.

Dissertation/Thesis Chair

Rongwei Yang

Committee Members

Alexandre Tchernev, Marius Beceanu, Anca Radulescu, Michael Stessin

Keywords

Douglas-Yang Metric, Dynamics, Grigorchuk Group, Infinite Dihedral Group, Julia Set, Projective Spectrum, Infinite groups, Banach algebras, Holomorphic mappings, Julia sets

Subject Categories

Physical Sciences and Mathematics

Abstract

For a tuple $A= (A_1, A_2, \ldots , A_n)$ of elements in a unital Banach algebra $\mathcal{B}$ its \textit{projective (joint) spectrum} $P(A)$ is the collection of $z~\in~\mathbb{C}^n$ such that $A(z)~=~z_1 A_1 + z_2 A_2 + \ldots + z_n A_n$ is not invertible. We call the complement of $P(A)$ the projective resolvent set, $P^c(A) = \C^n \setminus P(A)$. In this dissertation the primary focus will be on the infinite dihedral group $D_\infty = \langle a,t\mid a^2=t^2 =1 \rangle$ and the left regular representation $\lambda$ acting on $l^2(D_\infty)$ giving the tuple $ \big(I, \lb(a),\lb(t)\big)$. First, using the fundamental form $\Omega_A~=~- \omega _A ^\ast \wedge~\omega _A$ where $\omega_A$ is the holomorphic Maurer-Cartan type $\mathcal{B}$-valued (1,0)-form $\omega_A(z) = A^{-1}(z)dA(z)$, we define the Douglas-Yang metric on $P^c \big((I, \lb(a),\lb(t))\big)$. We demonstrate that $P^c (A)$ with respect to this metric is not complete. Next we turn our attention to the mapping defined by Grigorchuk and Yang in \cite{GY} from the self-similarity of $D_\infty$, $F(z) = \big(z_0(z_0^2-z_1^2-z_2^2), z_1^2 z_2, z_2(z_0^2 -z_2^2) \big)$ and examine Fatou-Julia theory of the mapping. Utilizing projective space to examine $F: \mathbb{P}^2~\to~\mathbb{P}^2$ allows us to define the Julia set of $F(z)$. Then we demonstrate a clear connection between the projective spectrum of $\big(I, \lb(a),\lb(t)\big)$ and the Julia set of $F(z)$.

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