All ETDs from UAB

Advisory Committee Chair

Alexander Ab Blokh

Advisory Committee Members

Vo Vl Liem

John Jm Mayer

Claudio Cm Morales

Lex Lo Oversteegen

Fazlur Fr Rahman

Document Type

Dissertation

Date of Award

2021

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

The goal of this thesis is to study two broad disciplines: \emph{One Dimensional Dynamical Systems and Complex Dynamical Systems} in the spirit of \emph{``Developing efficient machinery to make conclusions about the dynamics of a given map $f$ from reduced information''}. In \emph{One Dimensional Dynamical Systems}, we accomplish the above stated goal by studying the \emph{forcing relation} among \emph{over-rotation numbers} of points on the interval. We began by discovering peculiar \emph{patterns} named \emph{very badly ordered patterns}, having no \emph{block structure} over an \emph{over-twist pattern}. It has been proved that unlike \emph{circle maps} these patterns \emph{surprisingly} don't \emph{force} a lower \emph{over-rotation number} which demonstrates that dynamics of interval maps are far more complex than that of circle maps. Then we obtained an explicit description of the \emph{strongest unimodal pattern} that forces a given \emph{over-rotation interval}. The same has then been used to construct \emph{unimodal very badly ordered patterns} with arbitrary \emph{non-coprime over-rotation pairs}. Moving from \emph{unimodal} to \emph{bimodal}, we consider the family $\mathcal{B}$ of all \emph{bimodal interval maps}. For a map $g$ in $\mathcal{B}$, a simple algorithm for figuring out the left end point $\rho_g$ of its over-rotation interval was formulated. It is known that the single number $\rho_g$ epitomizes the limiting dynamical behavior of points of the interval reflected by their \emph{over-rotation numbers}. The \emph{combinatorics} of cycles of $g$ on which the over-rotation number $\rho_g$ is assumed , called \emph{over-twist cycles} is also explicitly described. Now moving to \emph{Complex Dynamical Systems} we developed \emph{combinatorial models} for \emph{parameter space of unicritical complex polynomials}. For this we used the idea of \emph{ $\si_d$-invariant Laminations} introduced by Thurston in \cite{thu85}. We show that the set of all \emph{chords} on $\disk$ which are \emph{minors} of some \emph{unicritical lamination} themselves form a lamination named as the \emph{Unicritical Minor Lamination} of \emph{degree} $d$ and denoted by $UML_d$ which serves as the \emph{combinatorial model} for the \emph{parameter space of unicritical complex polynomials}. We also verified the \emph{Generalised Fatou conjecture} for the \emph{unicritical lamination} by proving that the elements of $\uml_d$ associated with \emph{laminations with periodic Fatou gap of degree} $k>1$ are \emph{dense} in $\uml_d$. Finally we concocted an algorithm to construct $\uml_d$.

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