## 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 $\uml_d$.

## Recommended Citation

Bhattacharya, Sourav, "Topics in low dimensional dynamical systems: Interval Rotation Numbers and Laminations" (2021). *All ETDs from UAB*. 735.

https://digitalcommons.library.uab.edu/etd-collection/735