#### Advisory Committee Chair

Lex Oversteegen

#### Advisory Committee Members

Ragib Hasan

John Mayer

Lawrence Roberts

Nikita Selinger

Dongsheng Wu

#### Document Type

Dissertation

#### Date of Award

2023

#### Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

#### Abstract

The dynamics of polynomials is a topic of great interest for those studying complex dynamics. Polynomial dynamics is concerned with how collections of points behave under iteration of a polynomial P. The Julia set of P, which is the boundary of the collection of points which iterate off to ∞, is an interesting collection of points to look at because P is sensitive to small pertubations of points in the Julia set. Thanks to the work of William Thurston, we have a tool known as a lamination (a closed collection of chords in the unit disk D which do not cross) to aid us in studying the behavior of P on the Julia set. Given a polynomial P with a locally connected Julia set, Thurston showed us how to construct a q-lamination corresponding to P. In this thesis, we go the other direction by finding a polynomial that corresponds to a given q-lamination (with additional assumptions). We then explore the properties of such laminations which can be realized by a polynomial. We conclude by looking at an appropriate space of laminations. The space that we choose is known as the space of limit laminations which is the closure of the set of q-laminations. The space of limit laminations is used to understand the space of complex polynomials by giving us a way to assign a lamination to a polynomial without a locally connected Julia set. We focus on properties of limit laminations as well as the q-laminations which converge to limit laminations.

#### Recommended Citation

Carty, Adam, "Realization of Hyperbolic Laminations and Properties of Limit Laminations" (2023). *All ETDs from UAB*. 438.

https://digitalcommons.library.uab.edu/etd-collection/438