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