Advisor(s)

Lex Oversteegen
Alexander Blokh

Committee Member(s)

John Mayer
Lauren Wickman
Nikita Selinger

Document Type

Thesis

Date of Award

6-4-2026

Degree Name

Master of Science (MS)

School

College of Arts and Sciences

Department

Applied Mathematics

Abstract

Connected Julia sets of polynomials generally correspond to laminations, sets of chords of the unit disc that reflect the dynamics of the Julia set. If the circle is measured in revolutions and the polynomials studied are of degree $d$, then the dynamics on the lamination is given by the covering map $\sigma_d(t) := td \pmod 1$ where chords are mapped by their end points. Every lamination has at least one laminational invariant set, which is loosely an invariant complementary component of the lamination. That set has a significant impact on the shape of the Julia set. James Malaugh showed how to topologically transform a laminational invariant set in one degree into one in another degree, but he provided no way to execute these operations concretely. In this thesis, we show how to calculate these operations in terms of $d$-nary coordinates, which allows us to compute the exact locations of invariant sets. We also give a description of the space of certain quadratic invariant gaps of $\si_4$, which are relevant to the study of cubic and quartic polynomials. In particular, these quadratic gaps are relevant to cubic polynomials that have a cycle of Fatou gaps that return with degree 4.

Keywords

Complex Dynamics;Laminations

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