In this thesis we study the geometric limits under degree growth of Julia sets and filled Julia sets for complex polynomials with a unique critical point at $z = 0$. Specifically, for $c \in \mathbb{S}^1$, we are interested in the limit of the associated sequence of Julia sets $J(f_{n,c})$ in the Hausdorff metric where $f_{n,c} : z \mapsto z^n + c$ parametrizes the family of unicritical polynomials of degree $n > 1$. The geometry of $J(f_{n,c})$ depends on the location of $c$ in the parameter space of degree $n$, so the study of geometric limits of Julia sets as the degree tends to infinity is simultaneously a study of how the connectedness locus changes. We uncover a type of periodicity in the moduli space for unicritical polynomials which allows us to improve previous results obtained in this particular setting in 2015 by Kaschner, Romero and Simmons. We complete the description of subsequential geometric limits and the obstructions for existence for $\theta \in \rmathbb{Q}$. When $\theta \in \mathbb{R} \setminus \mathbb{Q}$ the question remains: For which $c = e^{2\pi i \theta}$, if any, does the geometric limit of Julia sets exist? We show that the limit fails to exist for a dense $G_\delta$ set of parameters. We use the periodicity in the sequence of moduli spaces and diophantine approximation to study conditions for existence of a subsequential geometric limit.It remains an open question whether there exists subsequential limits other than the extreme cases $\mathbb{S}^1$ and $\overline{\mathbb{D}}$.

Creator

• Jensen, Signe Emalia

DOI

• https://doi.org/10.21985/n2-h324-v312

Subject

• Mathematics

Language

• en

Alternate Identifier

• http://dissertations.umi.com/northwestern:15689
• etdadmin_upload_840531

Date created

• 2021-01-01

Resource type

• Dissertation

Rights statement

• In Copyright