## All Dissertations, Theses, and Capstone Projects

#### Title

Dynamics of the Family lambda tan z^2

9-2019

Dissertation

Ph.D.

Mathematics

Linda Keen

#### Committee Members

Yunping Jiang

Sudeb Mitra

Frederick Gardiner

#### Subject Categories

Dynamical Systems | Mathematics | Physical Sciences and Mathematics

#### Keywords

complex dynamics, quasiconformal surgery, Cantor set, dynamics of meromorphic maps

#### Abstract

We prove some topological properties of the dynamical plane ($z$-plane) and a combinatorial structure of the parameter plane of a holomorphic family of meromorphic maps $\lambda \tan z^2$. In the dynamical plane, we prove that there is no Herman ring and the Julia set is a Cantor set for the map when the parameter is in the central capture component. Julia set is connected for the maps when the parameters are in other hyperbolic components. In the parameter plane, I prove that the capture components are simply connected and there are always four hyperbolic shell components attached to a virtual center. The capture components and the periodic shell components of the period greater than one are bounded.

This work is embargoed and will be available for download on Thursday, September 30, 2021