Date of Degree

9-2019

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

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.

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