One parameter families of Schwarz reflection maps arising from Shabat-Belyi maps

This dissertation explores the field of complex dynamics, focusing on anti-holomorphic actions and the iteration of rational maps. It delves into Schwarz reflection maps, a class of anti-holomorphic dynamics, building upon previous work in the field. The research adapts the classical Douady-Hubbard straightening theorem to develop a semi-local theory for the dynamics of these Schwarz reflections, particularly those with parabolic behavior at the cusp point. The study also investigates one-parameter families of these maps, which arise from Shabat polynomials and are indexed by rooted planar trees. The findings demonstrate that the parameter spaces and escape loci for these families are connected and simply connected, implying that the connectedness loci are themselves connected. The research identifies similarities to the Mandelbrot set, showing the presence of many renormalizable parameters that lead to smaller copies of Multibrot and Multicorn sets within the connectedness loci. Furthermore, it is shown that non-renormalizable parameters are combinatorially rigid.

Curriculum

The dissertation is structured into several chapters, covering background material, detailed analysis of Schwarz reflection maps and their relationship to Shabat-Belyi maps, and the exploration of one-parameter families within this context.

1. 1
   • Introduction
2. 2
   • Background
3. 3
   • Schwarz Reflection Maps as Matings of Anti-Holomorphic Parabolic Maps with Kleinian Reflection Groups
4. 4
   • One Parameter Families of Maps Associated With Belyi-Shabat Polynomials

Careers

While specific career outcomes are not detailed, a PhD in Mathematics from Stony Brook University typically prepares graduates for advanced research positions in academia, government laboratories, and industry. Graduates develop strong analytical, problem-solving, and quantitative skills applicable to a wide range of fields.

• Researcher
• Professor
• Data Scientist
• Cryptographer
• Financial Analyst