Quasisymmetries of the Feigenbaum Julia set and transcendental dynamics

This dissertation investigates the group of quasisymmetries of the Feigenbaum Julia set, denoted as Jc. Quasisymmetry is a type of geometric mapping that preserves the qualitative structure of sets under distortion. The Feigenbaum Julia set is a non-hyperbolic quadratic Julia set, presenting a different challenge compared to hyperbolic sets. The research describes the group of topologically extendable quasisymmetric self-maps of Jc, providing the first example for an infinitely renormalizable map. The study relies on the topology and geometry of the transcendental dynamics of a related renormalization fixed point to fully determine this group.

Curriculum

The dissertation is structured to build understanding from foundational concepts to the specific analysis of the Feigenbaum Julia set and its quasisymmetric properties. It covers background definitions, geometric structures of the Julia set, transcendental dynamics, and the detailed proof of the main theorem.

1. Introduction & Background
   • Introduction and Organization of ChaptersChapter 1
   • Background: qs and qc maps, Quadratic-like maps, Tuning basilicas, Bottcher coordinateChapter 2
2. Geometric Structures and Dynamics
   • Structures within J: Notation, First properties, Limbs, Important Points, Little Julia sets, More properties of JChapter 3
   • Transcendental dynamics of the renormalization fixed point: Definition, Properties of J1, Limbs, Lakes, -points, Tree order, qs structure, Discrepancy between real and imaginary limbsChapter 4
3. Quasisymmetries of J
   • Quasisymmetries of J: Defining maps, No qs rotations, Deep little Julia sets, Lift to T, qc extension, Uniform ApproximationChapter 5

Careers

The dissertation focuses on theoretical mathematics, specifically complex dynamics and fractal geometry. Graduates with this expertise are well-positioned for careers in advanced research, academia, and potentially in fields requiring complex problem-solving and analytical skills.

• Researcher
• Professor
• Data Scientist
• Analyst