Quasiconformal surgery in transcendental dynamics

Abstract

[en] The goal of this thesis is to survey some results on quasiconformal analysis and quasiconformal surgery, today an essential tool for any researcher in the field of Complex Dynamics.

The first part of the project will be on quasiconformal geometry, to understand the necessary tools in analysis that lead to the Measurable Riemann Mapping Theorem, the main tool to perform surgery. These include several definitions of quasiconformal mappings, both analytic and geometric. The second part will consist on several applications to holomorphic dynamics, with emphasis on those in transcendental dynamics, showing the power of this technique.

A third part of the project will be dedicated to some original work on a particular family of meromorphic transcendental maps. More precisely, we study the family of transcendental meromorphic maps $$ f_\lambda(z)=\lambda\left(\frac{e^z}{z+1}-1\right), $$ and we prove, using quasiconformal surgery, that for certain parameter values the Julia set contains what is known as a Cantor Bouquet.