Topological regluing of rational functions

TL;DR

This paper explores the topological operation of regluing in rational functions, providing a framework that connects topological models with holomorphic interpretations, inspired by Thurston–Teichmüller theory.

Contribution

It develops a topological theory of regluing for rational functions and suggests directions for a holomorphic extension of this theory.

Findings

Topological models for rational functions are constructed via regluing.

A connection between regluing and Thurston–Teichmüller theory is established.

Potential for a holomorphic theory of regluing is outlined.

Abstract

Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional Thurston--Teichm\"uller theory. We will discuss a topological theory of regluing, and trace a direction in which a holomorphic theory can develop.