The Sullivan dictionary and Bowen-Series maps

TL;DR

This paper explores a new framework that unifies Kleinian groups and rational dynamics by combining Fuchsian groups with complex polynomials into a single dynamical system on the Riemann sphere, advancing the Sullivan dictionary.

Contribution

It introduces a novel approach to connect Kleinian groups and rational dynamics through a unified dynamical system framework, enriching the Sullivan dictionary.

Findings

Established a framework linking Fuchsian groups and complex polynomials

Demonstrated similarities between Kleinian groups and rational dynamics

Provided insights into the structure of combined dynamical systems

Abstract

The Sullivan dictionary between Kleinian groups and rational dynamics describes striking similarities between the fields, both in terms of the objects of study as well as the techniques used. We give an expository account of a recent bridge between the two sides of the dictionary by describing a framework for combining a Fuchsian group with a complex polynomial into a single dynamical system on the Riemann sphere.