Cut and semi-conjugate

TL;DR

This paper introduces a broad class of rational functions on the Riemann sphere, showing they can be semi-conjugated to critically finite hyperbolic functions through a process of cutting along specific curves.

Contribution

It defines a general class of rational functions and demonstrates their topological semi-conjugacy to critically finite hyperbolic functions via cutting along smooth curves.

Findings

Existence of a countable family of curves for semi-conjugacy

Construction of semi-conjugacy to critically finite hyperbolic functions

Generalization of rational functions with semi-conjugate dynamics

Abstract

We define a very general class of rational functions f:CP^1 --> CP^1 such that for every function f of this class, there exists a countable family of smooth curves \gamma_i and a critically finite hyperbolic function R such that the dynamical system obtained from f by cutting along the curves \gamma_i is topologically semi-conjugate to R.