On the classification of critically fixed rational maps

TL;DR

This paper explores the classification of rational maps on the Riemann sphere where each critical point is also a fixed point, integrating dynamical, topological, and algebraic perspectives.

Contribution

It provides a comprehensive classification framework for critically fixed rational maps, combining multiple mathematical approaches.

Findings

Characterization of critically fixed rational maps

Connections between dynamical and algebraic properties

New classification criteria for these maps

Abstract

We discuss the dynamical, topological, and algebraic classification of rational maps $f$ of the Riemann sphere to itself each of whose critical points $c$ is also a fixed-point of $f$, i.e. $f(c)=c$.