Tischler graphs of critically fixed rational maps and their applications

TL;DR

This paper investigates critically fixed rational maps on the Riemann sphere, introduces the Tischler graph as a key combinatorial invariant, and establishes a classification linking these maps to connected planar graphs, solving longstanding problems.

Contribution

It proves the connectedness of Tischler graphs for critically fixed maps and establishes a bijective correspondence between these maps and connected planar graphs, solving classical classification problems.

Findings

Tischler graphs are always connected for critically fixed rational maps.

A one-to-one correspondence exists between conjugacy classes of these maps and connected planar graphs.

The results solve classical open problems in rational dynamics.

Abstract

A rational map $f:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ on the Riemann sphere $\widehat{\mathbb{C}}$ is called critically fixed if each critical point of $f$ is fixed under $f$. In this article, we study the properties of a combinatorial invariant, called the Tischler graph, associated with such a map. We show that the Tischler graph of a critically fixed rational map is always connected, establishing a conjecture made by Kevin Pilgrim. This result allows us to solve two classical open problems in rational dynamics in the setting of critically fixed rational maps, namely the combinatorial classification problem and the global curve attractor problem. In particular, we prove that there is a canonical one-to-one correspondence between the conjugacy classes of critically fixed rational maps and the isomorphism classes of connected planar embedded graphs.