Thurston equivalence for rational maps with clusters

TL;DR

This paper studies Thurston equivalence classes of rational maps with specific cluster cycles, revealing how combinatorial data determines the class in some cases but not others.

Contribution

It establishes that for fixed degree and cluster, Thurston class is determined by rotation number and critical displacement, with exceptions in higher degrees.

Findings

Thurston class fixed by combinatorial data for degree d with fixed cluster

The result extends to quadratic maps with period two clusters

Higher degree cases show the classification does not hold universally

Abstract

We investigate rational maps with period one and two cluster cycles. Given the definition of a cluster, we show that, in the case where the degree is $d$ and the cluster is fixed, the Thurston class of a rational map is fixed by the combinatorial rotation number $\rho$ and the critical displacement $\delta$ of the cluster cycle. The same result will also be proved in the case that the rational map is quadratic and has a period two cluster cycle, but that the statement is no longer true in the higher degree case.