A decomposition theorem for Herman maps

TL;DR

This paper introduces a decomposition theorem for Herman maps, a class of postcritically infinite branched coverings, showing they can be broken into simpler components like Siegel and Thurston maps, advancing understanding of their structure.

Contribution

It develops a decomposition theorem for Herman maps, enabling analysis of their structure and rational realizations, and addresses a problem posed by McMullen.

Findings

Herman maps can be decomposed along stable multicurves into simpler maps.

The decomposition aids in proving Thurston-type theorems for maps with Herman rings.

The results provide a new framework for understanding postcritically infinite branched coverings.

Abstract

In 1980s, Thurston established a topological characterization theorem for postcritically finite rational maps. In this paper, a decomposition theorem for a class of postcritically infinite branched covering termed `Herman map' is developed. It's shown that every Herman map can be decomposed along a stable multicurve into finitely many Siegel maps and Thurston maps, such that the combinations and rational realizations of these resulting maps essentially dominate the original one. This result gives an answer to a problem of McMullen in a sense and enables us to prove a Thurston-type theorem for rational maps with Herman rings.