Rotation number and lifts of a Fuchsian action of the modular group on the circle

TL;DR

This paper characterizes the semi-conjugacy class of Fuchsian actions of the modular group on the circle using rotation numbers, revealing unique properties of 5-fold lifts and comparing rotation numbers with linear characters.

Contribution

It provides a new characterization of Fuchsian actions via rotation numbers and identifies special properties of 5-fold lifts.

Findings

Semi-conjugacy class characterized by rotation numbers

Only 5-fold lift admits similar characterization

Highlights differences between rotation number and linear character

Abstract

We characterize the semi-conjugacy class of a Fuchsian action of the modular group on the circle in terms of rotation numbers of two standard generators and that of their product. We also show that among lifts of a Fuchsian action of the modular group, only 5-fold lift admits a similar characterization. These results indicate similarity and difference between rotation number and linear character.