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

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.We denote by Homeo + (S 1 ) the group of orientation-preserving homeomorphisms of the circle. We regard the circle S 1 as the quotient R/Z and denote by p : R → S 1 the projection. Let Homeo + (S 1 ) be the group of lifts of orientation-preserving homeomorphisms to R, namely, homeomorphisms of R commuting with integral translations.