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

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