A note on semi-conjugacy for circle actions (Q2406612)

Summary: We define a notion of semi-conjugacy between orientation-preserving actions of a group on the circle, which for fixed point free actions coincides with a classical definition of \textit{E. Ghys} [Contemp. Math. 58, 81--106 (1987; Zbl 0617.58009); Enseign. Math. (2) 47, No. 3--4, 329--407 (2001; Zbl 1044.37033)]. We then show that two circle actions are semi-conjugate if and only if they have the same bounded Euler class. This clarifies some existing confusion present in the literature.