A note on semi-conjugacy for circle actions. (English) Zbl 1375.37123
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 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.
MSC:
37E10 | Dynamical systems involving maps of the circle |
20J06 | Cohomology of groups |
37E45 | Rotation numbers and vectors |
54H20 | Topological dynamics (MSC2010) |
20F60 | Ordered groups (group-theoretic aspects) |