A note on semi-conjugacy for circle actions

Abstract

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 Ghys. We then show that two circle actions are semi-conjugate if and only if they have the same bounded Euler class. This settles some existing confusion present in the literature.

39 pages. Improved version of the previous manuscript, incorporating further historical remarks and a discussion of the equivalence between semiconjugacy and monotone equivalence