A note on semi-conjugacy for circle actions

TL;DR

The paper introduces a new notion of semi-conjugacy for circle group actions, showing it aligns with classical definitions for fixed point free actions and is characterized by the bounded Euler class, clarifying previous ambiguities.

Contribution

It defines semi-conjugacy for all circle actions and proves its equivalence to having the same bounded Euler class, resolving existing confusion in the literature.

Findings

Semi-conjugacy coincides with classical definitions for fixed point free actions.

Two circle actions are semi-conjugate iff they share the same bounded Euler class.

Clarifies the relationship between semi-conjugacy and bounded Euler class in 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.