Theorems linking the Lie group of the circle and circle homeomorphisms

Question

This is more a question for literature: I'm currently working on my masters thesis about the classification of maps on $\mathbb{S}^1$. I recently started reading about the interpretation of $\mathbb{S}^1$ as a Lie group and found it to be really interesting. I found a few sources that examine the structure of the group of orientation-preserving homeomorphisms which seem to form kind of a Lie group (?). I'm wondering now if there is a central theorem and proof that I could present in my thesis. For homeomorphisms in general there's the Poincaré classification and for diffeomorphisms there's Denjoy's theorem. However I was unable to find some theorem of that kind that links homeomorphisms of the circle to Lie groups or circle maps to groups in general. Maybe someone knows about something like that, thanks!