Let $G$ be a Lie group whose lie algebra is $\mathfrak{g}$ with exponential map $exp:\mathfrak{g}\to G$.

For what kind of Lie group $G$ the standard process of definition of rotation number for circle homeomorphisms work well?

Namely for every homeomorphism $f:G\to G$ there is a homeomorphism $F:\mathfrak{g}\to \mathfrak{g}$ with $exp\circ F=f\circ exp$ and the limit of $$\frac{F^n(x)-x}{n}$$ as n goes to infinity exists?

This limit as an element of the Lie algebra would be called the rotation element.

What is a precise example for which this process work(other than $S^1$)?

Is it equivalents to the exponential to be a covering map?

The motivation: I was thinking to a possible generalization of Poincare Birkhoff theorem as follows: We have an area preserving diffeomorphism on $[0 1]\times G$ on the boundary we get two rotation elements. Now assume that the Lie algebra consist of matrices and we may asume that these rotation elements are invertible with opposite signe detrminant. Or we may assume that a reasonable linear functional , say trace, separate these two boundary rotation elements. Then we may state the Poincare Birkhoff theorem in this new setting and think to its possible validity.