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 works (other than $S^1$)?

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

The motivation: I was thinking of a possible generalization of Poincaré-Birkhoff theorem as follows: We have an area-preserving diffeomorphism on $[0,1]\times G$ where $G$ is a Lie group. On the boundary we get two rotation elements as described above. Now assume that the Lie algebra $\mathfrak{g}$ consists of matrices and we may assume that these rotation elements are invertible with opposite sign determinants. Or we may assume that a reasonable linear functional, say trace, separate these two boundary rotation elements. Then we may state the Poincaré-Birkhoff theorem in this new setting and think to its possible validity.