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](https://math.stackexchange.com/questions/3190787/when-is-exponential-map-from-lie-algebra-to-lie-group-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.