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