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)?