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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.

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    $\begingroup$ There's an existing notion of a "rotation set" which might be the generalization you're looking for. You can have more than one limit depending on your choice of $x$. It's already pretty interesting in the case of a torus! $\endgroup$
    – Martin M. W.
    Commented Sep 12, 2023 at 1:41
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    $\begingroup$ @MartinM.W. Thank you! could you please give me a reference. I googled "Rotation set" and Lie group simultaneously but I could not find materials in this regard $\endgroup$
    – Ali Taghavi
    Commented Sep 12, 2023 at 5:40
  • $\begingroup$ Here's a recent paper which has references to follow; note that this focuses on the torus case. (I'm not familiar with more general Lie group constructions, but a torus seems like the simplest entry point!) On the rotation sets of generic homeomorphisms on the torus, arxiv.org/abs/1901.00396 $\endgroup$
    – Martin M. W.
    Commented Sep 12, 2023 at 10:27
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    $\begingroup$ @AliTaghavi, I don't see how you can construct the map $F$ for an arbitrary $f$. The exponential map is not even surjective on certain Lie groups. On the other hand, you need to impose some restrictions on the map $f$ in order to guarantee the existence of the limit. For instance, in most of the cases $f$ must be homotopic to the identity. $\endgroup$
    – Alejandro
    Commented Sep 26, 2023 at 19:18
  • $\begingroup$ @Alejandro Thank you very much for your very interesting comment. I think about its details and also I try to remedy the post. regarding the surjectivity we may add extra assumption "compact connected Lie group" $\endgroup$
    – Ali Taghavi
    Commented Sep 27, 2023 at 18:30

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