I'm starting to understand some basics things about equidistributed sequences and i found my self asking this question on the basis of the example of the torus and Weyl equidistribution theorem:
Given a compact(real) connected Lie group $G$, is there always an element $g \in G$ such that $\{g^n\}_{n \in \mathbb{N}}$ equidistributes in $G$, with respect to the Haar-measure of $G$?
Is for example this property true for the rotation groups of Euclidean spaces $SO(n), n \in \mathbb{N}$ ?
Thanks for any suggestion
Edit: the answer received in the comment gives that for trivial reason the orthogonal group never enjoys this property for n>2. So my next dumb question is: what are the other examples where there is this property, apart from products of $S^1$ a finite number of times?