The orthogonal group $O(n)$ is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations.
http://en.wikipedia.org/wiki/Orthogonal_group
Any references for the ergodic properties of that group acting on the n-sphere?
Thank you
Ergodicity is ensured by the fact that the eigenvalues of an orthogonal matrix satisfy $\|\lambda_i\|=1$, hence $\lambda_i = e^{i\theta_i}$. By the Dirichlet box principle, it follows that for any $n\in\mathbb{N}_0$ there exists $m\in\mathbb{N}_0$, with $m\leq (2\pi n)^n$, such that: $$\max_{i\in[1,n]}\left\|\frac{m\theta_i}{2\pi}\right\|\leq\frac{1}{n},$$ where $\|\cdot\|$ denotes the distance from the closest integer. Ergodicity follows.