TITLE: What is known about dynamics on Grassmannians?
QUESTION [12 upvotes]: I have found myself becoming interested in dynamical systems given by homeomorphisms acting on $G(r,d)$, the space of $r$-dimensional subspaces of $\mathbb{R}^d$.  I tried to do a literature search and failed to turn up any useful references or papers.  I'd like to know what is currently known and/or where I can learn more about such things.
My motivation for studying such dynamical systems is the following: in general, a one-dimensional quasi-periodic family of self-adjoint operators can be thought of as one which fibers over an irrational rotation $ \omega \mapsto \omega + \alpha $ of the circle $\mathbb{T} := \mathbb{R} / \mathbb{Z}$.  This notion and its generalizations to operator families fibering over minimal translations of $ \mathbb{T}^d = \mathbb{R}^d / \mathbb{Z}^d $ are extremely well-studied, with many lovely results.
Now, instead of $\mathbb{R} / \mathbb{Z}$, it is reasonably natural to think of the circle as the real projective line $ \mathbb{R}  \mathbb{P}^1 = G(1,2)$, and, in this case, the rotations mentioned above now correspond to the natural action of $ \mathrm{SO}(2,\mathbb{R}) $ on $G(1,2)$.  In this case, it is well-known that the corresponding action is minimal if and only if it is uniquely ergodic if and only if the corresponding angle of rotation is an irrational multiple of $\pi$.  I'd like to know to what extent one can generalize dynamical statements like this to the action of $\mathrm{SO}(d,\mathbb{R})$ on $G(r,d)$.
More precisely, given $R \in \mathrm{SO}(d)$, $R$ naturally preserves a version of normalized Lebesgue measure on $G(r,d)$, which one could realize, for example, by (a constant multiple times) the pushforward of $d-1$ dimensional Lebesgue measure on $S^{d-1} \subset \mathbb{R}^d$ under the quotient map $S^{d-1} \to G(r,d)$.  It is then natural to ask when Lebesgue measure is ergodic with respect to the action of $R$, or when this action is uniquely ergodic (in which case Lebesgue measure is necessarily the unique preserved measure), or minimal.
Still more generally, are there interesting homeomorphisms on $G(r,d)$ which do not arise from the action of a matrix?
Obviously, there have to be some difficulties in higher dimensions. For example, it is easy to see that any $R \in \mathrm{SO}(2n+1)$ necessarily has a fixed point on $S^{2n}$, so the induced action on $G(r,2n+1)$ can never be minimal or uniquely ergodic.

REPLY [3 votes]: I suggest that you also look at the related frame flows, say on a compact smooth Riemannian manifold with negative sectional curvatures. The action (on the set of positively oriented orthonormal frames) moves the first vector of the frame according to the geodesic flows and the remaining ones by parallel transport. It was shown by Brin and Pesin that the time-1 map of the frame flow is a partially hyperbolic diffeomorphism, and later was shown by Brin that the flow is ergodic with respect to the product $\mu\times m$ of the Liouville measure $\mu$ on the unit tangent bundle and the Haar measure $m$ on $SO(n-1)$ ($n$ being the dimension of the manifold) for a set of metrics (with negative curvature) that is open and dense in the $C^3$ topology. Curiously, the dimensions $7$ and $8$ are apparently more complicated than all others.