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.

`What are interesting`

natural' ergodic transformations of higher Grassmanians?' (such as the RP(3) of the picture.) $\endgroup$ – Richard Montgomery Sep 16 '13 at 1:44