# What is known about dynamics on Grassmannians?

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.

• Beautiful answer. Nice question. Certainly there exist ergodic transformations of all Grassmannians. Maybe the question can be usefully changed to What are interesting natural' ergodic transformations of higher Grassmanians?' (such as the RP(3) of the picture.) – Richard Montgomery Sep 16 '13 at 1:44

I guess that most of your questions can be answered by looking at the reduction of orthogonal matrices.

For one, each pair of conjugate complex eigenvalues give you a fixed plane, and each real eigenvalue a fixed line, and you have enough of these to generate the whole $\mathbb{R}^n$. So, whatever $r$ you consider, a $r$-plane contained in the sum of some of these subspaces will stay there: neither ergodicity nor minimality can be true. The best you can hope for is a decomposition of your space into minimal and ergodic components.

Let's start with $R\in\mathrm{SO}(3)$. Then $R$ is a rotation, its conjugacy class being given by its angle. As you observed, its action on $G(1,3)$ has one fixed point (if $R$ is non-trivial, which I will always assume); moreover the complement of this point is partitioned into circles on which it acts by rotations (beware that this complement is a Möbius band, and one of this circle as rotation number half the rotation number of the others). Its action on $G(2,3)$ is the same by polarity.

So, the first interesting case is $R\in\mathrm{SO}(4)$. Up to a change of basis, you can assume it preserves the two orthogonal $2$-planes $(*,*,0,0)$ and $(0,0,*,*)$. Its action on $G(1,4)$ has two circular closure of orbits, corresponding to the two preserved planes. Moreover you can partition the projective space into these two circles plus a one-parameter family of closed surfaces (tori if I am right), each one of these surfaces being preserved (the parameter is simply the angle of a line with one of the preserved planes). You are in fact looking at the Hopf fibration, to which "equipotential" tori are added (see image below), quotiented by antipody.

To see all this, you should write $R=\begin{pmatrix} u & 0 \\ 0 & v \end{pmatrix}$ in complex coordinates, with $|u|=|v|=1$ and let it act on unit vectors before quotienting out.

If $R$ is generic, with rationally independent rotation angles, then the above decomposition of the space is probably the minimal decomposition, and ergodicity on each leaf should be clear.

Okay, things are a bit messy if you want all possible information, but I guess you can do quite a lot of things. Also, you might want to discuss this kind of stuff with algebraic geometers if you have one available, some of them know actions of matrix groups on Grassmannians quite well.

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.