## Ratner’s Theorems

Ratner’s theorems are a series of results concerning unipotent flows of homogeneous spaces. They have been applied to many different situations, notably in some number theoretical questions, such as Oppenheim conjecture on quadratic forms. In this post I present the statements of this theorems and sketch the proof of the Measure Classification Theorem in a special case. A good intuitive introduction to Ratner’s theorems, with lots of exercises is this book by Dave Morris.

Let ${G}$ be a Lie group and ${\Gamma be a discrete subgroup, and hence closed. Then the space of the ${\Gamma}$-orbits (or the space of all cosets of ${\Gamma}$), ${X:=\Gamma\backslash G}$ is Hausdorff for the quotient topology. A standard example is when ${G={\mathbb R}^n}$ and ${\Gamma={\mathbb Z}^n}$, and since ${G}$ is abelian, the quotient ${X={\mathbb Z}^n\backslash{\mathbb R}^n=:{\mathbb T}^n}$ is itself a group, the ${n}$-dimensional torus. Another example, more interesting in a way because it is not abelian, is when ${G}$ is the special linear group ${G=SL(2,{\mathbb R})}$ of ${2\times2}$ matrices with determinant ${1}$, with the discrete subgroup of those matrices with integer entries ${\Gamma=SL(2,{\mathbb Z})}$. This example is important for applications to number theory, because the quotient ${\Gamma\backslash G}$ is closely related to the modular curve (or more precisely its unit tangent bundle).

With this setup, ${G}$ acts naturally on ${X}$ by right multiplication: for ${x=\Gamma g\in X}$ and ${g'\in G}$ we set ${g'(x)=xg'=\Gamma (gg')}$ (this is more precisely a anti-action, to fix it we could use ${g'(x)=\Gamma gg'^{-1}}$ instead). Furthermore we can give ${X}$ a natural ${G}$-invariant measure, namely for a Borel set ${A\subset X}$ we can define ${\mu(A)=\inf\{\lambda(B):B\subset G, \pi(B)=A\}}$ where ${\lambda}$ is the right Haar measure on ${G}$ and ${\pi:G\rightarrow X}$ is the natural quotient map. It is not hard to check that this defines indeed a measure on ${X}$. We will say that ${(X,\mu)}$, or just ${\mu}$ is homogeneous, because ${\mu}$ is invariant under a transitive group action, namely the natural right ${G}$-action on ${X}$. We may also say that ${X}$ is topologically homogeneous in the sense that there is a transitive action (again by ${G}$) of homeomorphisms (but this may not be a standard term).

In our first example (${G={\mathbb R}^n}$, ${\Gamma={\mathbb Z}^n}$) we have that the measure ${\mu}$ constructed above for the homogeneous space ${X={\mathbb T}^n}$ is just the Haar measure on the torus (it is invariant under ${G}$, and hence under ${{\mathbb T}^n}$). In general, if the discrete subgroup ${\Gamma}$ is normal, then the measure on the homogeneous space will coincide with the (right) Haar measure on the quotient group. For that reason we will call the measure ${\mu}$ the Haar measure on ${X}$, even when this is not a group. If the Haar measure of ${X}$ is finite, then ${\Gamma}$ is called a lattice and we assume that the measure is normalized. In our second example, the homogeneous space ${\Gamma\backslash G=SL(2,{\mathbb Z})\backslash SL(2,{\mathbb R})}$ is not a compact space, but it has finite measure, and thus ${SL(2,{\mathbb Z})}$ is a lattice on ${SL(2,{\mathbb R})}$.

For a subgroup ${H and a point ${x\in X}$ we are interested in studying the orbit ${xH:=\{xh:h\in H\}\subset X}$. When ${G={\mathbb R}^2}$ it’s not hard to see that any non-trivial connected subgroup ${H is a one-dimensional subspace. In this case either the slope of the line is rational, and the orbit ${xH\subset {\mathbb T}^2}$ is one closed curve (for any ${x\in X}$), or the slope is irrational and the orbit is dense in the ${{\mathbb T}^2}$ (also for any ${x\in X}$). More generally, when ${G={\mathbb R}^n}$, a connected subgroup ${H}$ of ${G}$ is a subspace of ${{\mathbb R}^n}$ and the orbit of ${x}$ under ${H}$ is dense in a sub-torus of ${{\mathbb T}^n}$ (a sub-torus of ${{\mathbb T}^n}$ is a closed subgroup isomorphic to ${{\mathbb T}^d}$ for some ${d\leq n}$). In this example we see that the orbits are well behaved, in the sense that their closure, ${\overline{xH}}$, is the orbit of ${x}$ under a closed subgroup ${S that contains ${H}$.

Consider now the subgroup ${H}$ of ${G=SL(2,{\mathbb R})}$ formed by the diagonal matrices, this subgroup is isomorphic to ${{\mathbb R}}$ through the map ${ t\mapsto\begin{bmatrix} e^t & 0 \\ 0 & e^{-t}\end{bmatrix}}$. In contrast to what we saw above, the orbits under ${H}$ can be quite chaotic, in particular, there are examples of a lattice ${\Gamma< G=SL(2,{\mathbb R})}$ and a point ${x\in X=\Gamma\backslash G}$ such that the closure of the orbit, ${\overline{xH}}$, is homeomorphic to the cartesian product of ${{\mathbb R}}$ and a Cantor set. In particular ${\overline{xH}}$ is not topologically homogeneous, in the sense that it is not the orbit ${xS}$ for any closed subgroup ${S.

The first theorem of Ratner we will talk about gives a sufficient condition on the subgroup ${H so that the orbit ${xH}$ is topologically homogeneous. We recall that for ${g\in G}$, the adjoint Ad ${g}$ is the linear map defined as follows: let ${c_g:G\rightarrow G}$ be the conjugation by ${g}$, ${c_g:x\mapsto gxg^{-1}}$. Ad ${g}$ is the derivative of ${c_g}$ at the identity: Ad ${g=c_g'(1):\mathfrak{g}\rightarrow\mathfrak{g}}$ (where ${\mathfrak{g}}$ is the Lie algebra of ${G}$, which by definition is the tangent space of ${G}$ at ${1}$). We say that ${g}$ is unipotent if its adjoint Ad ${g\in End(\mathfrak{g})}$ is unipotent (and this means, as usual, that all eigenvalues are ${1}$, or equivalently that Ad ${g-I}$ is nilpotent where ${I:\mathfrak{g}\rightarrow\mathfrak{g}}$ is the identity map).

For instance, if ${G}$ is a subgroup of ${SL(n,{\mathbb R})}$, then an element is unipotent exactly when it is unipotent as a matrix (this is a well known fact that doesn’t seem very easy to prove). Ratner’s orbit closure theorem (or Ratner’s topological theorem) can be stated as

Theorem 1 (Ratner’s Orbit Closure Theorem, 1990) Let $X=\Gamma\backslash G$ be the homogenous space for some lattice ${\Gamma}$ on a Lie group ${G}$. Let ${H be a subgroup generated by unipotent elements of ${G}$, and let ${x\in X}$ be an arbitrary point. Then the orbit closure ${\overline{xH}}$ is the orbit of ${x}$ under some closed subgroup ${S containing ${H}$.

We remark that we need ${\Gamma}$ to be a lattice, not only a closed subgroup. This non topological assumption hints on the dept of this theorem.

In this case we have furthermore that the orbit ${xH}$ is equidistributed in the closure (not only dense). This result (which is Ratner’s equidistribution theorem) is technically easier to state only for one-parameter subgroups, i.e., the image of ${{\mathbb R}}$ by a ${C^\infty}$ homomorphism. However, according to Terrence Tao’s blog post on this subject, it can be formulated for more general amenable groups ${H}$. In the case when ${H}$ is a one-parameter subgroup, Ratner’s equidistribution theorem is:

Theorem 2 (Ratner’s Equidistribution Theorem, 1990) Let $X=\Gamma\backslash G$ be the homogenous space for some lattice ${\Gamma}$ on a Lie group ${G}$. Let ${h:t\mapsto h_t\in G}$ be a ${C^\infty}$ homomorphism from ${{\mathbb R}}$ to ${G}$ such that ${h_t}$ is unipotent for any ${t\in {\mathbb R}}$, and denote ${H:=\{h_t\}_{t\in {\mathbb R}}}$. Let ${x\in X}$ be an arbitrary point and let ${S}$ be given by theorem 1. Denote by ${\mu_S}$ the Haar measure on ${S}$. Then the orbit ${xH}$ is equidistributed in it’s closure, i.e., for any continuous function ${f:X\rightarrow {\mathbb C}}$ with compact support we have

$\displaystyle \lim_{T\rightarrow \infty}\frac1T\int_0^Tf(xh_t)dt=\int_Sf(xs)d\mu_S(s)$

As a (not immediate) corollary of this theorem we obtain a classification for the ${H}$-invariant ergodic measures on ${X}$. A measure ${\nu}$ on ${X}$ is ergodic under ${H}$ if the only ${H}$ invariant sets have ${\nu}$-measure ${0}$ or ${1}$.

Theorem 3 (Ratner’s Measure Classification Theorem) Let $X=\Gamma\backslash G$ be the homogenous space for some lattice ${\Gamma}$ on a Lie group ${G}$. Let ${H be a connected subgroup generated by unipotent elements and let ${\mu}$ be a probability measure on ${X}$, invariant and ergodic under ${H}$. Then ${\mu}$ is homogeneous, in the sense that there is some closed subgroup ${S containing ${H}$ and some ${x_0\in X}$ such that the orbit ${x_0S}$ is closed and ${\mu}$ is (the unique probability measure) ${S}$-invariant and supported in the orbit ${x_0S}$ of ${S}$.

We remark that all ${H}$-invariant measures on ${X}$ are the convex combination of ergodic ones (or a generalized convex combination, i.e. an integral over the space of ergodic measures), so studying ergodic ones is enough for most purposes. Historically the Measure classification theorem was proved first, then the equidistribution theorem and the orbit closure theorem follow from it.

— 1. Proof of a special case of the Measure Classification —

In this section we sketch the proof of theorem 3 in the case when ${H=SL(2,{\mathbb R})}$. We follow closely the approach in this paper by Einsiedler. Note that ${G}$ can be much larger than ${SL(2,{\mathbb R})}$, so this case is not so restrictive and indeed it is sufficient for some of the applications of this theorem, most notably to the Oppenheim’s conjecture.

We will need some preliminary results about ${SL(2,{\mathbb R})}$. One first step is to notice that a ${SL(2,{\mathbb R})}$ ergodic measure is also ergodic under the smaller subgroup ${U:=\left\{u_t:=\left[\begin{array}{cc} 1 & t \\ 0 & 1\end{array}\right], t\in {\mathbb R}\right\}}$. This result is an instance of the so-called Mautner’s phenomenon

Lemma 4 (Mautner’s Lemma) Let ${\{x_n\}}$ be a sequence converging to the identity (in any topological group), and assume all the terms in the sequence are conjugate to a fixed element ${h}$, say ${x_n=u_nhu_n^{-1}}$. Then for any continuous measure preserving action, say ${G}$ acting on ${(X,\mu)}$, any function ${f\in L^2(X)}$ which is preserved by each ${u_n}$ is also preserved by ${h}$.

In order to prove the claim that ${\mu}$ is ergodic under ${U}$ we use a variant of this lemma and create a sequence converging to the identity by conjugating a given matrix in ${SL(2,{\mathbb R})}$ with elements from ${U}$.

We will also need some facts about representations of ${H=SL(2,{\mathbb R})}$. For any given representation of ${H}$ on a finite dimensional real vector space ${V}$ (say the adjoint representation on the Lie algebra ${\mathfrak{g}}$ of ${G}$) and any invariant subspace ${W\subset V}$ (say the Lie algebra ${\mathfrak{s}}$ of a subgroup ${S}$) there is a ${H}$-invariant complement ${W'\subset V}$, i.e. a vector space invariant under ${H}$ and such that ${V=W\oplus W'}$. This also implies that any finite dimensional representation of ${H}$ is the direct of irreducible representations.

We are now ready to start the proof of theorem 3 in the case when ${H=SL(2,{\mathbb R})}$. We let ${Stab}$ be the stabilizer of ${\mu}$, ${Stab=\{s\in G: s.\mu=\mu\}}$ (where ${(s.\mu)(A):=\mu(As)}$) let and ${S}$ be the connected component of ${Stab}$ that contains the identity of ${G}$. Since ${H=SL(2,{\mathbb R})}$ is connected we have ${H.

We now claim that there is an ${S}$ orbit with positive measure. Since ${\mu}$ is ${H}$ ergodic and each ${S}$ orbit is invariant under ${H}$, this implies that ${\mu}$ is concentrated in a single ${S}$ orbit. Moreover, ${\mu}$ is actually supported on that ${S}$ orbit, because ${\mu}$ is ${S}$ invariant and this ${S}$-orbit is closed. This last step is formalized by the following lemma:

Lemma 5 In the conditions of theorem 3 with ${H=SL(2,{\mathbb R})}$ and ${S}$ as defined above, if for some ${x_0\in X}$ we have that ${\mu(x_0S)=1}$, then the orbit ${x_0S}$ is closed.

We give the main steps of the proof of this fact. The proof is based on the fact that if a group (in this case ${S}$) admits a lattice, then it is unimodular (the left and right Haar measure are the same). Using this, one can prove that if a sequence ${\{x_i\}=\{x_0s_i\}}$ in the orbit ${x_0S}$ converges to ${y\in X}$ then ${\{s_i\}}$ has a convergent subsequence to ${s}$ (say) and so ${y=x_0s\in x_0S}$.

We will denote the balls in ${S}$ around the identity ${1}$ by ${B_r:=\{y\in S:d(y,1), where ${d}$ is any fixed metric on ${S}$. If no subsequence of ${\{s_i\}}$ converges, then we can find a subsequence (call it still ${\{s_i\}}$) such that ${x_i\notin x_jB_1}$ for any ${i\neq j}$. Using unimodularity we have that the sets ${x_iB_r=x_0s_iB_r}$ all have the same measure (which is positive because open sets have positive Haar measure). Note that, writing ${x_0=\Gamma g_0}$, we have that the sets ${g_0s_iB_r\subset G}$ are pairwise disjoint (for ${r<1/2}$), due to the condition ${x_i\notin x_jB_1}$. Thus if ${r}$ is small enough and ${x_i}$ are sufficiently close to ${y}$ (and hence to each other) then the sets ${x_iB_r\subset X}$ are also disjoint, which contradicts the assumption that ${\mu}$ is a probability measure. This concludes the proof of lemma 5.

Now we need only to show that some ${S}$ orbit has positive measure. The strategy to prove this is roughly the following: For two points ${x,y\in X}$ we say ${g}$ is a difference between ${x}$ and ${y}$ if ${y=xg}$ (of course there are many differences between any two points). We want to create a set of positive measure such that any pair of points ${x,y}$ on that set have a difference that preserves the measure, i.e. is in the subgroup ${Stab}$. In particular we can make this difference be close to the identity ${1}$ of ${G}$, and so that difference is actually in ${S}$. This will imply that that set of positive measure is inside a single ${S}$-orbit. The next proposition follow this idea.

Proposition 6 There is a set ${X'\subset X}$ with ${\mu(X')=1}$ and such that if ${x,x'\in X}$ and ${x'=xc}$ with ${c\in C_G(U)=\{g\in G:gu=ug\ \forall u\in U\}}$ then ${c}$ preserves ${\mu}$. In particular if ${x}$ and ${x'}$ are sufficiently close, ${c\in S}$.

Using the Mautner phenomenon we have that ${\mu}$ is ergodic for the action of the upper triangular group ${U}$. Since this is a one-parameter subgroup we can define generic points for this action, thus a point ${x\in X}$ is generic if for any compacted supported continuous function ${f\in C_c(X)}$ we have

$\displaystyle \lim_{T\rightarrow \infty}\frac1T\int_0^Tf(xu_t)dt=\int_X fd\mu$

The set ${X'}$ will be the set of generic points for the ${U}$ action on ${X}$. That set has measure ${1}$ by Birkoff’s pointwise ergodic theorem. Assume that ${x,x'}$ are generic points and let ${c\in C_G(U)}$ be such that ${x'=xc}$. Then for any compacted supported continuous function ${f\in C_c(X)}$, let ${f_c(x)=f(xc)}$. We now have

$\displaystyle \int fd\mu=\lim_{T\rightarrow \infty}\frac1T\int_0^T f(x'u_t)dt=\lim_{T\rightarrow \infty}\frac1T\int_0^T f(xu_tc)dt=\lim_{T\rightarrow \infty}\frac1T\int_0^T f_c(xu_t)dt=\int f_cd\mu$

and this concludes the proof that ${\mu}$ is invariant under ${c}$.

This proposition requires that a difference between the two points ${x,x'}$ be in the centralizer of ${U}$. To get to such situation we will have two sequences of generic points with differences converging to some point in that centralizer. To make sure the limits are still generic points, we have to work with a big compact subset ${K\subset X'}$. Those sequences will be obtained by flowing two initial points by ${U}$ (this means consider the orbits under the ${U}$ action), and then we can study the behavior of those orbits using what we know about the representations of ${SL(2,{\mathbb R})}$.

Our task (to find a ${S}$ orbit with positive measure) is made easier if we prove it by contradiction. Thus we now assuming that no orbit has positive measure and we will find an element of the group close to ${1}$ and preserving ${\mu}$ but not in ${S}$, providing the desired contradiction. As we mentioned before, we want to flow two generic points ${x,x'}$ by ${U}$, and we want their orbits to be in a compact set ${K\subset X'}$ of generic points with measure ${\mu(K)>0.9}$ (say). Thus it will be convenient to require that the orbits of ${x,x'}$ are often in ${K}$. More precisely we define the set

$\displaystyle X_1:=\left\{x\in X:\frac1T\int_0^T1_K(xu_t)dt>0.8\text{ for all }T\geq T_0\right\}$

where ${T_0}$ is large enough so that ${\mu(X_1)>0.99}$ (say).

Assuming that all ${S}$ orbits have ${0}$ measure we can find two points ${x,x'\in X_1}$ close to each other and such that ${x'=x\exp(v)}$ where ${v\in \mathfrak{s}'}$, the ${H}$-invariant complement in ${\mathfrak{g}}$ of ${\mathfrak{s}}$. We now study the flow of those points under ${U}$: for each ${u\in U}$, we have ${x'u^{-1}=xu(u^{-1}\exp(v)u)=xu\exp(Ad_u(v))}$ (this last equality is the basic fact of Lie theory that ${\exp(Ad_g(v))=g^{-1}\exp(v)g}$). Thus the flow of ${x}$ and ${x'}$ diverge by ${h=\exp(Ad_u(v))}$.

By definition of ${X_1}$, for many ${t}$ we have that ${y_t=xu_t}$ and ${y'_t=x'u_t}$ are in ${K}$. What is left to do is to be able to make ${Ad_{u_t}(v)}$ be very close to a point ${w\in \mathfrak{s}'}$ fixed under ${U}$. If we can make that, then there is a difference between ${y_t}$ and ${y'_t}$ which is close to ${\exp(w)}$ (say distance ${\epsilon}$), and because ${w}$ is fixed under the adjoint action of ${U}$, we have that ${\exp(w)}$ is in the centralizer ${C_G(U)}$ of ${U}$. This is achieved studying the adjoint representation of ${U}$ in ${\mathfrak{g}}$.

By constructing such points ${y,y'}$ for decreasing values of ${\epsilon}$ (i.e. such that some difference between then is more and more close to ${\exp(w)}$, where ${w}$ can vary with ${y,y'}$) we get a sequence, and any limit point will satisfy our conditions, i.e., we find some points ${z,z'\in K}$ close together and such that ${z'=z\exp(v)}$ with ${v\in \mathfrak{s}'\setminus\{0\}}$ and ${v}$ fixed by the adjoint representation. Thus ${\exp(v)}$ leaves ${\mu}$ invariant but is not in ${S}$, which gives the desired contradiction.