 Thanks. All right, so let's continue. Let me just state somehow, the punchline was left as an exercise, so please do the exercise yesterday. But I'll just state the more general result, which is due to Margulis, and in the form I'm stating actually, it's due to Danny really, that is the following, with U be a one-parameter unipotent subgroup of SL and R, then for every point in the space of lattices, oh, we had a name for this, Xn, and every epsilon positive, there exists delta such that if you look at the measure of points in any in the interval, such that for any t equal to zero, if you look at the measure of times where your orbit, it's life in this compact set, then this is bigger than one minus epsilon times t. Let me just remind you what this was. This was the set of lattices such that the minimum vector, non-zero vector has a lower bound. But that's what the definition was, and we proved this for n equals two. As I said, the proof is substantially more complicated when you go to higher dimension, already dimension three. So if you can do it for dimension three, you can do it for all dimensions, but not two. And the reason was this star condition that I put on the board yesterday, that you cannot simultaneously have two short vectors that are linearly independent. This is of course not true in higher dimensions. In three space, you can have many vectors that are short without violating the covalent of the lattice B1 because they can all come from a plane that has many short vectors. The direction that's somewhat transversal to this can compensate for it. And you can have a long vector there and the determinant of this three by three matrix equals one. So you can have a very small plane. You can have a rational plane which has many short vectors. So this will have a small determinant, but then there's a direction, think about it coming out of the board. And this is big so that this three-dimensional parallelogram will have volume one. So you don't have this disjointness that we played with yesterday. And that was very crucial because all I needed to do was to look at individual vectors and control the time that each individual vector is short. And the simultaneous approximation would just fall. So that's all good. Now I am finally going to move to prove a major classification result. So usually when you wanna do this, you first start with a spherical and then you run out of time and say, okay, the other case is much harder so we won't do it. I am gonna do it differently. I'm not gonna prove major classification for your important flaws. I am going to, for unipot and more parameter groups, but I'm going to actually give complete proof of classification for measures which are invariant on their SL2R and then say the other case is much more complicated. So, but before doing that at least in any course with a title close enough to the title of this course, you need to state Ratner's theorem. So let me do that. It's already two lectures late. So, what is the setting? The setting is G is a connected B group and think about it really the most interesting case is the case of groups that they look like SLNR. Gamma is a lattice and U is a one parameter ad unipotent subgroup. So in concrete terms, at least one example that we essentially proved is a lattice and U is any group that is strictly upper triangular and has some, so think about this, zero every girls. That's one example. And so what is Ratner's proof? Let me all erase this here and write a more historically meaningful example. This was the group that Margules had to work with for the proof of Oppenheim conjecture which predates Ratner's theorem. So fundamental result of Ratner is that any new invariant ergodic probability measure on G mod gamma is homogeneous. It has an algebraic description in the following sense. That is there exists some connected subgroup L which contains U and is contained in G could be equal to either one. The generic case actually equals to G. The point X in G mod gamma such that LX is closed and has a finite L invariant measure. So in particular LX is a manifold and it's a manifold that if G mod gamma was compact this would be a compact sub-manifold of it. It's a closed sub-manifold, so it would be compact but maybe G mod gamma looks like S L and R mod S L and Z so the same way that you can go to infinity in G mod gamma you could in theory go to infinity in LX also. And mu is the L invariant probability measure on LX because it's a finite measure I can always normalize it to make sure it's probability and mu is precisely this measure. This is a very strong rigidity result. I start with a measure that for all I know it could be really very arbitrary. It's just invariant under one dimension of flow. Under a flow defined by this which is just action of R. And what Ratner tells me is that the measure is as nice as you could hope for it to be. It's supported on a sub-manifold and the measure actually has full support. Those are other parts. This is first part and the hardest part. There are second and third parts which I will state but before I start second and third part let me remark very different behavior for the geodesic flow. We talked about this the first day. This has positive topological entropy. It has Bernoulli factor. There are many measures. It's just impossible to classify measures. The only obstruction to the measure is dimension. Anything between dimension one and three you will have a measure of that dimension. So the measures can be supported on a cantor set. There's no manifold structure. You couldn't possibly hope. And the conjecture that we mentioned is that we hope such picture holds for action of higher rank diagonal groups. And this is times two times three. There's actually, you need to state this very carefully. It's not true for all quotients but this is believed to be true. The action of this on SL3R mod SL3Z. If you replace SL3Z with another lattice then this is not necessarily true. If you replace it with a lattice that comes from some Hermitian construction then you can have rank one factors. Whenever you have rank one factors bad measures could exist. So it really depends on the lattice. As I stated this is believed to be true. Any a invariant ergodic probability measure is nice and in this case actually you can describe the dimensions are small. So there are two possibilities for it. One is a closed A orbit and the other is the whole measure. So that's what the conjecture is. Yeah. Any simple example? I actually will in a minute but yeah of course. So the example that we did in the first day. So think about SL, well let me first start with a very low dimensional example. SL2R mod SL2Z. When G is R2, yes this is, yeah it's much easier. It's because it's a floor on the terrace. For if G is a nilpotin group this is a much easier theorem. If G is a soluble group still much easier theorem. The main difficulty is when G is semi-simple. That's, so but let's give examples of this. Okay, let's do it, anyway let's look at the first non nilpotin case SL2R mod SL2Z. Then what are the possibilities? We looked at this schematic picture and my group U is let's say this thing acting on this space. What are the possibilities? First of all you could have an orbit that goes and comes back to itself. It closes up because if you look at what happens to the identity coset and you look at its U orbit it will go and come back to itself. This is one possibility because the action is ergodic the other possibility is that you get the whole measure. We know actually that almost every point has this property and Ratner is telling you this is much easier and was known before Ratner's work but in particular Ratner is telling you you don't need to search any further. This is it. Any orbit is either closed or this. That's what Ratner is saying. Because you couldn't possibly have any intermediate group which satisfies this. There's only one other intermediate group and that is the group of upper triangular matrices but such group cannot have a closed orbit with finite other matter that you can see because it's not for example because it's non-urimodular and if you have a lattice in it it has to be in a much larger. So this is one example. The other example much harder is the application to open high. So if you could determine what the orbit closures of these things are in particular you could determine what the orbit closures of the isometry groups are because if you did the exercises you know that this is a particular group that appears as a subgroup of the group of isometries for a particular form. And so Margulis actually studied the action of this group and showed. So when this group U is horospherical which is the example that is here. Horospherical meaning that it's stable or unstable for some diagonalizable element. So everybody here knows stable or unstable. Let me use that. So if it is stable or unstable if you look at the foliation gives you stable or unstable of some flow then the question is much easier. There are other techniques to use. Harmonic analysis can be used. Mixing can be used. They're all the same flavor. And it was proved before it goes back to already Hedlund. Hedlund understood the action of this group on SL2R module or any lattice and he showed minimality when the lattice is co-compact. So if this is compact then you couldn't possibly have a closed orbit. So orbit of every point is dense. This is 1930s. Hedlund understood that. And then there have been other developments. First of all Marg, Beach, Danny. And Danny understood the orbit closures for such groups in general. So if you have a horospheric group he had a classification before Ratner's work. But if you go beyond these classes the classes of soluble groups or the classes of horospheric groups then the problem is just infinitely more difficult. The techniques that they were developed for those groups just do not work here. You need genuinely new ideas. So Ratner's papers, there's a series of papers that she wrote. One of them that contains the main idea is an active paper which is like 100 pages. So there are other more maybe conceptual proofs. The shortest proof is a proof that is by Margolis and Omanov. And this is an invention in paper. It's around 40 pages. So that's as short as you can get. But it's impossible to do this without introducing entropy. So that's why we are not doing it. You need to use entropy too. But before saying what we will do, let me state the other two parts of this which we'll follow with some extra work from the first part. This is the hardest part. The second part is that orbit closure of every orbit is nice. For every x in G mod gamma, we can describe the orbit closure and that's exactly with L as here. And the third part is that not only is the orbit dense in LX, it's actually equidistant with respect to the measure. And okay, I leave that to your imagination to write what it means. It's a one-dimensional flow. You average time t and it converges to space average, which is LX. So that's the theorem. Any other questions about the statement? So what this means is you average U tx dt, this one converges to integral of f with respect to mu where mu is this measure. Mu L invariant probability measure on LX. So the orbit is not only dense in LX, it actually has no bias. It goes everywhere, the amount that you want it to. All right, any other questions, remarks? So like I said, we won't have time to prove this. If we wanted to prove it, we should have started from one day and just talk about proof of this theorem. But we didn't do that for whatever reason. So, but I'm going to prove a very special, substantially simpler, but still interesting case. And this is the following case. Let me just state the example that I'm gonna be working. It's gonna be a working example of a classification result, but you see that nothing is special about this example. It just generalizes to the general case, but this is just much easier to work with. So I'm going to look at G, which is SL2C. Gamma is going to be a lattice in G and H is going to be SL2R inside G. I have its subgroup U. If I could classify U invariant measures on G mod gamma, I would be very happy. But as I said, that's ambitious. What we are going to prove is this theorem of retinas, which follows from that theorem, that any H invariant ergodic probability measure on G mod gamma is homogeneous in that sense, is one of the following. Because the group is so big, H is a subgroup that has half the dimension and it's maximal. There is no intermediate subgroup between H and G. If you just try to imagine what Ratner would say, you see that very quickly you run out of possibilities. There's one possibility, this is always possible, that your H somehow has a closed orbit and it's just a measure that's in there. So mu is the H invariant probability measure on a closed orbit HX. Second possibility is that you are the HAR measure. So this is what we will spend probably most of the next two lectures probing. The statement clear? I'm not assuming G mod gamma is compact. With a closed orbit, it doesn't have to be compact. If you look at SL2-R mod, SL2-Z adjoint I, then the modular group sits there. Doesn't have to be compact. So this is, thanks. So what François is hinting at is this is the setting of hyperbolic three manifolds. So gamma that I take here is a fundamental group of a hyperbolic three manifold because that's the isometry group of H3. And this is acting, the quotient is going to be, okay, up to finite index, is not gonna have torsion. But it's not gonna have a torsion, can you hear me? And then you get a hyperbolic three manifold. And the interesting cases are closed hyperbolic three manifolds. And the case one refers to totally geodesic subsurface. So you have hyperbolic three manifold and case one are totally geodesic subsurface. And case two is, okay, that's one of the reasons I chose this example. It has connections to, you don't have to assume torsion free, but you can always pass to a finite index subgroup and be torsion free. This is a theorem of Z equals, problem, I don't know. The other Z equals, I think Z equals. Any finite generated subgroup that you choose, you can pass to a finite index subgroup. So maybe I should give a concrete example of gamma also. So if G was SL2R, the first example you would think of for gamma would be SL2Z. An example to have in mind here is the Gaussian integers. This is gonna be a lattice in there. So SL2 of Z, a joint I. Example, so what is the general strategy? The general strategy which Ratner used and Margolis used also is that you're going to look at, we don't have much. We have the measure mu, that's all we have. We're gonna look at the support of the measure. So this is the space X, let's say. We're gonna look at the support of the measure. The support of the measure is sitting there. And then all of these are gonna start with a simple lemma. The lemma says that unless something very restrictive and algebraic happens, you can always find two points which are both in the support of the measure, X1 and X2, which are both in the support of the measure so that the displacement, so you can write X2 as GX1. And this G is not in the group that you already know you're invariant on there. And G is not in H. This you can get relatively easily. Basically, it's because if every point, a neighborhood of a point was just coming from H, then you can just take this neighborhood and push it to everywhere with H and everything would be supported in an H orbit. This is, okay, we'll prove this. This will be the first lemma that we prove. That unless something like case one happens, you can always find two points which are both in the support of the measure, in the support of the measure, so that something like this happens. And then, okay, that's good, but you don't know anything about the relationship between this point G and the measure. So if you change your point and go to another point, X1 prime and X2 prime in the support of the measure, you don't know anything about what G does to this. G might just leave it outside the support. You don't know anything like that. But this is a good start. And then, what you're going to do is to apply, so you have this, you have X2 which is GX1, and then you're gonna try to apply parts of H to these two points. So you're going to maybe apply some subset H in H. Two point X1, and do the same thing. Some particular subset which we will specify. Two point X2, and study how these two points diverged. And you're gonna try and choose this subset so that this piece of the orbit has two properties. First of all, it mimics the measure, meaning that this one goes everywhere in the space. So it equidispers with respect to the measure. And same up here. That this piece also equidispers with respect to the measure. And this is dynamically. So the dynamical input, which will be ergodic theorem, work of ergodic theorem, we will choose it, we will choose subsets of H so that this is, this sees all of the measure. The algebraic input is that we will choose this set so that the displacement, this divergence of these two orbits is controlled. So we'll choose subsets where divergence is controlled. And if I can find subsets of H, where these two happen simultaneously, then we will celebrate. Because this will tell me that, okay, this was going everywhere in the space. This was going everywhere in the space. And there is just one element that takes me from here to here. That means my measure is not feeling the displacement by this element. And this is gonna be some new element that is leaving the measuring variant. And you add that to your H and you open the bottle of champagne. That's what's gonna happen. But we need to, we need to get to that point. It's, I mean sometimes, some work. All right, so this is the general strategy. And unipotents are going to be used to guarantee that times that these happen can be actually, we can find times that both dynamically we are happy and algebraically we are happy. And this is this polynomial-like behavior of unipotent flaws, which was the exercise that I put up there yesterday. Where is Sasha? Of course he's not here. So it's the polynomial-like behavior of unipotents. Okay, let's get to work. But I started at 10.15, right? Okay, so first thing is first. Let's try to prove something like this. Maybe a little more information about G. Not only G, not in H, maybe really transversal to H. So I need to introduce some coordinate system. So the algebra coordinate. G is going to be the algebra of G. This is going to be the two by two matrices with trace zero, A, B, C in C. Which is, we all know, gothic SL2 of C. The le algebra of H is the same object with that C replaced by R. And I have the decomposition. There's a decomposition that G equals H plus IH. These are real subspaces, not complex subspaces. It's where they have this decomposition. And what is good about this decomposition is that if I look at conjugation action of the group H, it leaves the space invariant. So this is a decomposition that's invariant under conjugation action of H. Is invariant. That's one thing that is convenient about looking at SL2R. You have naturally coordinates that are adapted to the action of SL2R. So this is not only a vector space decomposition. If you want, it's an H module decomposition. It's the action of H leaves this decomposition invariant. And just to not keep writing IH, this I will try to write it as a gothic R. So that's how I'm defining it. So there's a decomposition of H and R. And this is, of course, a general fact whenever you have a finite dimensional representation of SL2, then you can decompose it into invariant subspaces. So it's completely reduced. But here you can see it by hand. Okay? I'm proving the first part, hopefully. There's no, no, no, no, no, no. This, that I can find some transfer. But right now I'm just going to introduce notation. There will be a lemma stated, which looks like this. Yeah. As an appeared, we'll appear in a second. Thanks. But let me, let me maybe make it appear here. Okay? This is, there was a reason we did Mountner phenomena. What do we know about the measure mu? Mu is H-ergodic. Hence, Mountner phenomena tells me it's u-ergodic. And that's how we will use it. These parts that I said we will choose from H are actually going to be part from the unipotent group. So, note. All right. Now I have the exponential map. I have the exponential map which goes from G to SL2C. And I will choose delta small enough. So that if I look at the exponential map of ball of radius delta around G, and okay, you need to ask me what norm you'll fix, fix whatever norm you want, but then you need to bear with me that I am thinking of max norm. So, delta ball is a sum of two delta balls. That's what it is. This into G is a defumorphism. So, exponential map restriction to this is a defumorphism. This you can always do because exponential map has the Jacobian identity, so you can always find a small neighborhood so that this happens. And then I'm going to think about this as ball of radius delta in H times ball of radius delta in R. So, what just happened? In G, I have this product structure. There is H, there is R. And using exponential map, I put this in G. So now my G also has product structure. This is exponential coming from the le algebra H. This is exponential coming from the le algebra R. So at least I made sense of what transversal means. Two points being transversal means that the displacement comes from exponential of this complement. That's what it means to be transversal. So I'm going to find two points X1 and X2 so that this displacement is coming from the transversal direction. There exists some epsilon positive such that if omega inside G mod gamma satisfies this condition. If it has almost full measure, measure is a probability space. Well, space is a probability space. If I have a set that's almost the whole space, then one of the following holds. Either there exists some X in G mod gamma such that the measure of this orbit equals one. Compare this to case one. Either you're just completely supported on one orbit. And actually you can post this up and show that this is really a closed orbit and mu is the H invariant measure because it's one orbit. And it has a probability space that basically means you are lattice. Secondly, or I mean, or there exists sequence Xn in omega, a sequence Rn in R. None of them equals zero, converging to zero such that X pop Rn Xn is also in omega. So if you give me a set that's almost the whole space unless something very algebraic happens, think about one as a condition that's algebraic, then not only can I find two points with this, I can find a sequence of points that they differ in the transversal direction and this distance actually gets arbitrarily small. That's what this lemma is going to prove, right? The statement clear? Yeah? Oh yeah, sorry. Thank you. That's what I meant to write. Uh-huh. Rn's R in R, yes, this is not Y, it's R, sorry. And yeah, Rn's R in R. This was supposed to read like that, but. So let me read it. I'm closer to it and I know somehow my handwriting is not. Okay, so Xn is a bunch of points in omega. Rn's are some vectors in the complement, in IH, in R. Rn's are going to zero, but they are not zero and if I move Xn with X of Rn, I still hit omega. That's what the statement is. This is R and that's, this is Gothic R. This is not Gothic R. What is the proof? The proof is Fubini. The proof is Fubini and finding a density point. That's basically what you need to do. You need to find a density point for this. If the density point does not satisfy something like this, that means you are in one leaf of H, here that. That's, let me choose any point P that's in the support of the measure. Because of definition of the support, I know that small neighborhoods of P are going to have big measure. A very good question. It's a very good question. Okay, then maybe I won't prove it. Let me say Y omega, thank you. Y omega, the reason I have omega here is that I said I'm gonna apply ergodic theorem. I said I'm going to apply ergodic theorem. What does ergodic theorem do for you? The measure is U invariant. The ergodic theorem is going to have the following form. That there exists a set X prime inside X, a full measure such that for every F, which is say continuous compactly supported in G mod gamma, whatever, whatever. The limit as capital T goes to infinity integral from zero to T F of U T X DT converges to FDM. Okay, this is a slightly different formulation than we are used to for ergodic theorem. Ergodic theorem usually give me an L1 function. I give you a full measure set. But here I'm choosing a full measure set that works for all continuous functions. Sorry, what's it? One over T, oh, yeah, yeah. Okay, however, even though this holds, I can have points for which this convergence is very, very slow. I don't like those points. I'm going to throw out those points and rather than working with X prime, I'm gonna find a subset of X prime that has almost full measure and this convergence is uniform. No, it's a stronger statement. I could find two points that are in the support and this happens. Yeah, this is a stronger statement. Any set that has almost full measure will have two points. There exists some epsilon so that as soon as your set is bigger than one minus epsilon, I can find two points in it for it. Yeah, it's a stronger statement. And I want stronger statement because I'm gonna apply this. Rather than proving that, let me prove something that's fun. Which would be the next lemma. So the proof of the lemma, you go and ask Ali. Maybe tomorrow and the morning we'll prove it. So let me prove the following which then you see how I can find elements that they will leave the measure invariant and this can be done actually in five minutes. So notice that the centralizer of U in my group, the matrices that they commute with you. So this is G such that UTG equals GUT. This is precisely the group of upper triangular matrices. Now lemma two, which is a baby version of what the argument will be to get extra invariants is the following. Suppose something not very generic but still. Something happens. This X prime that I gave you here. Suppose all of a sudden you can find a point in there and some G in the centralizer, such that GX is also in X prime. Then mu is invariant under G. So here I said if you have two points that the displacement you just know is not from H, you can't say anything more. But now suppose something that is rather odd happens. You have a point. So this is your space X, that's X prime the support and you find two points in this set that they are got a theorem holds where the displacement comes from the centralizer direction. Then every point has a friend which is translated by G. The measure is invariant energy. Why is that? This is the point X, this is the point GX. Let me flow this with UT. Let me flow that with UT. What do I know about G? G comes from the centralizer. That means these two orbits are really parallel. There's no divergence. So this equals G UTX. All right? Now I'm gonna use this and finish. GX is in X prime. What does that mean? That means for any F you give me in the space of continuous compacted support of functions then I can average my function a long time and this will converge to integral of FDB. Now let me use algebra. I'm allowed to use algebra. Switch these. This one is nothing but F of G UTX DT one over T. But now X also was in X prime and this is a new function F translated by G. This is, I will write it as zero T F of G UTX DT where FG by definition is F of GZ for any point. Now this is another continuous compacted support of function. So Ergodic theorem tells me this converges to integral of FGD mu on the space. These two are equal for all and every single continuous compacted support of function. That means the measure is invariant. So the measure is invariant under G as soon as you can find one unusual displacement. But this is not, okay, this is too good to hope for. You can't find something like that. But this you can find and then you try to flow and control how these two points diverge.