 So up here is a picture of Plex, of a foliation of the torus, or of the square, if you like. I'll say something more about this later on in the lecture, but I'll leave it up there. It's a pretty picture to look at. So last time, in Kadim's lecture, he showed that if G is the diffeomorphism of the torus, and the distance from the cat map to G in the C1 topology, which is defined to be the soup over all points of the distance between the values in the torus, plus the soup over all points. I guess we could put these together. But the difference in the norms of the matrices representing, so this is just a, the derivative of f sub a and the derivative of G. So if this distance is sufficiently small, less than some epsilon, then G is a Nosov, or uniformly hyperbolic. So there exist cone fields, P and Cs. So at each point, P, we have a cone inside the tangent space to P of the torus. And he had a parameter alpha. I'm just going to leave that out, that are invariant in the sense that under the forward derivative, the cone at P is taken strictly into the cone at G of P, unstable cone. And under the inverse derivative, the stable cone is taken into the stable cone at G inverse of P. And there exists some constant lambda bigger than 1, such that if V is in Cu of P, a vector in Cu, then the norm of the derivative of G at P applied to V is greater than or equal to lambda times the length of V. And similarly, if V is in Cs, inverse derivative expands length. So this is some kind of expansion condition, or if you like, expansion contraction, since one is expansion under the inverse. And clearly two such cones have to be disjoint because you can't have both of these conditions holding. And our cones do not contain the origin. They were cleverly designed to exclude the origin. So the picture on the torus, I'm going to draw two copies. And maybe we have a few points here. Maybe here is P. Here is G of P. Here is G inverse of P. And similarly over here, it's going to be interesting to try to work around this. I was trying to avoid colorblind issues by picking the right colors, but I've just caused a big mess. OK. So for example, it might be Cu at G inverse of P. Maybe the cones actually don't have to. The cones don't actually have to be sort of constant. They will vary continuously. So for example, if this is the cone here, Cu of P. And I apply the derivative, where I look at what G does very close to P, which is essentially the derivative. And I go over to G of P. So here's Cu of G of P. Just copied it from the other side. But then if I apply G to this cone, it's going to be sent strictly inside by the derivative. And if I took an individual vector here, its length will actually be expanded by a factor of lambda. And similarly, if I look at the stable cone, and I put a stable cone over here, maybe it looks like this, I copy the stable cone over at, maybe I better do this at G inverse of P. And here's G inverse of P. And it's stable cone. Well, if I apply DG inverse derivative at G, sorry, here's P, if I apply DG inverse to the cone here, I will contract the cone but expand the vectors. And this holds for all points on the manifold. So we saw that these two conditions, you can even, by the way, you could even add that there exists a lambda bigger than 1 and some constant and not bigger than or equal to 1, such that the expansion occurs after and not iterates. These two conditions are equivalent to the Anosov condition he gave with ES and EU, the bundles. And I won't restate that. Let's just take this to be the definition of Anosov. And now he also explained, and he did this in his proof of mixing for perturbations of the cat map. His proof, if you remember, he took two open sets and he wanted to show that if you iterated one long enough, it would stably cross the other. And so he took one open set and took a stable curve and iterated it backwards. So the backward iterate contained a long stable curve. And the other set, he took a little unstable curve in that open set and iterated it until it was very long and that forced an intersection between the two open sets. So an important ingredient of this is that if I take a curve, gamma sub u, something c1, is, he used the term tangent to cu by which he meant simply that the derivative at every time t, the derivative vector, lies inside the cone at the image point on that curve, gamma u sub t for all t in 0, 1. Then when you iterate such a curve forward, the curve g composed with gamma u, here might be the original curve, gamma u, and everywhere, if I were to use the same colors, I would use orange, everywhere the tangent lies inside this cone. And if I apply g, then, again, the tangent will continue to lie in the cone because the cone is contracted. And the length will be expanded. So in words, it is even more tangent to cu. So meaning that if I take the derivative of the composition, then by the chain rule, this is the derivative of g at gamma u, like t, times gamma u prime of t. This is inside the cone. Now I've taken something that contracts the cone. So this is inside the unstable cone at g of gamma u of t. But it's inside, I'll just write it this way. This is a slightly different notation than Kadeem was used. But it lies in a cone whose width has been contracted by a factor 1 over lambda squared. So all that I mean is that it's even more inside the cone because it's inside the cone. And then roughly speaking, well, maybe I'll make this kappa just to be safe. And this is some kappa less than 1. So there are these two cones. This direction is contracted going forward. This is expanded. So once this cone is sent inside to itself, it's actually going to, and these vectors are expanded, it's actually going to be contracted by a definite factor. And similarly, oh, that's part 1. And 2 is longer by a factor of lambda, by which I mean literally the length of the composition is greater than or equal to lambda times the length of gamma u. OK, so we're going to use these two facts that he established last time to construct foliations that are invariant, stable and unstable foliations for an Osof diffumorphism on the two torus. Then we're going to examine the properties, one of the key properties of these foliations. So recall that for F of A, the cap map, again, A is 2 or 1, if we take through any point P in T2, we take in the universal cover in R2, we take the line that's parallel to the eigenspace. This is parallel to the expanding eigenspace for A. We project it down to the torus. We get this line that densely winds around the torus, et cetera. And we call this line, so we could also go backwards and so on, whatever, dot, dot, dot. And we call this line w u of P. For the map, I can put this down here, for the map F sub A. And that's called the unbelief. This line is called the leaf of the unstable foliation w u F sub A. And similarly, if we take a line that's parallel to the stable eigenspace and we project it, we get a line called w s F sub A of P. And that is the leaf of the foliation w s F sub A. So w u F sub A is the partition formed by taking the collection of all w u F sub A of P for P in the torus, where there's an obvious equivalence relation if two points are on the same leaf, we consider those the same set. So it's a partition. It's a partition into leaves. And w s is a partition into the stable leaves. And recall that these two foliations were invariant. First of all, why were they called foliations? Well, because at any point, well, first of all, it's a decomposition into one-dimensional sub manifolds, in this case, lines. So these are nice. The elements of these partitions are nice. And locally, both either one of these foliations, if I put the correct coordinates, for example, w s, this is called w s, I'm just going to leave off the F sub A. Locally, we have what are called plaques. And these are connected components of intersections of leaves with this little neighborhood. And locally, these plaques, well, under a homomorphism, in this case, just by rotation, these little plaques just look like coordinate disks. So these are coordinate disks, or disks in coordinate plane, in this case, coordinate line. So parallel to the x-axis. And similarly, for w u-loak, OK? So these foliations, so this is what makes them foliations, these foliations have the property that if I take the leaf of w s through a point P, and I apply F sub A, I get the leaf of w s through F sub A of P. So it's invariant. And similarly, w u is invariant. So these are invariant, and they are uniformly contracted, respectively, expanded by F in length. OK. So I'll give you the general theorem. I'll state it on the torus, because in higher dimensions, these foliations will have higher dimensional leaves. So in the three torus, if I have something in Ossoff, one of my two foliations is going to have two dimensional leaves. It'll look like surfaces. The other will be one dimensional. But so if G is in Ossoff, it only has to be C1, in fact, then there exist foliations w s of G, w u of G, by smooth curves as smooth as G is. So if this is C infinity, then the curves are C infinity. So by smooth leaves, which are curves, so the leaves are smooth, and they are curves, such that if I take two points, q, q prime, w s of p, that implies that the distance from G to the n of q and G to the n of q prime tends to 0 as n goes to infinity. And in fact, it does this exponentially fast at an exponentially, exponential rate. This distance is less than or equals to a fixed constant times lambda to the n, basically. And if I take similarly two points in the same unstable leaf, so this is for G, then similarly the same is true, but now I have to take G to the minus n. So we have two foliations like that. Locally, they look like a collection of smooth curves that partition the square. And what's the idea of the construction is the following. It's really quite simple to prove all the properties that it has is a little bit more technical, but the construction itself is quite simple. So I start with T2. And again, I'll draw two copies of T2. Maybe I want just one copy for now. I'm just going to start with a collection. It doesn't have to be a foliation. I just need, through each point, a little unstable curve. So this is construction of w u. I start with a collection gamma u p, p in, I probably should have used a superscript before I was using a subscript, gamma u p for p in T2 of curves lying inside the unstable cone. And they can be short. They're really short. In fact, they can't be arbitrarily short, but fix an epsilon and choose them all to have, say, length epsilon, where epsilon is small. So there's our unstable cone as before. And now I can just reach down and get some chalk. So through the unstable cone, I'll just take a little curve. It could just be a line segment if you want. There's no need to get fancy. So if this is a point p, this is a curve actually in the manifold, so it stays inside. It's not, I have to make sure that when I move the cone, it still stays inside the cone. But OK, this is gamma p. And now what do I do with these curves? I just apply g. I apply g to the m, where m is very large. And what happens to these curves? Well, they move around, so let me just draw a picture around one point. So for p and t2, I want to construct the leaf w u g of p. So what I'm going to do is I'm going to take the curve where I go back to my fixed initial gamma u, but at g to the minus m of p. And then I apply g to the m to this curve. Now each time, so this is some curve through p. It looks like this. So if here is g to the minus m of p, here's my little starting curve. Let's just make it a little line. What happens if we apply g once? So g composed of gamma u of p is now a curve through g to the minus m plus 1 of p. It's still tangent to the unstable, but it's even more tangent to the unstable. In other words, the cone field that it stays inside is contracted, and it's longer. And now if I go all the way up to the point p, well, since I'm still in the unstable cone, I can apply g again. It'll be even more inside the unstable cone. And it'll be, because it's still in the unstable, it will continue to be expanded by lambda. So at p, I get a very long curve. And in fact, if I really want to prove things, I just restrict myself to some fixed neighborhood of p. And I'm not going to worry about this for now. But in fact, it is the case. So this curve, that g to the m composed with gamma u, g to the minus m of p. This is very tangent to, well, instead of saying it's very tangent, let's just say it lies inside the intersection from j equals 0 to m minus 1 of, you don't even have to say the intersection, it lies inside of dg to the m at g to the minus m of p applied to the unstable cone at g to the minus m of p. This curve everywhere of this derivative, sorry, at t, this is at gamma u, g to the minus m of p at t. So for each time, it actually lies in the image of this cone under the derivative of g to the m. And as you saw, I hope, in the exercise yesterday, that if I let m go to infinity, this converges to what we call e u of p, the unstable line through p, the unstable direction. And so this curve, I've kind of obliterated, is going to be very long, and it's going to be tangent to e u. And this limiting curve is the unstable. And because it was constructed by this iterative process, once you can show that that limiting curve exists, and it really is a curve, but I hope you're convinced, visually, it gets longer and smoother and more and more closer to this ideal unstable direction, that as m goes to infinity, we get, if you like, applied to 0, 1, that this limit equals, some suitable sense, the unstable curve of g through p. And that's the leaf of the unstable voliation. Well, similarly, we can construct WS g of p using stable now, stable curves, some family inside of the stable cone. And taking the limit is then, now we go forward and pull the curve backwards. And again, it gets longer and more and more tangent to skinnier and skinnier cones. So this is the limit as, again, m goes to infinity, but now we do g to the minus m. So this is the idea. And it shouldn't be surprising that these limiting curves, well, they certainly lie inside the unstable cone. So if I iterate a curve inside of w u, if I iterate it forward, it grows in length by lambda. And if this limit exists, then clearly the image of a piece of unstable manifold now is the image piece of the unstable manifold, because I got this one, unstable manifold. I got this one by going forward and pulling back to here. And I got this one, well, I could go forward to the same point and just pull back once more. And if the two limits exist, then the preimage of this would better be this. So the invariance kind of follows immediately once you've shown that these limiting curves really are curves. You also need to show that it really forms a foliation, even though your initial curves might have intersected each other. When you take the limit, those intersections basically go off to infinity. And so you actually get a foliation. Two points are either their leaves are either disjoint or the two points are on the same leaf. And because a piece of unstable curve is in the unstable cone, it has to be expanded when I go forward in length, because all the tangent vectors are expanded. And so therefore, if I take an unstable curve and go backwards, it gets contracted by a definite amount in length. So this is how you establish these properties. Do you have any questions about W, S, and W, U? Just how they're constructed? Yeah, how do I what? Yes, how do I show that they're smooth? Because what you show once you have these curves, but in the limit, it's possible that these smooth curves, maybe they're not smooth. How do you know they're smooth? Well, what you do is you show that you have a tangent vector at each point. Or tangent space. But what will that tangent space be? It will be the intersection of all these cones. And the intersection of all these cones is the unstable direction. So that's how you show that you have this continuous tangent space, roughly speaking. That's why they're C1. Now to show higher smoothness, I won't discuss. Are there other questions? So yes, I've left out a lot of steps because I want to talk about a key property. So Hannah is going to prove that if G is C2 and a NOS off, then it's ergodic. If it preserves Lebesgue, it has to preserve Lebesgue. And I don't know if you recall, when she gave the Hopf argument for 2, 1, 1, 1, she said something about the foliations being good enough for Fubini, because she used Fubini's theorem at some point. And now what I want to try to convince you is that the foliations that I've just constructed are good enough for Fubini. So let me state precisely what good enough Fubini means. OK? So theorem. If G is C2, this is important because it's false. If G is C1, it can be C1 with plus holder. So it could be C1 and the derivative satisfies some holder type of condition, if you don't know what that means, don't worry. But C1 alone is not enough for this theorem because we're going to use distortion estimates. So then the plaques, W, U, or S are good enough for Fubini. The technical term that's used is absolutely continuous. And what that means is the following. It means two things. So consider what's called a foliation box, or let's just call it a square in the torus. So maybe it could be very small, but it has to be arbitrary. In other words, anywhere in the manifold in the torus. Let me draw a picture. So here's S. And here are the plaques of W, U, loke. That's an example of plaques of a foliation, but I'm turning this on its side. So these are W, U, loke, G. And this should be of G, of course. And let's denote, now I'm going to change notation a little bit from Hannah's lectures. So for P in the square, denote by mu sub P super U. This is the Lebesgue measure on the plaque restricted to the plaque W, U, loke of P. So if you like, I could use arc length in the torus to send this to the interval, each one of these curves, to send it to some interval, maybe the 0L, where L is the length of the plaque. And I can take Lebesgue measure here and just push it forward here. But it's just the measure on that leaf induced by length. So Hannah I think was calling this M sub U. But I want to emphasize that each point has its own measure. And that measure only sees that little plaque. Then we have two properties. And similarly for mu sub SP with W, S, loke. The idea is that these W, U plaques behave well with respect to Lebesgue measure. So Lebesgue measure on the leaves gets along well with Lebesgue measure in the square. So then if Z inside of the square is any set with Lebesgue measure 0. So this is the first property. So then for almost every mu, almost every p in the square, the measure or p in the square, the unstable measure of Z is 0. Or if you like, if I intersect Z with the plaque through p, so I have some 0 set, Z. And then I take almost any point in the square and I intersect Z with the plaque. I get a set that has measure 0 in the plaque. So 0 in the ambient space restricts to 0 on plaques. And the second property kind of goes the other way. It says that if I have, and in fact these are kind of, this is if and only if. So if Z is any set, if this is a measurable set, then mu of Z equals 0 if and only if for almost every point, that intersection. So it's really an if and only if. And then the other is a sort of transverse kind of form. So if I take any set Z sub S p contained in a stable plaque. So really this stable plaque is just playing the role of some curve that crosses all of the leaves inside of the square. So I take any set and I take Z sub S, a set that has measure 0 here. So if mu sub S of p, so the inside that mu sub S of p of Z sub S p equals 0, then if I what's called saturate, so I take each point in Z sub S and I take the whole plaque like that, that has Lebesgue measure 0. So the measure of the union of overall p in Z sub S of these unstable plaques is also 0. And that is also an if and only if. So if I have the saturate as measure 0, then the intersection with any plaque has measure 0. OK. That there is a picture of a foliation box of plaques. It's actually a foliation of the torus, but does not have this property. It has very nice smooth leaves, but it is not the unstable foliation for an anossof diffeomorphism. In fact, that foliation is quite pathological. It has the property that I can pick a subset of this square. There exists a subset of this square that has full measure. It has the same as the measure of the square. Its complement has measure 0. But if I take almost any point in the square, or I take any point in the square and I intersect the plaque with the set, I get finitely many points. So it doesn't have full measure at all. In fact, it's finite on that number. So just to give you an idea why this is not sort of an automatic property of a foliation whose leaves look nice, I present you that picture. And I can give you an article that explains pictures like this. OK, so I have just a few more minutes. So I just want to quickly say why this is true. Well, there's no way. I can say really so much why this is true. Certainly why this is true uses C2, and so it uses distortion estimates. Let's look at property two just for a second. So let's look inside of a square. And imagine this is a very small square, very small. And let's take a WS loke plaque of some point p. And let's take some set z sub s of measure 0, z sub s of p. And recall that a set that's a 0 set can be covered by a collection of intervals whose total measure is arbitrarily small. OK, so to show, so what I want to show is that if I change the colors of my plaques, it's terrible. So I want to show that if I take the union of the unstable plaques, this is a 0 set. It's enough to show that if I have an interval of length less than epsilon, then when I saturate it, the measure is at most some constant times epsilon. If I, some interval inside of WS loke, is tiny. So to show, there exists a constant c greater than or equal to 1. So if the length of i is less than epsilon, then, I just have two minutes. I really can't say more. Then when I intersect, let's just pretend this is just a vertical line. I'm not even really talking about WS. I'm really talking about vertical lines. So here's my i. It's some interval in the vertical direction. Then, after saturation, any line here has length less than or equal to c times the length of epsilon. So these are just vertical curves. These are not. I did a nice change of coordinates so that this stable leaf is just a vertical line, a little vertical line segment. So it's enough to show there's a constant so that if this is tiny less than epsilon, then everywhere I go in this box is less than epsilon when I saturate after saturation, after taking the union of plaques. So that's a measure theory exercise. And I simply have a picture to illustrate how you do this. And the picture just goes like this. Here's my vertical lines. So imagine just vertical or horizontal in that picture. Here's my teeny, teeny tiny because maybe this is small, but this is incredibly small. I apply f to the minus m. These vertical lines, well, they don't stay vertical, but they stay inside the stable cone because they started out inside the stable cone. And these lines, because they were unstable, they stay unstable, but they get contracted. And so I choose m so that when I look at this little i, after m iterates, this length, oops, wrong color. Here's, sorry, this should all be g to the minus m, not f to the minus m, all of this. This length, these get very short. I want so that this length and this length are approximately the same and on the order of epsilon until the length of g to the minus m of i is approximately epsilon where epsilon is fixed, is least epsilon. And we've chosen this so small so that the length of g to the minus m of i is going to be greater than the length of g to the minus m of any one of these plaques. Thank you, i. OK? So I iterate till this minuscule thing becomes kind of size epsilon. And then these are going to have to get shorter. And since this is so minuscule, I've picked it ahead of time so that when I iterate backwards, this can only be smaller. OK? So this is possible. And now, how do I know when I'm on this scale? Now I know that the length, because now we're just on a scale where this distance and this distance are comparable, I know by the triangle inequality that the length of this is no more than three times the length of this. Because the distance from here to here is less than or equal to this length, but this length plus this length. And so on this scale, all these lengths are comparable. But notice that here, this is very small and it only gets smaller. This is extremely small. This is exponentially small, and it gets to a small size. So when I take the derivative, when I take g to the m and bring the picture back like this, I can use distortion estimates to say that the derivative is basically constant in this direction. It's basically constant for all m. The ratio of derivatives at two points of g to the m at two points is bounded by a constant. That's enough to tell me that if this and this are comparable in length down here, then this and this are comparable in length up here. So I use distortion at that crucial point restricted to pieces in the stable cone or in the unstable. So roughly that's the idea. A set that has small measure here, I cover it with intervals of very small measure. And then when I saturate in this direction, I get something whose measure is multiplied at most by some constant that's fixed. And so zero sets saturate to zero sets. So that's kind of how absolute continuity works in general. And that's the role of distortion estimates in the proof.