 Okay, thank you. It's always a pleasure to be here at the ICTP. I thank you very much the organizers, especially Stefano, for the invitation. Okay, so let me state the setting in which we are going to work. We are going to prove that center and stable foliation do not have compact leaves. But we are working on this setting. We are working on a three-dimensional manifold, and we are considering a partially hyperbolic defumorphism, which means that the tangent bundle splits into three invariant bundles such that this is strongly contracting, this is strongly expanding, and this has an intermediate behavior. This means that it is not as contracting as this, nor as expanding as this. And three bundles are one-dimensional, so they are non-trivial. Okay, what do we know about these bundles? In general, we know it's classic result that the strong bundles, the expanding and the contracting and the expanding ones are integrable. These are integrable, and this is integrable. But in general, it is not known whether the center bundle is integrable. For 35 years, it has been an open question whether you could integrate this to a foliation. And in, I guess, 2009, it is now in print, Federico Raúna and I proved that there exists an example, an open set of defumorphisms on the three torres such that if the center bundle is not integrable to a foliation. Let me explain better what we would like to have as an integrable foliation. We will say that a defumorphism is dynamically coherent. We will say that f is dynamically coherent if there exists an invariant foliation tangent to the center and stable bundle. And if there exists an invariant foliation tangent to the center stable bundle. Of course, the existence of these two foliations imply the existence of an invariant one-dimensional foliation tangent to the center bundle. So what we proved here is that there exists an open set of defumorphisms, of partially hyperbolic defumorphisms such that f is not dynamically coherent. This example consists, has a torus, a center and stable fixed torus which is attracting and in these torus, if you cut this, consider this as a center and stable torus. You will have that. It is uniquely integrable outside the torus, but the leaves are like this. So you cannot obtain a foliation, a global foliation. So when we have this, after we build this example, we conjecture, what about if this is the general case? And we have a still open conjecture, a non-dynamical conjecture, non-dynamical coherence conjecture with Federico and Raúl which states that the only possibility for a defumorphism to be non-dynamically coherent is that it has a torus, either tangent to the center and stable bundle or tangent to the center and stable bundle. So if f is not dynamically coherent, then either there exists a torus, tangent to the center and stable bundle, or there exists a torus, tangent to the center, stable bundle. And this conjecture so far has been proven true for manifolds that have a solvable fundamental group by Andy Hammerlidl and Rafael Potrie. They have proven that conjecture is true if the fundamental group is solvable. Okay, so this is the state of the art so far. Let me mention, even though it has nothing to do with this, that we have a kind of symmetric conjecture with respect to ergodicity. We have conjecture that if it is non-dynamically coherent, then there is a torus, tangent to the center and stable or a center stable. But we also have a conjecture that if, let me call it non-ergodic conjecture, which is for conservative setting, this is for any setting. But in the conservative setting, we have kind of symmetric conjecture that if this is non-ergodic, then there exists a torus, tangent to es plus eu. And this has so far proven true only in nil manifolds. So we don't know, but it has some kind of, strange that it has some kind of symmetry that the only obstacle to non-dynamical coherence is the existence of this tori. And the object of obstacle to ergodicity is the existence of this tori. It's kind of symmetric. Okay, so for solvable groups, we have this situation. But let me tell you that we have a result that states that if there exists a torus, tangent to either the center stable or the center unstable or the stable and stable bundle, or, or. Then the manifolds can only be the three torus or the suspension of the minus, how to say mapping torus. It's the mapping torus of minus, plus minus the identity on a, yes, minus the identity. Or this is the mapping torus of a hyperbolic map on the torus. In particular, if there exists a torus, tangent to either es plus eu, es plus es, or es plus eu, the only possibility is that m, the pi one of m be solvable. So, in particular, conjecture, the non-dynamically coherent conjecture, now is that if pi one is not solvable, then all partially hyperbolic defiomorphisms are dynamically coherent. Okay, because if it is, if there is a torus, then we are in this case where we already know the situation. Okay, so I haven't mentioned so far the result. So, this is the conjecture of when the, what we are presenting here is that in general, almost everybody will be dynamically coherent. Okay, well, so we wanted to present as kind of converse and the theorem here is kind of converse. So, this is the main theorem. We conjecture that if it is non-dynamically coherent, then there exists a torus. And what we are stating here is the converse. If there exists a torus, there cannot be a foliation containing it. So, this is main theorem with Riko and Raúl. It says that if there exists an invariant foliation tangent to, yes, to EC plus EU, then this foliation, this foliation FCU does not have compact leaks. Okay, so I'm not meaning that there cannot be tori. There could be tori, but in fact, in the example we built, we built an example also of a dynamically coherent example that has a center and stable torus. But this torus cannot be part of the foliation. It can be a torus tangent to the center and stable bundle, but cannot be part of the foliation. You cannot complete the torus into an invariant foliation. Okay, so let me tell you what the strategy will be. First, a remark. I don't know if everybody knows what a red component is. In our setting, a red component will be a torus, a solid torus, foliated by planes inside. I mean, there will be like paraboloids, one inside the other. I don't know if you can see, it will be planes like paraboloids, one inside the other. And in the boundary, you will have a torus. This is a red component. One first thing that we must know is that if a red component is transverse to the one-dimensional foliation, then once the one-dimensional foliation enters transversely here, it cannot get out. So it must have a loop, yes, a closed curve. In particular, this implies that there cannot be red components transverse to a stable flow, to a stable bundle. Because if you had this transverse to a stable bundle, this would imply that you have a closed loop, a closed stable loop. So in particular, if there exists an invariant foliation tangent to a center unstable bundle, it cannot have red components. Okay, so first remark is that the center unstable foliation cannot have red components. So our strategy will be to produce a red component. Assume that we have an invariant foliation tangent to the center unstable bundle with a compact leaf and produce a red component. If a compact leaf of this must be a torus because it is foliated by a one-dimensional field which has no singularities and no closed loops. So if it has a compact leaf, it must be a torus. So the strategy will be to produce a red component. And how we will produce this? We will leave this to the universal cover. And how do you produce a red component? If you get a closed loop transverse to the center unstable foliation in the universal cover by Novikov, you get a red component. Okay, so this is what we are going to do. We will get, in the universal cover, we will find a closed loop transverse to the leaf of the center unstable foliation. And this will produce a red component in M. And that will show it is not possible. So our strategy will be to produce a closed loop up. Okay, so how we will do this? First of all, we have to prove that the existence of a compact leaf implies the existence of periodic torus with hyperbolic dynamics. Center unstable torus will produce a periodic torus with hyperbolic dynamics. So if there exists a torus, a compact leaf, which will be a torus, this implies the existence of a periodic torus. How do you prove this? Well, by head leader, we have that in a foliation, the set of points belonging to a compact leaf is compact. So you take all the points belonging to tori and this is compact. In particular, the limit of tori will be tori. And so this will be, you will take all the tori in the foliation and there will be a recurrent torus. Okay, but the torus, the center unstable torus is transverse to the stable leaf. It's transverse to the stable bundle. So you can produce a tubular neighborhood of a center unstable torus by stable leaves. And then when you get a recurrent torus, you will by iteration be able to get fat torus inside itself. Then you iterate and then you produce a periodic torus. Okay, is it clear? So this implies there exists periodic. So now we want to show that the induced action on pi one is ergodic, is hyperbolic. And how is this? Well, this will preserve the unstable direction. Okay, so the action of f star on the torus will preserve the unstable direction. So you will have an eigen space which is irrationally. Okay, so this has, this leaves us only two possibilities. Either f star is hyperbolic or f star is identity. Okay, but if f star was the identity, then we would have that f star is the identity plus some constant. Pardon? Yes, it's some plus something. I'm sorry. This is plus five with five bounded periodic. And in particular, yes, in particular, we will have that the diameter, you take an unstable leaf, a small piece of unstable leaf. And then the diameter of this will be bounded by the diameter of this plus mk, where k is the bound of this. So on one hand, we will have that the diameter grows at most linearly. And on the other hand, we have that this, the lengths of this, of this grows exponentially. Okay, so we have on one hand that this grows, the diameter grows at most linearly and the length grows exponentially. So we will have very long unstable segments which are with endpoints at distance smaller than epsilon. Then by Poincaré-Vendixson, this produces a closed loop inside this or singularity and this cannot be. So what we have is that the existence of a compact leaf implies the existence of a periodic torus with hyperbolic dynamics. Okay, and now we are closed. Now that we have a periodic torus with hyperbolic dynamics, we will produce this, we will have this torus to produce a closed loop transverse to the center and stable leaf. Okay, so by Frank's, then cut the manifold along this torus, cut the manifold along this torus, and you will have that the manifold is the torus times the identity. Okay, if you cut it along the torus, you have the manifold and you can trivially extend the diffeomorphism to this. And by Frank's, you have a semicontregacy of this into torus times zero by a linear hyperbolic map. Okay, so this is H0, I never remember. Yes, H0f equals a... Okay, so it is not hard to see that on the torus, H on the torus restricted to this torus takes center leaves into stable leaves of A. Center leaves of Df into stable leaves of A. It has to be like this, because if H wouldn't... I mean, for our purposes, it's enough to look what happens with a periodic point. What I mean is that H minus one, H of P, intersection T is contained into a center segment P. And you can take it to be in a local center segment. If it had the points of different center leaves, then you would have points that go very far away. With respect to the unstable, you can join it with an unstable segment. And then you can have that this grows very far away. But this, I haven't said, this is homotopic to the projection. So in particular, H, the diameter of H minus one of H of I is uniformly bounded for all... Yes, for all Y in T. So this is uniformly bounded. So for a periodic point, this will be contained in a center leaf. It is not hard to prove that in general, if you do not restrict to T, if you state you just take this and you intersect this with a small ball, this is contained in the center stable local leaf of P. Then it is not hard to see that because you take a point here, then you project it into the center leaf. You project along this table into the center leaf, and then this commutes with this and takes center into center, stable into stable, so you will have that this preimage of this image is always in the center stable leaf of P. Once we have this, we are almost done. So we have that the diameter of all these points are uniformly bounded. So in particular, for each small epsilon, there exists some N such that for all set of N points, there are at least two of them that are epsilon close. You always have this. So there are systems such that if you have N points, then there exist two of them that are no more than epsilon apart. So take in X1, XN inside H minus 1 H of P, which is contained here, but take them so that in different center leaves, so you have P here and take X1, X2, in different center leaves. You can take this in different center leaves. Now you iterate this backwards many times, as many times as you want, and then you will have for each N always a couple of points that are epsilon apart. In particular, there will be a pair of points, the same pair of points that infinitely many times is epsilon apart. So you will have, there will be two of them. Let's assume X1, X2 such that distance of F minus Nj of X1 and F minus Nj of X2 is less than epsilon for all j. You see, since this is always at the distance, they are here and this is invariant by an iterate, we can assume it is fixed, so this is invariant, and so all iterates of this belong all the time here and the backward iterates by our choice, we can always assume that there are a pair of them and iterates infinitely many iterates of them such that they are always epsilon apart. We are almost there, with this we are almost there. So we will have this picture, we will have X1, this is the center of X1, these are all centers, this is X2 and then we can do the following. We can join X1 until we go by the center leaf until X2 and then we join X2 with a stable segment. So we have for infinitely many iterates, these two are epsilon close. Now I iterate this picture backwards infinitely many times. So at infinity, very close to infinity, I get this drawing. I will get X1 and X2 very close. A huge stable loop because this goes to infinity when I iterate backwards, this goes exponentially to infinity and this center leaf that can go or may not go to infinity, but I don't care. This is in a stable, in the center unstable, you may assume it is in a center unstable leaf, but now this whole loop is transverse to the center unstable. This is transverse to the center unstable. I may assume that I have done this in the universal covering. This is transverse to the center unstable. Now this is a straight segment contained in the center stable leaf. This closed is horizontal, but I can perturb it because this is very near, this is epsilon apart. So this is horizontal, this is very close, so I can make, I can take a two-wheeler neighborhood here so that I can join this point and this point by a segment which is transverse to the center unstable leaf. This is super classical. So I have built a loop which is transverse to the center unstable foliation and I complete it, close it by a segment that is transverse to the center unstable leaf, to the center unstable foliation. So I have got in the universal cover a closed loop transverse to the center unstable foliation and so that is a red component and this implies that an absurd contradiction. Okay, and this is it. I'm finished.