 It's good to be here. ICTP is a very nice place. And, okay, that's what I'm going to talk about. And I'll try to give some motivations for the final theorem. Okay. So, let me start mentioning this classical result from the 70s, 60s. A nozz of proof that a nozz of, what is now known as a nozz of film or fissures, are periodic. I'm assuming things are regular now. And, okay, talking about the ergoress today, I mean volume preserving, okay. And the, one of the main ingredients of the proof is the, what we call absolute continuity of the stable and unstable fallations. Let me just make sure we are, just quickly remind, I think, up to now we all are familiar with the definitions. But remember the nozz of is the tangent bundle has this splitting. The derivative contracts in this direction, expands there. And since I'll be using partially hyperbolic, let us recall that for the partially hyperbolic, it's similar, but we have that center direction, okay. Also, we have associated for the stable and unstable directions the stable fallation and unstable fallations. And for the context I'll be working, if I say center fallation, it's because it will exist in my context, okay. It has been discussed in some previous lectures that it doesn't always exist, but it might, in the context I'll be working, it does. So, okay. What else? The absolute continuity, what is absolute continuity of a fallation is, I want to give a definition, not using, I can't use like, halonyms, but I want to give the definition which is close to what I'm going to use, which involves this integration of a measure. So, briefly, let me tell you what is this integration. Well, recall that, for instance, if we were, consider the square and consider the horizontal lines. If I give you a set A and I ask you to calculate the volume of A, that's how you do it. You consider, I'll call lambda X as the, the back measure on the, so it's the length, okay. I'll use like Y here, Y. So, you compute the length of A, each horizontal, and then you integrate, that's how you do it. Okay. It turns out that we can do a similar thing when we're working with falliations that are, you know, doesn't need to be as regular as this horizontal. It's not a, a forbidden thing. We call it, it's, Rockling does integration, Rockling is integration, which is the following. So, consider a foliated box, similar to that. And then, now the leaves are a little bit like this, okay. And then I want you to calculate the volume of a given set, okay. So, the Rockling does integration theorem states the following. That we have similar to that. We have on each, I'll use C now. We have for each leaf, we have a measure which I'll call like mu C. Volume of A will be. I will integrate with respects to this measures. The set A, and I will integrate with respect to, I'll use like this. Volume hat, where volume hat is the projection of volume. So, what is pi? Pi is from the square to sigma. So, it's just the projection through the, the leaf, okay. So, Rockling does integration theorem. So, you can see that it's, okay, very similar. The thing is that now we're doing what we do with Fubini, but for general falliations, okay. And by absolutely continuous, now we can define. So, I say that a, that a falliation F is absolutely continuous. If, so when I disintegrated on the falliated box, the, those measures which are the conditional measures, they are equivalent to the, so I'll keep this notation for the Lebesgue measure. They are equivalent with the Lebesgue measure of the leaf, okay. So, that means in some sense that when a falliation is absolutely continuous, it's kind of like a, has a similar behavior as the horizontal. For instance, a set of, a set of full measure will intersect almost all the leaves in a set of Lebesgue full measure of the leaf. Okay. So, the spirit, okay. So, these are the definitions so far. Well, what's the spirit of the talk is, it's, it's from this theorem. What happens is we used some knowledge of the behavior of the invariant falliations to obtain some dynamical consequence in the case ergodicity. So, we want to see how the invariant falliations interact with the dynamics. That's one example. A more concrete, this is, you know, like a, a vague, that's the spirit, the spirit. And then another example is a conjecture. An example in this spirit is a conjecture. I'm not so sure who to give it credit for, but it's around. It's that if the stable and unstable falliations of an anaus of different amorphous are smooth, the Diffel F is smoothly conjugate to its linearization. As you might know, the, so an anaus of Diffel is conjugate to a linear one. So, the conjecture is if you have enough conditions on the stable and unstable, you can make the conjugacy very good. So, the conjugacy is in fact smooth. If that would imply, I quite get it. You want to meet? Oh, I don't know. So, as far as I know, I could just mention the theorem for the conjecture for on the torus. And as far as we know, people just proved for T2 and with some conditions on T4. That's all they have. So, the conjecture we can say, oh, it's on T4, it's on Tn on the torus. Okay. And, but the thing is just to mention, okay, do some conditions on the falliations, give some information on the dynamics. And, all right, but in our case, we are, let me give you another thing to, you know, inspire the philosophy, which is now let's go to the partially hyperbolic context, which is, let me state a theorem from Avlovianna, Wilkinson, which states the following. If we consider F, a perturbation of the time one of your desert flow on a surface of negative curvature, then only two things can happen. So, the thing is, the first observation is that F is a partially hyperbolic, because it's the perturbation of the time one. It also has the center fall, a center falliation. And the thing is, so if you consider F, well, everything that will appear here will be volume preserving. Then only two things can happen. The first one is, if the center falliation is absolutely continuous, you have, I'll call it rigidity, then if you have this information in the center falliation, you have a very strong thing, you know, a very strong thing that F is indeed a time one of another flow. If the center falliation is not absolutely continuous, then you have some, if you see it for the first time, it seems like a weird behavior, which I'll call it Fc has atomic disintegration. Atomic disintegration is, I'll just say, because I want to talk about the rigidity, I'll just say it in, I'll put it here like this. Atomic is just, if you look at a foliated box, and then there is a set A, I'll say a full measure, okay, things normalize. So there is a set A of full measure, which intersects each leaf in, let us say, two points. So that's strange, you have a set of full measure, but intersects each leaf in, let's say, two points, or one point. Okay, that's atomic disintegration. Okay, but the thing is, it's a good description for this, for this dynamics concerning the, knowing the behavior of the center falliation. Then we can ask, well, does this kind of behavior happen for some other types of partially hyperbolic? And so another important class is what we call derived from an OZF. So I'll say that F is, I just would like derived from an OZF if F is partially hyperbolic, and so we have the linear OZF and F is homotopic to a linear OZF. This is what we call derived from an OZF, derived from a linear OZF. Okay, these, they have a good relation, they are semi-conjugate. So known as Frank's Manning semi-conjugacy is that there exists H such that, okay. So I want to investigate the rigidity concerning this kind of partially hyperbolic. And while somehow here it's appearing, you know, to talk about the rigidity, to relate it somehow to, for instance, to some dynamics. And I want to see how strong some conditions to see how strong or how good the semi-conjugacy is. That will be the goal. But let me mention first that the, thus these partially hyperbolic, they have a richer behavior when compared to that one. So derived from an OZF, they have a richer behavior because what can happen is that the center foliation can be, well, it's a, easy thing is that it can be the disintegration, I'll say disintegration can be equivalent to the bag. This is just a simple case, for instance, the linear OZF. These defoliations are lines. But they also have that behavior from there, which is the atomic. I just put atomic disintegration. This was done with Gabriel Ponzi, Alitazibi. It also has another one, which I'll just call it neither, which is, in fact, the disintegration can be, it's neither atomic and it's singular with respect to the bag. Okay. But the thing is, today I want to talk about things concerning the rigidity. And if you allow me some prejudice in the talk, I will say that the good measure is the bag measure, just in this talk, to say that, okay, this, the difference that I have this behavior, the defoliations, you know, they are not as well behaved as we want. And maybe the derived from an OZF that have good disintegrations, they, you know, might, let's say, these are good defoliations. So in the same spirit as that theorem, I might ask, does, you know, does good behavior might imply any information in the semi-conjugacy, for instance? See, if you compare here, we could still ask, for instance, does, if the centrifugation for the derived from an OZF is absolutely continuous, then do we have, I don't know, smooth semi-conjugacy or something like this? The thing is, the, the A have, it has a richer behavior for the center defoliation and also has a different behavior concerning the, this rigid way of approach. So let me state a theorem which is, says the following, that there exists F, once again, everything volume preserving, volume preserving. But then, now, on the three tours, volume preserving the A with the center defoliation being the C1 defoliation, but C1 conjugate to its linear part. Okay, so if I have a C1 defoliation, when I look at the defoliated box and I disintegrated the leaves, then the disintegration is, because the defoliation is C1 actually, so the disintegration will have a density with respect to the, the back measure of the leaf and in particular the, the density is very good. It's a, it's C1 and okay, even if with a very good, this is much better than only absolute continuity, even if with a very good disintegration, we do not obitain, for instance, you know, smooth conjugacy or something. That's, that's what I'm looking for. Let me also mention a property of this defoliation is, now, consider this C1 defoliation. The, when we fix a defoliated box, the four, I'll say, so we fixed the defoliated box, then the densities, they are, you know, fixed this, there exist a constant and the densities, they are limited from below and above. But a priori, it, it doesn't mean that with the same K, we, or we could find a K, it doesn't mean that we can find a K, that it works for, for all, for, for any defoliated box, okay? So there is, now I will follow here a definition from the, they define the uniform version of the absolute continuity, which is, I'll call, so the uniform, so I'll say that definition. I'll say that F has the uniform bounded density property, the following happens. There exists, so there exists K, such that, and then now it depends for, for all defoliated box disintegrate. I disintegrate volume on those leaves and the disintegration S has a density with the, the bag measure. I'll just put hat here to emphasize, when I disintegrate, the disintegration, this is always a probability for the way I use the broccoli disintegration, so probability, so I, I'm considering the normalized Lebesgue measure. So the definition is, defoliation has the UBD property if, it doesn't matter the, the box, I consider there exists this K, such that the densities are uniformly bounded. So, this is, the, the, so this is the uniform, we can see it as the uniform version of the absolute continuity. Okay, it turns out that in the end, this is just the right condition for rigidity in the context of derived from an OZF, defiolomorphisms. So the theorem I wanted to state is, goal theorem is this one. So F in the three torus volume preserving the derived from an OZF, then F is smoothly conjugate to A, I'll put A, okay, A, the linearization, if and only if the center foliation has the UBD property. So that is the condition that finally gives the, the rigidity, the type of rigidity we're looking for. And so it's very important to have this example in mind. So even if with a good foliation, we do not obtain conjugacy. This here, the UBD property, it's a, it's a measurable property. C1, it's, you know, something that holds everywhere. This we substitute as, you know, similar to the absolute continuity. And the difference with also the C1 is that it is not guaranteed, although, although at first it does seem that it would be, but it is not guaranteed that we can make the densities uniformly bounded. So this, if you see with the, what Avalovianow-Wilson did for the geodetic flow, that is the, the similar result for these partially hyperbolic. Like one minute. I'll just state something that the, concerning the proof, the steps of the proofs has to do with, we have to construct good conditional measures. I'll just say like good conditional measures such that they have the following characteristic. The push forward of them, it's the, there's relation, but we can construct this conditional measures in such a way that they have uniform bounded densities. Now this is the Lebesgue of the whole measure. They have uniform bound densities with the Lebesgue measure. So if you see, if we iterate, we do, we got this. Here we are getting the exponent of the center direction. If you consider the derivative of this measures, some Jacobian will be appearing and then we will be able to obtain Lyapunov exponents. And from Lyapunov exponents, we will obtain Lyapunov exponents for every point. And from that, we will have periodic data. And from that, we can use Google Live's result and prove the smooth conjugacy. Okay, that's it. Thanks.