 OK, good afternoon. So let me start by recalling a little bit what was the purpose of today's lectures. And so I mentioned a theorem due to Bonati and Krovisie, which says the following. It says that essentially you can split the space of C1 difumorphisms as a disjoint union. These are open, C1 open and disjoint. In such a way that difumorphisms which are here have a trapping region. And difumorphisms which are here are almost all of them, or at least a dense subset of them, are transitive. So we had this dichotomy. So if F belonged to O1, we knew that there existed a trapping set that meant a proper open subset of the manifold, which is compactly mapped inside itself. And we discussed at the beginning that this had certain implications. And otherwise, we could approximate difumorphism by one which was transitive, meaning that it has a dense orbit. So if there exists C inside O2, C delta dense, such that for every F in C, F is transitive. So this was quite nice, because in principle, all we knew was that the non-existence of a trapping region only implies a very weak recurrence that we called chain recurrence. But transitivity is a much more interesting recurrence property. However, the downside or what we would like to do is try to remove the condition of having to restrict ourselves to a residual subset. So we were wondering about the abundance of robustly transitive difumorphisms, meaning is it possible to change this C delta subset of this by an open and dense subset of O2? And so what I wanted to explain in this afternoon lecture is one partial result in this direction, which makes certain assumptions on partial hyperbolicity. And so it fits very well in the subjects of this mini course. So this is a theorem shown with Flavio of denoure, Sylvain Crovisier, and Martin Sambarin. And the point was to show that the set, I will make some remarks afterwards, the set of partially hyperbolic difumorphisms with central dimension 1. This meant that the dimension of the central bundle was equal to 1 can be split into the joint open subset. So it's essentially the same statement. But instead of asking for a residual subset here, we knew that every difumorphism here will be transitive. If F belongs to U1 and trapping set, or if F belongs to O2, then F is transitive. And so what we did in the morning was to discuss some possible approaches that allow one, on the one hand, to show that certain transitivities are robust, are robust under perturbation, and also how to promote weak recurrent properties into stronger recurrent properties, such as transitivity. And we mentioned that there were two ingredients. One had to do with certain robust geometric conditions on the invariant manifolds. And the other one was to try to apply arguments similar to the case of the anosol difumorphisms. But so let me first make some remarks. So here, I'm saying partially hyperbolic C1 difumorphisms with central dimension 1. I will focus mainly on the case where ES and EU are both different from 0. In our definition of partial hyperbolicity, we allow one of these two bundles to degenerate. But I will make arguments that make use of both foliations at the same time. So we will assume that both foliations, both bundles are non-zero. In any case, if you have an extreme bundle, which is one-dimensional, there are other arguments, mainly due to Pujales and Sambarino, that allow you to obtain the same result. So it's no loss of generality to work in this context. And so let's recall a little bit what were the ideas we were working with. And so let's start with the context. So f from m to m will be a globally partially hyperbolic diffio with a splitting like this and this dimension equal to 1. And so we are going to say that disappeared in the morning lecture that the strong stable foliation has property SH, SH as a short hand for some hyperbolicity, if it verifies the following property. If for every sufficiently large strong stable disc, if there exists r bigger than 0 and lambda bigger than 1, such that every disc of the strong stable foliation, which has sufficiently radius, has the following property. So rc has a point x such that for this point, the center direction behaves as a stable direction, as a true stable direction. So this I can write it like this. So the minimal expansion of the derivative of f to the minus n in the center direction. So let's do only one iterate. It's larger than lambda, which was larger than 1. OK, forget about it. So this is a property that we checked in the morning for a specific example. So we have this manias example where we made a local perturbation around a fixed point, making that the center direction, which originally was a stable direction, turned into a neutral or non- hyperbolic direction. But that happened only in a small neighborhood. So if we took a long arc of stable foliation, we would find some points where the center direction was expanded. OK? So in a certain sense, this seems kind of a tough property to have. But one of the points of our work is to show that this property is quite abundant. It's not as tough to check as one may expect. So some comments, which are exactly the same as we did in the morning, so remarks. The first of all is that the fact that this foliation varies continuously with the diffeomorphism, particularly this of a fixed radius, and the fact that the diffeomorphism, if you make a C1 perturbation, this number does not change very much. It means that this is a robust property. Maybe it's hard to obtain, but once you get it, all perturbations will have this property. OK, this is a C1 open property. And the second remark is that, OK, so you have a disc over here of large radius, and you have at least one point x where the center bundle is expanding for backward iterates. So this happens in a neighborhood. This is continuous. And so as you iterate backwards, this neighborhood will again contain a disc of radius r. So because as you iterate backwards, this disc is getting larger and larger, and so you can apply the same property once and again until you get a property that holds for every iterate. So let me write it like this. The stable manifold has the SH property. Then for every disc in the stable, there exists a point in the disc such that the derivative to the minus n in the center grows exponentially fast. Maybe you have to change it and get this lambda a little bit smaller. But essentially, this is the property you get by applying this property once and again. Iterate backwards, and you find this open set eventually contains another one, and you get this to apply this again. So you need to change this lambda to a certain sigma, but this sigma is larger than one. Now, why was this interesting? Because we have this argument that we use for a lot of diffeomorphisms, and we have an open set here, another open set here. You take a disc of certain radius. The property you demand is that every disc of larger radius has at least one point with this property. By continuity, it has an open set of points with this property. So as you start iterating backwards, this set of points which expand will have another small open set here which expands in two iterates. So let me assume that you are expanding more than three times in the stable direction as you iterate backwards and that this is like two or something. So you have a point here, it's expanding in one iterate, and then this ball contains another ball of radius r, and then it expands again, and then it does again. So maybe it's not an open set of points that have this property, but at least you will get an intersection of closed sets which have this property. So for those who know, this is a property which is very related to what they are called blenders or some foreign legal policy. In any case, what we need is some property that allows us to do this argument we did in the morning. We had a neighborhood here. We started iterating forward this neighborhood, and then it became close to the unstable direction. We iterated backwards this set, and then it became close to the stable direction. But now if we are in higher dimensions, these two sets might not intersect. You have a machine during the whole talk. You can imagine ES has dimension one and EU has dimension one. And therefore, you have this open set is becoming close to a line. This open set, as you iterate backwards, is close to a line so they can miss. You might have this figure. So we need a certain property of expansion in the center direction so that we can make this set grow as we iterate backwards. That's the reason we need this kind of property. Unfortunately, it's not so easy to finish only with this property. But let me explain a little bit more what type of argument we are going to do. Yes, this sigma is larger than 1. So I'm iterating backwards, and this is N. Sorry. OK, of course, you can say the same for the unstable manifold and whatever. So in the morning, we did an argument to show that Manier's example was robustly transitive, which essentially can be summarized in this proposition, which is due to Puschal's Sambarino and says the following. Suppose that the unstable foliation is dynamically minimal, and that the stable foliation has the SH property. Then F is robustly transitive. So how was this argument? Well, I haven't defined what the dynamically minimal foliation is, but a minimal foliation is a foliation for which each leaf is dense. A dynamically minimal foliation is one for which the orbit of each leaf is dense. So the idea of the proof, so let me recall the idea, was that we have something like this. We have the open set U, the open set B. We started iterating U in the future. So this is for F is robustly transitive. So you took C1 close to F. And so the minimality of the unstable foliation was not preserved by perturbations, but at least we could manage to have epsilon minimality. So the unstable foliation of C by continuous variation is epsilon minimality. And so as we iterate this set U, it becomes at least epsilon dense in the manifold and very close to the unstable direction. And so now all we want to do is to try to iterate backwards the set B and try to obtain a disk transverse to the unstable direction, which has radius at least epsilon. So we want to iterate backwards so that it contains a center stable disk of radius larger than epsilon. If we get that, we are done because it will intersect the set U. So we will have the existence of an iterate such that fn of U intersects B, which is equivalent to transitivity. And to do this, what we use is exactly this property. Because inside B, we do have a stable disk. And as we iterate backwards, we know that at least for one point here, we have expansion of the derivative. And this expansion has a uniform behavior. It holds for every sheet which is close. And so we get that the stable direction is getting larger and larger. And around this point, at least, we have this disk. It's a center stable disk. Now we have the stable direction. So as we iterate backwards, we get that the stable direction now is very, very large. But at least at one point here, x, we know that the image of this center is more or less of size, say, 2 epsilon. And now the pre-image of the disk we had here will be something that may be very small here, but it contains something which is big, at least somewhere. And this is enough to intersect the forward iterates of u. So this was the argument we did this morning. And in fact, this is the argument we use to obtain this result. But for the purposes of this lecture, let me explain a small generalization of this condition that also allows you to get robust transitivity, which allows us to make a symmetric argument. In fact, we need to do two arguments. One to show that one of the two fallizations is minimal, and then to show that the other one has this property sh. So instead of doing this, I will show that every fallation has property sh. And I will forget about this. And this is just an extension of this, says the following, so if, and this is related to a work of alien error and a tertiary, it says that if f is transitive and both the strong stable direction and the strong and stable fallation have the sh property, then f is robustly transitive. So it's a similar criteria. It's a little bit more complicated to prove, because it's in the same spirit as what we did this morning with anosovetheomorphisms. For this morning, for anosovetheomorphisms, we said if we are transitive, we have an epsilon dense orbit. Then if you are anosov, you have uniform size of stable and unstable manifolds. So you iterate this forward, you get something, let me do another drawing, again the same drawing, u, v. So this morning, we used anosov property, we iterated this forward, we get some uniform size, we iterate these backwards, we get some uniform size in a transverse direction, and then we used an epsilon dense orbit to take this set here, so that they intersect. But if I wanted to do this here, I would be cheating, because I know sometime it will get bigger, but then it could become smaller later. And so it's a little bit more delicate, but it's possible to prove this, the spirit is the same. So that's why I'm asking for the SH property for both variations, so I gain dimension in both sides, and it's a little bit strong. That's all I wanted to comment. And now let me try to explain why this property, or this hypothesis, are abandoned in this context. So now is the moment where I will try to obtain a certain geometric property of the fallations that will guarantee that we have this condition. So unfortunately, this argument uses some things which are not so related to the things we are doing in the course. So I will only explain the things which are related to what we did, and only comment on the rest. So explain the geometric property we use. You always forget. Let me make some definitions. So lambda inside them, this was defined by Sylvain yesterday, is a strong and stable. All the definitions I will make are symmetric. So now I will stick to unstable lamination, but everything is the same. Lambda in M is a strong and stable lamination if it is compact, F invariant, and saturated by WUU leaves. So we are going to be interested in minimal laminations like this. So as you have inside the set of partially hyperbolic diffeomorphisms, so the unstable fallation is over there, and it accumulates at certain parts. So we are taking the minimal sets of accumulation. And we will say that lambda inside them, if it is a minimal strong and stable lamination, if it is non-empty and every street strong and stable lamination inside lambda is empty. You have a compact saturated set of saturated by unstable leaves, and if it has a compact subset which has the same property, I will take this one, and then I will make a minimal procedure. Every strong and stable lamination will contain at least one minimal strong and stable lamination. Notice that for strong and stable lamination, I'm also asking invariance. So this is kind of related with the dynamically minimal. It's not minimal as lamination, but minimal dynamically. So the remarks, the remark is that every strong and stable lamination contains a minimal one. And now I will express a geometric property of unstable lamination. We say that a strong and stable lamination is transverse if there exists a number r sufficiently big such that for every unstable disc inside the lamination of radius bigger than r. OK, so I said that you could imagine that the unstable direction was one-dimensional. This means you take an arc which is large enough. If it's higher dimensional, you have to take a disc, but it's the same. You have, in each of these discs, you have two points which have this behavior. So let me draw it first. And then I write. So you take a very large disc. Since the manifold is compact, this large disc will pass nearby. And what I will ask is that it has this image. So you will have two points, x and y, which are connected by a strong stable direction. And here is the place where I use the fact that the center dimension is one to speak about transversality. What I will do is once these two points are connected by a same stable manifold, I will project one leaf into the other one. So I have here a center unstable disc. And when I do the projection, I will see something like this. I draw it like this because this projection is only continuous, but what I want is that the projection is topologically transverse. So for every unstable disc, there exist points, x and y, in D, U, U, which belong to the same local stable manifold. And such that the projection along stable manifolds of the local unstable manifold of y is transverse to the local unstable manifold of x, meaning that it will intersect both connected components here. So why do I say this is a geometric property? Because in principle, this foleyation has a certain co-dimension, and it could be that this projection would fit exactly here. This is the case. For example, in the linear and also the thermomorphism, when you take the linear and also the thermomorphism, essentially the strong and stable manifold and the strong stable manifold form an eigenspace for the torus. And so when you saturate, you never see this geometric property. But now that I draw this, I hope you believe that this is more or less an abundant property. So the fact that this will not project tangentially is more or less a good property, abundant property. That said, let me make the disclaimer that it's not so obvious that this will be the case. Because for example, when you look at the unstable and the center, when you project, they really fit together. Because the center stable bundle usually is more integrable than the stable and stable. OK. So let me now explain the key steps in the proof. The first result we prove is a perturbation result, essentially saying that this property will hold with quite some abundance. So let me write this theorem. There exists a xi delta dense subset of partially hyperbolic diffeomorphisms with one-dimensional center, such that for every f in xi, xi delta dense subset, and lambda a strong, a minimal, or no, let's put minimal, strong and stable lamination, we have that lambda, there are two possibilities. Either lambda is transverse, which is what we want, or lambda is a quasi-attractor, which we defined the other day, implying the existence of a trapping region. So let me just, I won't explain the proof of this theorem, but let me just say a few words on the ingredients. So the proof has essentially two components. The first component is a perturbation result, which can be seen as a localized version of Dolgopiat-Wilkinson result I mentioned this morning. OK, so if you look this carefully, this is quite related to the property of accessibility we were talking about today. So if you go with an unstable direction, then you move along the stable. You move along the unstable, and you go back along the stable, then you come back in another place. And this is related to being able to go everywhere by making shams along stable and stable direction. The thing is, to prove this accessibility, it's enough to make some places where you can move up. And then you can just go from one place to the other using some density or whatever. Here, we need this result to hold inside a minimal set. So we really need to control the continuation of this set. We have to make the perturbation, but as we make the perturbation, the set might move, and we still want this property to hold. But unfortunately, we are not able to obtain this property. All we get is a certain non-integrability result. So to get a non-joint integrability condition on lambda. But then what we're able to do is to show that if we don't have this transversality, then we can show that the set has to be an attractor. So we study carefully this to show the dichotomy. So I don't say anymore. So this is the main result, and this is the perturbation result that allows us to obtain this transversality. And then the second result we are able to show is the following, which is a strong and stable lamination. Then there are three possibilities. One, either lambda is the whole manifold, so the unstable foliation is dynamically minimal, which is more or less good. Or lambda satisfies the SH property, which is the thing we wanted. Or essentially the idea is that if you're, so very, very briefly, the idea is that if your lamination does not have many points which expand in the center direction, then you should attract. And you get that lambda is an attractor, a quasi-attractor. Have a trap instead. So now I wonder, why did I state this theorem? So in fact, what we need is this. But this was just so to really prove this theorem, we need to use the transversality, but I could have forgot about that. I don't have time to explain this. So let me just finish the argument using this result. Still with another perturbation, we get this following consequence. Let F be a partially hyperbolic with center dimension one, transitive, and in a G, a G delta dense subset, then every minimal WU saturated set has property SH. And now I remember why I explained the transversality. So my point, this follows by a further perturbation. So this one is not a possibility because I'm assuming it's transitive, so it doesn't have an attractor. This one would be good. It gives you SH property. And finally, you have this other condition. But then there is a further perturbation that allows you to create what are called blenders and generate this SH property for the minimal set. And so the point is that now what do we have? We have F is transitive. We have that every WU minimal set verifies SH property. And every applied, this applied to F minus one, you get that every WSS minimal saturated set has SH property. And we want to do the same argument we did before. Only that now, we only know that this SH property is verified in minimal sets. And that is why we will use the transversality. So let me make a drawing. The same drawing we did today, we want to show that a perturbation of F is still transitive. So we have the set U, we have the set B, and we want to iterate this set forward to see it grows both in the stable direction and in the center direction. And we want to do the same when we iterate this backwards. So what happens when we iterate U forward? This becomes closer and closer to the unstable foliation, OK? However, we don't know whether the unstable foliation has points which are expanded in the center. What we do know is that every minimal set has this property. But we know that minimal sets have this structure here. So as you iterate the unstable manifold, it will become closer and closer to the leaves of the minimal sets of the foliation. And this transversality forces the intersection with the stable manifold of some point of the minimal set. But then, since in this foliation you have points which are expanded in the center, the same will happen for some points that are here. So that's a brief reason why after some iterates which depend on the open set, you will get some big open set in the center stable direction. And then you will do the same for the path for v. And you will get the intersection with a similar argument. So I'm sorry it was a little bit messy. I will finish with the idea of the proof now. And to end, I would like to pose some questions with respect to this robust transitivity which we still don't understand. Even in the center dimension equals to 1. So the first problem, which has already quite a long time, is that there exists a robustly transitive, partially hyperbolic diffio with one dimensional center such that the derivative of f along the center is an isometry or is bounded. So as we said, the mechanism, the geometric mechanism we used to obtain robust transitivity was to, by some perturbation, create some expansion and some contraction along the center direction. And that's why we managed to iterate things and obtain big disks in every direction so that they will intersect. However, when you have a partially hyperbolic diffio where the center direction is an isometry, we don't know whether there are or there are in the examples of robustly transitive systems. So I should mention that there is an indication that this might not be the case by some work by Ishii, which is also based on some previous work by Bonati and Gellman. And finally, another problem, which I like a lot, is whether there exists some isotopic class of diffeomorphisms of three manifold, partially hyperbolic diffeomorphisms with one dimensional center such that isotopic class of diffeomorphisms such that every partially hyperbolic diffeomorphism in this isotopic class is transitive. So this is another thing we don't know the answer. And a natural candidate for this would be the isotopic class of anosov in T3. But for the moment, we have no idea. And so thank you very much.