 So, thank you, and so today we are going to speak about one aspect of partial hyperbolecity, which is robust transitivity, okay? So on the first day, Sylvain explained a certain dichotomy that was useful to decompose the dynamics. And so there was this dichotomy which said that if f from m to m is a diffeomorphism, in fact, this dichotomy doesn't require the map to be smooth, it's enough to do it for homeomorphisms, but since this is the context we are working in, so there are two possibilities. Either there exists a trapping set which was an open set such that the image of its closure was contained in itself, the dynamics is chain recurrent. This meant that for every pair of points x and y in m, there exists, and for every epsilon bigger than 0, there exists an epsilon pseudo-orbit from x to y, okay? Remember a pseudo-orbit was you had x here, you looked at f of x and you were allowed to make a small jump, you got x1, and you continued, okay? So something like this. So you could go from wherever you wanted to wherever you wanted via small jumps on the dynamics. In the first set of exercises, the very first exercise was to show this dichotomy, okay? So essentially what you showed is that the set of points that could be reached from a given point x by epsilon pseudo-orbit was a trapping set, and so if either you had a strict trapping set, so let me say you an open set which is non-empty and it's not the whole manifold, and so either you had such a trapping set or you could reach everywhere by epsilon pseudo-orbit for every epsilon, okay? And so this is a quite nice description or dichotomy regarding the dynamics of points for a homeomorphism, but it has certain downsides, okay? So let me write some downside, and to explain the downside this definition has, let me give one example. So let's consider a dynamics of the circle which makes the following. So I don't know if you can understand what this means, I'm meaning it's a homeomorphism, a diffeomorphism of the circle which has one fixed point, and every other point is moving in this direction, in a counterclockwise direction, okay? So these dynamics do not have any strict trapping set, okay? So if you took an open set here, so the closure won't get mapped inside itself, and whenever you take some point here, it's moving in this way, so you won't get any trapping set. So this dichotomy means that this dynamics is chain recurrent, but you can also check it directly if you have any pair of points here, x and y, to get from one to the other, you just iterate until you are very close here, you have the right to jump epsilon, and so you can go to the other side and arrive here. So this example is just to show that this is a very weak recurrence property, because as you see here, if you take a point here, it's never coming back if you iterate it really. So you don't have a nice recurrence property. So let me try to explain a stronger recurrence property which is more interesting to us, so definition, which we already talked about this, but let's say f is transitive if it has a dense forward, meaning there exists a point x in m such that the set of future iterates effects is dense in m. This is much stronger than this property and as you can see here, here this dynamics is clearly not transitive, whatever point you choose, you make the orbit and the closure is its orbit and this point. So the question is whether we can improve this dichotomy at least for most dynamical systems. So let me state one important result in this direction, which is a theorem by Christian Bonati and Sylvain, which states the following dichotomy. So there exists a disjoint open set of the space of the morphisms of a manifold, open sets, let's call them OT, sorry, O1 and O2, such that the union is dense and you have the following property. If f belongs to O1, then f has a trapping region and then if f belongs inside O2, which is a g delta dense subset, then f is transitive. So what we are saying is that we can improve this result at least in an open and then dense, in a dense set where we can promote the recurrence from chain recurrence to transitivity. So the goal of today's lecture is to try to explain a joint result with, it's a mixture of joint results, Sylvain, Flavio Abdenour, Sylvain and Martin Sambarino. Where the idea is to try to improve in the partially hyperbolic setting to improve this g delta dense set to an open and dense set. So to get what we call robust transitivity. So let me state the theorem which I won't prove in the morning, I will prove it in the afternoon, but just to know where is that we are heading. Ph1c equal to 1 be the set globally, partially hyperbolic with one dimensional center. So the theorem is the following. Essentially the same statement here only that instead of working in the whole space of diffuse we are working in the set of partially hyperbolic diffuse, but we can remove this g delta dense union such that if f belongs to O1, then there exists a trapping set and if f belongs to O2, then f is transit. So now in the rest of this lecture I will try to explain some motivations for this result and some ideas on the difficulties to show it. So in the afternoon I will try to show how to solve these difficulties. So first let me explain an easy case. The easy case is the case of anosov diffeomorphisms. This meant that f from M to M admits as the composition of the form ES plus EU. It does not have a center direction. And here I am saying the easy case because we are looking for this weak type of result. So it is an notorious open problem to know whether every anosov diffeomorphism is transitive. However, here we are allowing ourselves to make a distinction. So either there is a trapping set or the anosov diffeomorphism is essentially robustly transitive. So essentially let me state this as an easy proposition. f anosov and transitive then there exists a neighborhood such that for every shi in EU shi is transitive. Of course and here there is a one line proof which is not the one I am interested in is Sylvain told the first day that hyperbolic diffeomorphisms are structurally stable. That means when you make a perturbation they are topologically conjugate to themselves. So if it is transitive you make a perturbation it is still transitive. But I will try to explain another proof which does not use these shadowing properties just to get a taste of how do we need to use the stable and unstable directions and why is it a problem to have a center direction. So we have here m and since f is transitive let me first say the following. Since f is anosov and this follows from what Sylvain did yesterday it has strong stable and strong unstable manifolds. And this strong stable and strong unstable manifolds but I continuously as you perturb the diffeomorphism. So there is a neighborhood such that for every shi in EU zero one has uniform sized stable and unstable manifolds. This is just the stable manifold theorem that Sylvain explained yesterday. Now what do we want to do? We want to apply the hypothesis that f is transitive to get one point that at least visitors and so this have uniform size let's call this size delta. So since f is transitive there is a dense forward orbit. So you can get a finite orbit which visitors every neighborhood of size epsilon with epsilon much smaller than delta. So fix epsilon much smaller than delta and there exists a finite orbit equal one to k which is epsilon dense. But now being transitive needs not be a robust property. You can make a small perturbation of a transitive diffeomorphism an example is irrational rotation of the circle so that it's not transitive. However if you fix a finite segment of orbit and you fix a size of perturbation you can keep this property for perturbations. So this is a robust and now how do we conclude? So I should have stated this criteria before. Exercise and this will be in the exercise session. So f is transitive if and only if for every u and b open sets there exists an iterate such that fn of u intersects b. This will be discussed in the tutorial session but it's a quite simple equivalence with having a dense orbit and the point is that now we want to show for a diffeomorphism which is in this neighborhood but also has this property that it's still transitive. So how do we do this? We have the manifold here and we pick any pair of open sets u and b which may be very, very small but as we iterate forward the set it will eventually contain an unstable manifold of size delta because you have an unstable manifold you start iterating and this unstable manifold gets bigger until it reaches this uniform size delta. So I iterate fn1 and I have something which is like this and this is of size delta. If I iterate backwards this open set it contains a stable manifold and stable manifolds get bigger in the past and so eventually it contains a stable manifold of size delta f minus n2 get something like this and I'm drawing this like this because stable and unstable manifolds are transverse maybe in different places you cannot compare but the idea is that one is getting bigger in the unstable direction the other one is getting bigger in the stable direction and now it's the important point of being stable and unstable complete the whole dimension. So when I apply this segment of orbit x that passes from here and goes near here I can transport this manifold to here and get an intersection. So this is fn3 and so finally what you get is that fn1 plus n2 plus n3 of u intersects b. So this is the easy proof of this proposition and now I hope it's clear that this argument will not be available in the partially hyperbolic setting because when I do this I get something which is large in the unstable direction something which is large in the stable direction but when I transport this to here I don't get any intersection. So now let me explain the heuristic idea or previous results that motivate what we will do. So the idea of the proof I will try to summarize it in two key points. The first point is to obtain a certain robust geometric property of the invariant manifolds by perturbation. In a certain sense what we have here we have this invariant fallation, the unstable, the strong unstable. Typically they can be very bad for our purposes but we will make some perturbation putting us in a certain dense set where we have a certain robust property of this invariant manifolds. Which is this geometric property I will explain later. First let me explain the idea behind this and then the second key point is to explain how this property allows us to do perform essentially the same argument. So try to perform a similar argument. So before I enter into the proof let me explain a quite related result which has the same spirit. So what we are trying to do here is to promote a recurrence property into a better one. So let me explain. So we try to improve chain recurrence to transitivity. So let me first start with a stronger recurrence property. So let's call the non-wondering set of a diffeomorphism is a set of points such that for every epsilon bigger than 0 there exists an iterate such that fn of bx epsilon intersects bx epsilon. And so a recurrence property which is stronger than chain recurrence is the following. Is that the non-wondering set is a whole manifold. Which if you know something of ergodic theory when your diffeomorphism preserves a volume form this condition is always satisfied. So this implies chain recurrence. This is an exercise. And so we can wonder how to promote this non-wondering condition into transitivity. So why is this worse than being transitive? So consider identity on M verifies this. This is a recurrence property but it's a little bit weak. So the identity has this property. It's not very transitive. And so let me explain a geometric property of invariant manifolds which allows to promote this condition to transitivity. This condition the geometric condition is called accessibility. So now I won't assume that the center dimension is 1. Now the center dimension can be whatever you want. And so take a partially hyperbolic diffeomorphism, a globally partially hyperbolic diffeomorphism. We say it is accessible if for every two points x and y in M you can find a sequence such that x, y, x, y plus 1 belong to the same stable or unstable manifold. So let me make a drawing. This is a very classical drawing. These are stable manifolds. These are unstable manifolds. And this is y, this is x. The idea is that you can show any two points by going from stable manifold and stable manifold and stable manifold. Okay, so if you are in the ANOSOV case this is a trivial property because going from unstable and stable is something that fills up an open set. But here the unstable and the stable dimensions sum up to be less than the total dimension. So this is a non-trivial property. Okay, you need to go from stable and stable and raise the dimension of the points you can reach. But in a certain sense this is a geometric property. Okay, so let me explain a beautiful argument due to green. Okay, so if we have a partially hyperbolic diffeomorphism which has this property and then a wandering set is the whole manifold, then we can promote this recurrence property to transitivity. So I will draw the proof. And so the idea was that you have to start with an open set u and an open set b. And by the exercise we said to prove transitivity we need to show that a future iterate of this one intersects this one. So let's make this drawing. We pick a point here, we pick a point here and we can start making the drawing. So we have an unstable manifold here, a stable manifold here, an unstable manifold here, finally a stable manifold here that gets you inside the open set. Okay, so now what we are going to do, we have the point x here, the point y here. And we wish to show that this open set u starts mimicking as you iterate forward this unstable manifold. Then as you iterate backwards you want to like iterate backwards when you have reached here so that you can follow this direction and then continue doing this until you reach this place. Okay, what's the difficulty is that if this were a fixed point this would be very easy because you take an arc of unstable manifold, you iterate forwards it gets bigger but the point is still here. But if this point is not fixed you take this unstable manifold and it goes here. So you don't have any reason to expect that this open set will accumulate on this point. So that's the reason we ask for this property. Okay, so what will happen? We have a sequence of points which have arbitrary large returns to this ball. And so as you iterate forward you get that the open set here becomes larger and larger in this direction. And now you could say well but now you cannot do this starting from this point. But you don't care because you take any open set here and as you mimic this you can push the manifold over here. So the idea is you continue, you iterate backwards and it goes here and then you iterate forwards and you get that U is over here and eventually you enter the side network. Okay, so the more detailed proof is in the notes. But it was just to explain how a geometric property of the fallations can help you to promote the recurrence properties you were looking for. And so here what I explained, this green argument plays a role of this part of the strategy. Okay, so trying to perform an argument like in the Anosov case using this geometric property. So let me explain about this. Okay, so it's not something I will do in neither of the two cases. But I would like to state a theorem by Dolgopiat and Wilkinson that shows that there exists an open and dense AC1 open dense subset of partially hyperbolic dynamics in M. And this open and dense is also open and dense when you restrict it to volume preserving ones. I write it down here. Consisting accessible different morphisms. Okay, so putting these two results together and using the fact that if you preserve the volume, you have this property, you get as a consequence, as a nice consequence, is that among volume preserving partially hyperbolic diffuse, transitive ones are open and dense in the C1 topology. Okay, I won't discuss about the regularity of these perturbations, but just let me say that BNC1 open is great, but BNC1 dense is not so great. Okay, it's easier to be C1 dense than to be CR dense, but BNC1 open is great. And there are some results in the direction of improving the topology here for the density. I just mentioned the one by Herz, Herz and Urez, but you can look for more references in the notes. In the case of center dimension one. So now in the time I have left, I will try to talk about robust transitivity. Okay, so what does it mean to be robustly transitive? So the theomorphism F in this one of M is robustly transitive if there exists a neighborhood, I see one neighborhood of F consisting of transitive diffuse. Okay, and Sylvan explained in the, so what is the relationship between robust transitivity and partial hyperbolicity? So Sylvan explained in the first lecture, a result by Bonati, Diaz and Puschels, which says that robust transitivity implies a weak form of partial hyperbolicity, volume hyperbolicity, which as the dimension goes down looks more and more like partial hyperbolicity. Okay, and this is continuation of a work by Manier and Liao, many others, but let me mention this one show they characterize robust transitivity in dimension two. So in surfaces, robust transitivity is equivalent to being an also just to state before I continue an important problem. I think so we do know that robust transitivity implies some form of partial hyperbolicity, but we don't really understand robust transitivity as we do in surfaces. So let me pose a question which we don't know the answer is, does S3 admit robustly transitive? This is just a discretion, so let me come back to explaining robust transitivity. So we've seen that transitive and also difumorphisms are always robustly transitive. So the question is do we have more examples or more importantly do we know examples which are not hyperbolic? Okay, and there has been a long list of examples, but I would like to come back to the example we explained the other day by Manier. Okay, because here we will get a taste of the techniques that we will do in the afternoon. So I will explain very quickly how we did this example. So we started with FA from T3 to T3 linear and also we say eigenvalues say lambda 1 smaller than lambda 2 smaller than 1 smaller than lambda 3. Okay, so let me draw here. This is the space of difumorphisms of T3. We start with FA which is a linear and also a let me draw the space of an also difumorphism. This is a very big simplification and what we did we started to change the dynamics near a periodic point so that we could leave the space of an also difumorphism but more or less doing all the perturbation in a small region and trying to keep what happened outside of this region. So we did something like this and here over here we had Manier's example. Okay, and the drawing we did was something like this. We started in the strong stable direction, in the stable direction, sorry, we had this weak eigenvalue associated to, eigenline associated to lambda 2 and this strong eigenvalue associated to lambda 1 and we started to move and in the end we had something like this. So since the perturbation was made in a small region here, the dynamic outside does not change. So from here you still see something coming in. We didn't touch the fixed point so we keep this fixed point but we had made it repulsive in the center direction. So we had something like this. Since things are coming in from here and this is repulsive, this forces the appearance of two new periodic points, fixed points, which will be attracting. Okay, and let me draw the stable manifolds, something like this. And so because I'm kind of lazy, I will not show that this one is robustly transitive, I will show that this one here, the one when we leave, and as of the thermo-theism, is robustly transitive because once we show that this one is robustly transitive, we will have a neighborhood of these ones which are all transitive and this implies the existence of non-hyperbolic, robustly transitive thermo-theism. Okay, I have ten minutes. I think it's enough. And so let me credit this argument to Pujals and Sambarino who gave a general criterion that we will try to reproduce in the afternoon to show robust transitivity. So why to stop here is because when we are here, what do we have? We have exactly the same picture, okay? Only that now the eigenvalue here, it became equal to one. Okay? But the dynamics is the same. We still have a attracting periodic point here. Okay? And so we can claim that this one is transitive and moreover that the unstable direction, the unstable fallation is minimal. Okay? So let me say some things, some words about this thermo-theism here which I call f of T0. T0 verifies the following. The first thing that it verifies is that it is partially hyperbolic, which is the context we are trying to work in. So it is partially hyperbolic. We center dimension one. So it preserves a splitting of this form. And now when we were here, it was an osso, so this center direction was uniformly contracted, but now in the boundary, it's no longer uniformly contracted. But the non-contraction, you can see it only at the fixed point where you have derivative equal to one. But now this became a genuine center direction. And then since we are in this boundary here, I claim that this implies that the unstable fallation is minimal. What does this mean? It means that every leaf of the fallation is dense. Okay? So this is not obvious, but it is an exercise. So you need to use that whatever neighborhood you want in the manifold, if you iterate forward enough, you will get there. And this follows from the fact that every point has a big stable manifold, okay? So exercise. And here is the really key property that we will use is that for every Y in the strong stable fallation of length equal to one, okay? Equal to one, what does it mean? So this neighborhood has size epsilon, very small. So equal to one means much larger than the size of this neighborhood. So if you take any strong stable leaf, it might intersect this neighborhood where you made a perturbation, but it also intersects some place outside this neighborhood. And this means that it will contract in the center direction, okay? So for every arc of length one, there exists a point X in Y such that the derivative of F minus one is zero, along the center direction is larger than or equal to lambda two, okay? So which is to the minus one, which is bigger than one, okay? And I should have written here the minimal norm of the derivative, but since it's dimension one, we don't care. So what is the argument now to get robust transitivity? What happens when we perturb the diffeomorphism F t zero? So when we perturb the diffeomorphism F t zero, this property is robust, okay? Because we have shown that being partially hyperbolic is an open property among diffeomorphisms. And the key thing is that this property is also an open property. Why is it so? Because as Silvan showed yesterday, as you perturb the diffeomorphism, the stable manifolds remain close in compact sets. The center direction remains close to the previous center direction and the derivative is close because you are C1 close. And so this property will still hold. Maybe you have to change one by two and maybe you have to change lambda one is for something a little bit smaller, but this will still hold for perturbation. So this is robust. So now the problem is that this is not robust, okay? But as in the case of transitivity, we cannot guarantee that you can keep transitivity, but you can keep epsilon density of the leaves. Again, by continuous variation of the stable foliage and the unstable foliage, we get that for every epsilon bigger than zero, there exists a neighborhood U epsilon, such that for every she in U epsilon, every strong stable leaf is epsilon dense, okay? This is just the continuous variation of the strong manifolds. But now how do we conclude? We repeat the argument for the Anosov case, okay? We have the open set here, U, the open set here, B. And on the one hand, if we iterate forward the open set B, U, it contains an unstable arc, so as it iterates forward, it becomes epsilon dense, because every unstable leaf, which is sufficiently large, it's epsilon dense. So it has an iterate, it's very large. I don't continue drawing it, but it's kind of epsilon dense. But now this B, I draw it very big, but it's smaller than epsilon. So how do we know that if we iterate it backwards, it will intersect here? So to do this, we will try to use this information we have in the center that gives you an expansion in backward iterates, okay? So as we iterate backwards, this open set B only grows in the stable direction. But the stable and unstable direction, as we said, don't complete the dimension, so we could miss the open sets. But since we have a lot of points where the center direction is expanding, it's contracting for the future or expanding for the past, as you iterate backwards, you gain size in the center direction, okay? So using property three, eventually we get a center stable disc of size epsilon, which will intersect this manifold. Okay, so in the afternoon, I will try to generalize a little bit this argument to try to show that something like this will hold for an open and dense set of partially hyperbolic digital models. Okay, thank you.