 OK, thank you. So now I will continue with this series of examples of partially hyperbolic systems. But as I announced this morning, now we will switch to give examples of partially hyperbolic systems in a weaker setting that I recall again. So we said that F was partially hyperbolic if there existed a filtration. A filtration was something like this. And all of these open sets were attractive neighborhoods, meaning that F of UI closure is contained in UI such that the maximal invariant sets in this neighborhood is partially hyperbolic, splitting this meant that d lambda i splits as a stable plus a center plus an unstable direction. OK, so you might say, and these filtrations were something like this. You have this open set, first is the whole manifold, then you have an open set over here, an open set over here. Each one is mapping inside itself. And what you're looking at is this lambda i corresponds to the set of point that remain here in this region over here. And now what we are asking is that the dynamics might not be globally partially hyperbolic, but inside each of these sets, we have a partially hyperbolic splitting. And essentially, this, in principle, provides a larger class of examples. So let me tell you that, in fact, it provides a much larger class of examples. And by this, let me just recall a very old theorem by Schubert's Mail that says that hyperbolic diffeomorphisms are dense in the C0 topology among diffeomorphisms. So why is this result interesting for us? Because this morning, we had to work a lot in order to obtain examples of partially hyperbolic systems. And each one was kind of difficult to get. So we had these algebraic or geometric examples, which were the starting points. And then we had to start with them to construct new ones. And we have no other way of constructing examples. And in principle, as in the example I mentioned this morning, and Silvan also said that this theorem by Franks and Newhouse says that being hyperbolic in the whole manifold gives a lot of constraints on the topology of the manifold. However, here, this theorem is telling you that whatever the manifold is, and whatever the isotopic class of diffeomorphism you're looking in, you can approach it by a hyperbolic diffeomorphism. And so this local property, the fact that we are asking hyperbolicity only in the recurrent part, gives a lot of liberty to create examples. So now we will look more to localize kind of examples. So the first type of examples I will do is the ones I promised in the morning, which are skew products. Let me say what a skew product is. So you have F from M to M, a diffeomorphism. And you have K inside M, a partially hyperbolic set with splitting TKM. OK, so now we are not asking the whole manifold to be hyperbolic or partially hyperbolic. We just need a compact invariant set, which is partially hyperbolic. So for example, you can choose the horseshoe that Silvan explained the other day, a hyperbolic attractor, or even a periodic orbit or whatever. And so now we will choose she to be a function from M to the space of diffeomorphisms of another manifold, N. But we will require that the contraction and expansion that this diffeomorphism have in the direction of N is smaller, is dominated essentially by the expansion here and the contraction here. So how do we write that? We write it like this, that for every x in K, if you are outside K, we don't matter. It's OK. So we have that the ES is smaller than the minimal contraction of she here. So she is a diffeomorphism of N, so we don't have to restrict to anywhere. This is always smaller or equal than the she, x, and the norm here, and we require this one to be smaller than the lowest expansion here. We wrote like this. So now, what are we going to look at? We're looking to look at this diffeomorphism times N such that f of x, y equals f of x, she, x of y. For example, an example we did in the morning, so you just take she to be constant equal to the identity, and so you will always have this condition. And the point is that you have the following proposition that tells you that the set K times N is partially hyperbolic FBF with splitting PK times N, N times N. And we can say a little bit more. We can say exactly who this bundle is, and we can say more or less who these bundles are. So what we know is such that the bundle EC hat is exactly the bundle EC that we have here times the tangent bundle to N. And these bundles, we don't know exactly where they are. They need not be the same bundles that we had for F, but at least we know that they will project into these bundles. And if pi from M to N to M is x, y maps to x, then the derivative of pi sends each of these bundles into the correct one. So when you multiply here by the identity, it's an exercise to show that the new bundles are exactly the stable or unstable bundle. You have times 0, so there's no contribution into the new part of the tangent bundle. However, when you make something more complicated, which is mixing, then it's possible that the thermomorphism changes the angle of this. But by a simple cone criteria, you can show that this is partially hyperbolic. So the proof of this is really easy, and we are leaving it as an exercise. So before I continue to explain why I wanted to do skew products over general sets and the motivation for this, let me first give an example of a skew product, which is not so nice in the sense that the manifold is not a product. So it's a very beautiful example, due to Bonati and Wilkinson. So the example is older than this, but the way to see it appears in a paper by Bonati and Wilkinson. So let me explain this, because it shows how you can construct non-trivial skew products. So the idea is you start with this manifold, T2 times the circle. And this is a trivial product manifold. But now you want to construct a circle bundle over the torus, which is not a trivial one. So how do you do this? You cut. So it's related to the surgery thing we talked in the morning. So cut T3 along a circle. We have this T3 over here. So this is T2, and here we have S1. And we cut here along, we make like a tube here, and we remove this tube we have here. And we glue it again by pushing fibers when you do a circle. So what we are doing is remove a solid torus and glue again, preserving the fibers. So you have now T2 minus a disk times the circle, and you remove D times the circle. Minus, sorry. You have these two manifolds now. The boundary of each of these manifolds is a torus. This one, the boundary is equal to a torus. And this one, the boundary is a torus. And each one of these torus in this boundary is a circle. So you have a foliation of this circle. Let me, this represents the boundary of D, and this represents the circle. So I cut these torus away, and then I glue back. But now, instead of using the identity to glue back, what I do is I start rotating in each of the fibers. So this fiber, I glue it correctly. But then this one, I glue it a little bit pushed away. And when I make one turn, I did one turn of the rotations. So if you like a formula, you glue, like if X, let me use another S, parametrizes this, and T parametrizes the circle. So you glue with S, T, moves to S, T plus S, or M times S. When you glue this, you can show. So exercise, the new manifold is not T3. In fact, it's what's called a nil manifold. And now, what we are going to do is to construct a partially hyperbolic diffeomorphism here, which is, OK, but this is still a circle bundle over the torus. You have a projection, you have not changed the basis. So you can project to the basis, and you get a torus. And so now, you wish to construct a partially hyperbolic diffeomorphism, which projects to an anose of diffeomorphism. So let me explain how to do that. If you don't believe in this construction, so let me say what do we have? We have n, which is a manifold, which is a circle bundle over the torus, meaning that the preimage of each point here is a circle. And this manifold is not a torus. And what we want is to define a function f from n to n, which is partially hyperbolic, and such that if you project, so if you do f composed with p equals 2, 1, 1, 1 composed with p. We want to make this example. And so to have it partially hyperbolic, a nice way to do this is to preserve the circle fibers. So p goes from the manifold to the torus. And I'm assuming this is acting on the torus. So essentially, what this means is that this f will preserve the fibers. And it will move them as an anoso. And the easy way to guarantee that this will be partially hyperbolic is to know that the action on the fibers is an isometry. If we manage to do that, it's easy to see that it's partially hyperbolic. It's essentially the same idea that this construction. These are also called SQ products. So let me explain the trick. So choose U and B in T2, such that you have the following design. So this will be U, for example. And B will be something like this, the outside part. So from the way we constructed the example, we can choose trivializations of the neighborhood. This is a circle bundle. So you can trivialize. So let's choose trivialization phi 1 from U times S1 into N. And phi 2 from B times S1 into N. So these are diffeomorphism, preserving the fibers, or something like this. So that if you look at phi 1 composed, phi 1 minus 1, composed with phi 2, which goes from U intersected with B times the circle to itself and fixes the fibers, then these are all rotations. It's exactly the way we constructed the manifold. So you are cutting, making a turn. Here you are just making translations. And then you are gluing back like that. So you can choose these trivializations. OK, if you don't believe me, that is the argument. Otherwise, you can believe that this is possible. And now what we are going to do is to define one diffeomorphism, which is supported here. One which is supported here so that the composition will be nice. So exercise show that 2, 1, 1, 1 can be written as C1 composed with C2, such that C1 is the identity where C1 restricted to B is the identity, and C2 restricted to U is the identity. How do you do this? So that's why I draw U very big and B very small. So you just choose a diffeomorphism, which is equal to a nozzle here and glues to the identity outside. And then you compose, you take the inverse and you compose with this. So you have the identity in each one. And so once you have this, you can define C1 from N to N, such that C1 equals identity on B, and equals, how do I write it? Phi 1 composed with C1 times identity, composed with phi 1 to the minus 1. And C2 equals the identity on U, and equals phi 2 composed with C2 times the identity, composed with phi 2 to the minus 1 on B, on the rest. So doing this, you get that the diffeomorphism F obtained as C1 composed with C2. This one has exactly this property, because in the basis, it's the composition of C1 and C2, which have this property, and it acts as isometries on the fibers. Because here it acts as isometries on the fibers, here it acts as isometries on the fibers, and the change of coordinates are isometries. So it's an exercise to show that F is partially hyperbolic. So this was just to give an example of a non-trivial skew product, so that the topology might change when you do this construction. So let me just quickly mention some other reasons why studying skew products may be interesting. So this will be just a list of places in mathematics where skew products appear, but maybe some of you are interested in these other things. So in general, there are several people which are interested in what's called iterated function systems, which essentially require, if you have a finite number F1 Fk of diffeomorphisms of n, then you want to understand the dynamics of the semi-group generated by these functions. And a nice way to study these objects is to construct a skew product, which is partially hyperbolic. So how do we do that? You take a horseshoe with K different legs, and you make a product which is equal to F1 in this leg, F2 in this leg, F3, et cetera. So when you make this skew product, what you obtain is that the dynamics on this set K times n is, resembles a lot, the dynamics of this iterated function system. And so you can always choose, and this is a very interesting remark here, that independently on the strengths of these functions, you can always create a horseshoe for which the contraction and the expansion are stronger. So you can always embed this into a partially hyperbolic system. So OK, this is partially hyperbolic. And essentially, this idea appears in many other contexts in mathematics. For example, when people look at random dynamics, which is like choosing at random a diffeomorphism and applying it to a manifold, usually the notion of randomness is very related to the notion of a shift in a certain space, which is a horseshoe. And you can always recreate this type of study by making a skew product over a horseshoe. The same goes for co-cycles of diffeomorphisms. And this is how a very typical hypothesis appears, which is sometimes called fibered manching. This hypothesis that appears a lot in the study of co-cycles has to do with the fact that a skew product will be partially hyperbolic. So now in the rest of this lecture, I would like to focus on one thing that will be very important for us, which are attractors. So let's speak about these attractors. So when we discuss these filtrations, so if we have a partially hyperbolic diffeo, we have these filtrations, which are mapped inside themselves. And this separates what we call the chain recurrence classes. But sometimes understanding all chain recurrence classes might be a very hard thing to do. And sometimes we focus on a particular class of chain recurrence classes, which are interesting because they are attractors. So at least a lot of points, we know that a lot of points go to those classes. And so let me just introduce a notion that will reappear mainly on Friday's lecture, which is the notion of quasi-attractor, definition. Q is a quasi-attractor if two things. On the one hand, it is a chain recurrence class. And on the other hand, it admits a decreasing basis of trapping regions. This means that you can write Q as an intersection of Un such that f of Un is contained in a particular class of quasi-attractors. Attractor, if not only it is, but the same open set, you can obtain the set A. So if it is a quasi-attractor, and A equals the intersection of fn of U. You can find the particular trapping region is just one open set. But why do we focus more on quasi-attractors than attractors? There are mainly two reasons today. One is because it's easier to guarantee that something is a quasi-attractor than an attractor. In general, we don't know when we take an open set, we send it inside. If it will stop somewhere and have new points, or it will continue forever towards our attractor. And this is related to the fact that these ones exist, and these ones not always exist. These sets always exist. And they have very similar properties to attractors in a certain sense, so we prefer to study these ones. No, the example would be something like this. So take the dynamics on the line, and take a sequence of periodic points converging to this point, and make something attracting, repelling, attracting, repelling, attracting, repelling. So this point here, on this side, you put attracting. And so this point here has a basis of trapping neighborhoods that defines it. But it's not an attractor, because whenever you take an open set and you iterate forward, it stops somewhere. This is, of course, very non-stable. But in dynamics in higher dimension, this phenomena can be quite persistent. And so it's important to treat these kind of examples. So just to mention some examples, let me explain generalized solenoid, which is an example due to Bonati, Mingli, and the way Yang. So maybe you already know the solenoid. In any case, I can repeat it. You consider a solid torus, and you make it C-mash so that it makes two turns around the circle. And so now the usual construction of the solenoid makes this to be a hyperbolic set, a hyperbolic attractor. If you know this, you can construct this in order to preserve the foliation by disk here. And so you have an expansion in this direction, but and a uniform contraction in this direction. However, you can imagine to do whatever you like here as long as you preserve these disks. And as long as you don't expand in this disk more than this in this direction. So if you want, you can think about this example as a skew product over the doubling map in the basis. You can think about it as a skew product. So each disk maps to another disk. But the mapping depends on the point. And you want that the expansions inside the disk are less than the expansions than two, for example. So this map, you can write it in d times s1 as a mapping of the form x t maps to f t of x to t. I'm seeing the circle as 0, 1 quotient. And so what we require is that the derivative is smaller than 2. And if we have this, we will have here an invariant cone field that will give a partially hyperbolic splitting for the maximum invariant set here. So what I would like to explain is an argument due to Bonati, Li, and Yang, which shows that if we do this correctly, we can ensure that there's at most one quasi-attractor here. If we do it freely, we can make things here to be quite wild and that many classes appear here. In fact, what they do is they construct examples which have infinitely many chain recurrence classes inside here. But the point is to show that there's at most one quasi-attractor. So let me explain that. For the moment, no restriction. Now I will put one hypothesis that will guarantee the existence of a unique quasi-attractor. The derivative of f of t is less than 2. And another condition is that f t of the disk is mapped inside the disk so that we have this drawing. We require this to be an embedding. So we are not working in a global manifold. We are just defining an attracting region. So the map is a map big f from here to here, which has this form. And what we require is that the disk is mapped inside the disk. But it changes one disk to the other with the rule of the doubling map here. And then we ask for this so that the maximal invariant set here will be partially hyperbolic. So no other hypothesis. In fact, that's the reason they do this, to be able to play with this. And then Bonati and Shinohara have continued to play with these examples to get more interesting properties. But the only property I will show is that under some assumption, you have only one quasi-attractor just to explain a mechanism. So proposition, assume f of 0. So the map f of 0, 0 is a fixed point of this map. And we will ask that the map f of 0 is a contraction. It's a real contraction. It has a unique fixed point, and everything goes there. It's a uniform contraction. Then f has a unique quasi-attractor, the proof of this. So the reason we are asking this hypothesis is to know that f has a unique fixed point, p in d times 0. The unique fixed point is given because this disk is invariant, and it's a contraction, so you have a unique fixed point. And the idea is to show any quasi-attractor contained in d times s1 will contain this point p. And since chain recurrence classes are disjoint or equal, this completes the proof. So we will show that if q is a quasi-attractor, then p belongs to q. And this is enough for our purposes. So to do the proof, I will explain it. It's very easy, but I will explain a little bit. So because this type of argument will be very important, mainly on Friday's lecture. So why? So choose any trapping u which contains q. And now, if we want to show that p belongs to q, it's enough to show that p belongs to u for any trapping u containing q because we have this property. q is the intersection of every trapping set. So it's enough to show that p belongs to u. So how are we going to do this? So let me make a drawing here. So this here is d times 0. And we are choosing an open set u which is a trapping region. So in principle, it does not intersect here. But well, I won't draw u because we don't know where it is. But let me take any open set inside u which is inside this set. So take this ball b. And so what we know, since u is a trapping region, is that every iterate of b will be contained in u because it's a trapping region. But now, if you look at what's the dynamics here, you get that the projection by 2 of b in the second coordinate contains an interval. So if the projection contains an interval, as we start iterating, we are iterating the doubling map. So eventually, we will intersect this region here, probably over here. This will be fn of b. But as we intersected here, at this point here, as we iterate it, it will converge to the point b. So what we have proved is that p belongs to the closure of the future iterates of u. And the future iterates, we can start by the third iterate if we want, so n larger than 1. And so this is contained in u as we want. So this proves this proposition. And I think just to finish, I don't have time to explain the rest of what I plan. The other thing I wanted to do is to show other types of attractors, which can be constructed more or less in the same way as we did Manier's example today. They are called derived from Anosov examples. But then it's possible to play with which eigenvalues you change in order to have different types of interesting attractors. So since I have no time, in the notes, there's some explanation on how to do these examples. So I finish here. Thank you.