 OK, hi. So today, I am continuing the course that Silvan started yesterday. And in the two lectures of today, I will present some of the battery of examples of partially hyperbolic dynamics we have. So in the lecture notes, which are now in the web page, there are more examples and more explanations than the ones I have the time to give here. But let me just tell a little bit about this. So let's start by recalling the two kinds of partial hyperbolicity that Silvan defined yesterday. So on the one hand, we have globally partially hyperbolic dipheos, which were diffeomorphisms such that the whole tension bundle splits as an invariant. It has an invariant splitting where this bundle was contracted. This was expanded. And there was a domination between these bundles. And then there was another definition of a partially hyperbolic diffeomorphism, which only required this splitting in the set of points which were recurrent in a certain sense. So it was partially hyperbolic diffeos. This meant to be a diffeomorphism f admitting a filtration that was the chain recurrent classes. There were some trapping regions, a finite set of trapping regions, such that each one of the sets was a trapping region. And the sets lambda i equal the maximum invariant set is partially hyperbolic set. So this meant that for this set, we had a splitting like this above this set. So these were two different kinds of definitions. This will be the one that we will be more interested in because it's more general. But today, in the first lecture, I will give examples of this particular class of partially hyperbolic systems. And in the second lecture, we will focus on this kind of examples. So let me see if I have another remark is that in general, we want to have these bundles, this center bundle, as smaller as possible. Because in general, we have much better control of these bundles than of this one. So we will try to give examples and try to provide the best kind of the compositions we can obtain. So today, this morning, we will talk about globally partially hyperbolic examples. OK, and these ones are the harder to obtain. Because one expects that if you have a splitting like this of the tension bundle, this will give some constraints on the topology of the manifold, which admits such an example. So yesterday, Sylvain gave one example of a result in this direction that was an osso-difemorphisms, which were the ones that don't have a center bundle, for which this bundle has dimension one only live in the torus. So there are several restrictions to admit this kind of difemorphism. So before I give some example, let me tell you the only known, at least to me, ways to construct examples of globally partially hyperbolic system. So there are the first kind of examples, which are kind of the building blocks of other examples, are archaic or geometric constructions. So these are examples that come to partially hyperbolic dynamics from outside. So this is the field called homogeneous dynamics. There will be a course on this next week. And this is by geometric reasons. I will try to explain a little bit about this. And then once you have these building blocks, you can make new examples out of the old ones. And there are essentially three ways I know to do this. One way is skew products. So let me give an example in each one. So this is an example that Silvan gave yesterday of an archaic example. It's a matrix acting in RD that descends to the torus. A skew product will be another example that Silvan gave yesterday, which is this times the identity, for example. So if you have this, if you have a partially hyperbolic system already, then you can multiply it by the identity of some manifold. And you get, again, a partially hyperbolic system. You are getting a higher dimensional center. So this kind of example I will explain in the afternoon, because it's also useful for these kind of things. And then there are two other mechanisms, which we don't fully understand, which are surgery methods. And here let me say the keyword is an answer flows. And finally, there is another method, which is kind of particular to partial hyperbolicity, which is the formation or composition, which I'll try to explain at the end of the lecture. OK, so as I said, none of these methods has been exhausted. We still have questions in each of these methods, but I will try to give some examples on each of the classes, not this one. This one I will speak in the afternoon. So let's start by al-Shefraic examples. To start with, I will try to give a generalized version of these examples. So consider A, a matrix with integral coefficients and determinant equal to 1, and B, a vector in Rd. So one can consider the following diffeomorphism of Rd, which is FAB of x plus Ax plus B. So notice that this is a diffeomorphism of Rd. You can compute FAB to the minus 1 equals F of A minus 1, A minus 1, B. This is an exercise. And of course, since this matrix has determinant equal to 1, this is also an integral coefficient matrix. And so as you compute what happens if you add an integral, you get that this is Ax plus B. And this, so if this belongs to CD, this also belongs to CD. So this diffeomorphism descends to a diffeomorphism of the torus. And notice that if you look at the composition by eigenvalues of this matrix, so let's write the composition like the short end of the composition, but we are grouping in each bundle all the eigenvalues of the same modulus. So if you apply the derivative of this diffeomorphism, of notice FAB is equal to the matrix A. And if this is the composition of this matrix, this will be an invariant decomposition by the derivative. And the fact that eigenvalues here are different in modulus implies that this composition is dominated as Sylvain explained. So this big introduction of an example you may already know is because I wanted to explain the fact that here you have plenty of ways to decompose in this partial hyperbolicity, to obtain this decomposition into three different sub bundles. So one way to decompose is to choose by modulus eigenvalues smaller than 1, equal to 1, and bigger than 1. But this is by far not the only way. And sometimes it's useful to decompose this in different ways, to get different partial hyperbolicities. So I forgot to remark. So if k is bigger or equal than 2, then A is F is globally partially hyperbolic. So this is a well-known example. It's just an initialization of this example. Notice that also this example, if you take the identity on the circle, also belongs to this class of examples, because you can put a 1 over here. But now let me try to generalize these kind of examples to a really larger class of algebraic dynamics. So essentially what you're doing here is you're thinking you can think these manifold are d as a group, as a d group, which has a product, which is the usual sum of vectors. And this is an automorphism of this d group. So essentially the key point of homogeneous dynamics, which is essentially apparent here, is that if you have homogeneous manifold, that means that it looks the same in every point. And you have a map, which preserves, in a certain sense, this homogeneous structure, then to check partial hyperbolicity, it's enough to check it in one point, because the rest of the points will behave exactly the same. And to have partial hyperbolicity, it's enough to have two different eigenvalues, which seems not so hard. So in general, if you have g, a d group, and if you don't know what this means, just think about the matrix subgroup, a set of matrices that is close under multiplication. And you take gamma inside g, a close subgroup, such that g by gamma. So we are mainly interested in compact manifolds. So we will require that there's this d group that you take. It has a compact quotient by a subgroup. OK, in general, the most classic examples, this subgroup is just a discrete subgroup, like cd in rd. But you can think that this subgroup is larger. It has positive dimension. For example, here, we could quotient, if one of these bundles were invariant under cd, we could quotient by one of these bundles, and it would be OK to take the quotient. So in rd, the subgroups are always of the form rk times cd for something. Here, it might be more complicated. So I will give examples. But first, let me present the general construction. And so the idea is choose, and when I say choose, this is the difficult part in general, is choose phi, an automorphism of g, such that it preserves this close subgroup. This in this setting is just taking a matrix with integral coefficients and determinant equal to 1, so that the matrix and its inverse preserves the group. And choose g in g, which will play the role of this translation. Notice here that when we took the derivative of this function, which was an automorphism plus translation, this translation didn't play any role here. However, when the group is not abelian, then translating from one side and translating from the other side, it might be different, so that translations in a group may have non-trivial action in the derivative. So that said in fancy words, the derivative, if you take the function, so I will start on the other side, so if you take the function f equals g times phi of x, which is exactly what we did here, then the derivative of f in the identity will be isometric or essentially the same as what's called the adjoint representation of g times the derivative of the automorphism phi. So what we have to check to get partial hyperbolic is that this matrix is partially hyperbolic. So if phi preserves the lattice or the subgroup gamma, then you have that f of x times gamma will be g times phi of x gamma. But since this is an automorphism and preserves gamma, this goes out and you have this. So you can define f in this homogeneous space. And checking partial hyperbolicity here is equivalent to checking that this matrix has one eigenvalue of modulus different from one. So just to say a word about this thing for those who don't know what this is, so essentially what you have is that in this group, the small f, I'm writing like this the points in g over gamma. So the classes is like writing in td is a point like x plus cd. And here I was just checking that this map preserves these classes. So it descends to this quotient, which is compact by requirement. And so with this matrix, which was the derivative, I claim that this is the derivative. If it has a eigenvalue different from one, you have an example of a partially hyperbolic difereomorphism. And so to explain a little bit why a translation may be partially hyperbolic, just notice that when you have a Leigh group, you have its Leigh algebra g, which is the tangent space of the identity of the group. And then if you pick an inner product here, you can translate it to every point of the group by right translations. OK? And so right translations become isometries of the group. And that's why we quotient this group from the right. But when you act from the left, there's no reason to be an isometry. And essentially this matrix measures how far from being an isometry you are. And that's why this matrix may have eigenvalues different from one. OK, so we would do an example now. There are many examples in the notes, but we are choosing now only one example. It's also a very classical example, which is the Shadesico, the frame flow. So let me do parallel to two examples, which are nice. So let G be one of these two groups. This one is the isometry group of the hyperbolic plane. And this one is the isometry group of the hyperbolic space. And OK, you can think about this. These are two by two matrices with determinant one, coefficients in R. This is quotient by minus the identity. This is the same, but with complex entries. And it's not trivial to prove, but it's known that there exist subgroups of she, which are co-compact and discrete. In this case, this is called the uniformization theorem for surfaces, for Riemann surfaces. So it says that if you have a surface of Sheen's larger than one, then there exists a metric of curvature minus one. And when you leave this metric to the hyperbolic plane, you get a subgroup of isometries, which has compact quotient. Here, the existence of co-compact discrete groups is much harder, because constructing hyperbolic tree manifolds is something which is not so easy. But there are many theorems that allow you to show that there exist hyperbolic tree manifolds, and so you can choose such a subgroup. And so now what we will do is we will forget about this automorphism part and just take a translation. So we choose AT to be the following matrix and consider identity, AT of x to be AT of x. OK, so this is one way to show that this is exactly, OK, we obtain she over gamma, the sheuristic flow, in the case PSL2R, and the frame flow, in the case PSL2C. So what does this mean? So the thing to check, which I don't have much time to do here, but it's explained in the notes, is that if you take any point in this group and you multiply by this one parameter subgroup, what you get in the projection to the hyperbolic plane or the hyperbolic tree space are sheuristics of these manifolds. And so in the case of the hyperbolic two space, then this corresponds to the isometric group. When you fix one point, you have all the directions to choose, and this corresponds to the sheuristic flow. But when you go to hyperbolic three space, the stabilizer of one point is the set of frames in one point. So you have the manifold here. And if you look at the isometrics of the hyperbolic three space that fix this point x, you get not only the tangent vectors, but also a perpendicular vector to choose. And when you act but this one parameter subgroup, you are moving this frame by parallel transportation. So this was to have a relationship, because I said are sheuristic geometric constructions. And by this, you can see this construction as sheurometric if you take M, a manifold, Riemannian manifold, negative curvature. Then the time one map of the sheuristic flow, the frame flow, are partially, globally, partially hyperbolic. This is also a classical result. So this is it about algebraic or sheurometric examples. And now let me speak a little bit more about examples that arise inside the field of partially hyperbolic dynamics. So once you have each one of these examples, algebraic or not, you can make products or even skew products, which is something I will explain in the afternoon. But there are also another ways to obtain new examples of partially hyperbolic systems by different methods. So the first one, which I won't speak a lot about, is surgery. This is a very general type of construction, but we don't know how to perform it in many cases. So this is well known to work for anosov flows. This we have not defined. So an anosov flow, phi 1, is globally, partially hyperbolic, with one dimensional center. So when you have an anosov flow on a manifold, then sometimes it is possible to cut this manifold along a certain sum manifold. So you have an anosov flow in M. And you can sometimes cut this manifold along by a certain sum manifold at all, for example. And then you can glue another manifold here or glue the same thing you have cut in a different way and obtain a new flow in a new manifold. So this surgery allows change in the manifold partial hyperbolicity. However, this is a mechanism we don't know very much, beyond the realm of anosov flows. And since I want to explain another thing, so let me point to two papers, which are quite easy to read and provide a very nice example of this surgery construction. One is a very old paper by Franks and Williams, which you can look at. It's a short paper. And essentially, you can follow the arguments by looking at the drawings, which are very nice. But they take some time. And then there's another kind of surgery construction, which was performed later by Handel and Thurston. There are other examples. But this one is particularly easy, and it applies the cone field criteria that Silvan explained yesterday to show that the flow, once you glue again the manifold, is still an anosov flow. So in the time I have, I will try to explain the last mechanism to construct examples of partially hyperbolic system. So yesterday, Silvan explained why the set of partially hyperbolic diffeomorphism is an open set in the C1 topology. So the idea. So essentially what he showed, that you have a globally partially hyperbolic system, if you like. So then you have a cone field, which is preserved by the derivative. And so if you make a C1 perturbation of your partially hyperbolic system, then the derivative is very close to the original one. And so if the cone fields were sent inside themselves, this C1 perturbation will still preserve these cone fields. OK, but essentially this is one way to guarantee the preservation of the cone fields. So one could imagine that one can compose by any diffeomorphism, which preserves the cone field, and still get partial hyperbolicism. So for the moment, we know essentially two types of way to guarantee this cone field preservation. So I will try to explain in detail an example due to manier. And if I have some time, I will explain the other one. OK, so I have to write the matrix. So we will start. OK, so the idea here is to start with a globally partially hyperbolic system. In fact, it will be an OSO. We have all eigenvalues different from one. And we will compose it with a new diffeomorphism and still get partial hyperbolicity, but changing some of the behavior of the dynamics. For example, changing the index of a periodic orbit. So we start with some partially hyperbolic diffeomorphism F of the torus with eigenvalues 0, more than lambda 1, and lambda 2, more than 1, more than lambda 3. So Sylvain found an example. This one is that's the shove. And so what we have here is a linear, in R3, we have a linear map of the torus. So let's look at what happens in a neighborhood of 0, which is a fixed point of this. So choose a chart, for example, of this form, where x, y, z maps to lambda 1, x. So we have the torus here. We have the zero, which is the point for which zero projects, which is in the middle. And we choose a small chart here, and we linearize. It's just diagonalizing this matrix, and then renormalizing so that we are in this neighborhood. So don't confuse. This is not the coordinate in R3. It's just a chart we are putting around 0. And so we can look the dynamics this way. So now the idea is to change. So choose a function, c, minus 1, 1, to R, such that c of x, y, c equals lambda 2y in a neighborhood of the boundary of minus 1, 1, 2, 3. And secondly, we will ask that the derivative of this function with respect to y is between these two other numbers. So the idea is to change what happens in the y direction by this new function, which is gluing well in the boundary. But now we will choose something here in the middle, which is more or less arbitrary, except that we require this function to be dominated by these two values. So this derivative belongs to lambda 1 plus epsilon lambda 3 minus epsilon. It's separated from the values here. And so we consider F. Let me, I will use a parameter because it's easier to show partial hyperbolicity. F A is y sub, which is equal to F A outside the chart. And equal to lambda 1 x, c of, right, like this, A or 1, A, c, x, A minus 1, y, c. So we are keeping what happens outside this chart. And we are changing inside the chart by this new function. Here what says it's A minus 1, y, and A here. OK, this is just technical to prove partial hyperbolicity. What you have to think about is the following. So you have the chart here. And you have the function she, you have the find. But for technical reasons, what you will do is you define not she, but you will try to make the deformation only in a small band here, changing with A, and then renormalizing this. So it's just a trick. So if you want to think that I'm putting here, she only. But now to show that there are cone fields, it's easier if I do this. No big deal. So let me show partial hyperbolicity. So we have to show the existence of cone fields. And so let me write the derivative of F. So DFA looks like this. Well, it looks the same outside the neighborhood. And inside the chart, it looks like this, lambda 1, 0, 0. Here, when you derivate this by x, you get A times the derivative of she with respect to x. When you derivate with respect to y, so this A and A minus 1 cancel. So you get she, y. And when you derivate with respect to c, you get A times she of c. So why did I put this so that I can choose these two derivatives to be very small? So that this is more or less diagonal. And here, I have lambda 3, 0 and 0. So the point is that if A is small, then we will find cone fields around each vector. So let me write the cone field. So B equal Bx, By. So we have this. And we will choose the cone field to be, see, it's, say, Bx smaller. And this is the final cone field. I will choose alpha right away. So what you get when you do DFA of B, you get x prime, Py prime, this prime. And what you have is that Bx prime equals lambda 1 Bx. And what happens if you look at the sum of these two? So you have to multiply by this. So you only get this matrix. And since A is very, very small, you can choose A very, very small. You can assume that you are multiplying by Xi y and lambda 3, each one of the vectors. So you take the smallest one of them, which is this one, but which is still larger than lambda 1. So it's larger or equal than lambda 1. And so if this was smaller, this is still smaller. And you get the contraction of the cone field. And you can do the same by inverting C and x plus y. And you get the other cone field. And you show that this example we have constructed for A, very small, is partially hyperbolic. So let me make some comments about the function Xi. What can it make? I will come back to this in the afternoon because we will make another examples with this idea. But essentially, you may have seen what are a lot of derivatives. Essentially, if you take Xi, the original Xi, lambda 2 times A y, is the following function. That's y. Here's the diagonal. And the original function was like this, this lambda 2. But now Xi is allowed to change this for the fixed point 0 the way we want. So for example, we can do something like this and create a repellent fixed point around 0. So this has the following drawing. So originally, I'm drawing the stable manifold of the linear map here. And it had here one eigenvalue lambda 1. Here, the weak eigenvalue lambda 2. But after we do the perturbation, we get the following drawing. So this is something to remark. So notice that the lines of the form something 0 y 0 here are preserved. So this still preserves the vertical foliation like this. But now, when you look at the fixed point here, it's no longer attracting. Now it's repelling. And it has this loop. Of course, when you have this, you have to create new fixed points. These appear over here, and they are attracting. And the new stable manifold now modifies. Notice that the direction of the stable manifold, which was asterisk 0 0, now it changes because it gains some coordinate here. Even if there is a stable manifold, because of the config criterion, the direction might change. And so the drawing is something like this. I have seen this drawing. So in the exercise session, one of the proposals will be to start with any linear analysis on any torus and try to make this kind of construction to play with the partially hyperbolic splitting. If you have this, here we still have the same decomposition. So for this one, we have the decomposition t of t3 equal ES plus EC, strong stable, stable, and stable. And what we have done is to change this stable direction to a neutral one. Now it's still an invariant direction. It's a central direction, but it's no longer stable. It has points where it expands. So the same you can do, for example, to break a domination. If instead of changing only this coordinate, you change these two coordinates and you create a complex eigenvalue, for example, then you won't have this splitting anymore. And you can kill this kind of. So I will explain this in the afternoon, but it's also part of the exercises for the tutorial session. So since I have some minutes, I will explain another way of creating partially hyperbolic examples, which is the same spirit as this, in the sense that what we need is to try to preserve some cone fields. And this is a construction, which is showing with the embonati, amnesia parwani. And so the idea is very similar to this. So suppose f from m to m is partially hyperbolic, with splitting s plus e c plus e u. And assume there exists h m to m, ifomorphism, such that the derivative of h applied to the bundle e u is transverse to the bundle e c s. And the derivative of h applied to the bundle e s is transverse to e c plus. And so the proposition is almost an exercise, is there exists n, such that f n composed with h is partially hyperbolic. And so the proof of this is very simple. Essentially what you have is that the unstable direction is mapped by h transverse to the center stable. So as you apply f many times, you get that the unstable direction come back to the same position. And so take c u, c unstable cone field, so that the h of c u is transverse to e s plus e c. You can do this because you can always choose the cone field as narrow as you want. And so as you apply the h, you get something which is still transverse to e c s. So for large n, f n of dh of c u is contained in c u. And so you get the cone field criteria. So why is this proposition interesting? And it's the same spirit as this construction. You can look this construction as making a composition by something which preserves the cone fields. And the interest of this proposition is that now you can more or less expect h to be whatever you want. All you need is to have a little bit of control of which are these bundles. And maybe you can construct a diffeomorphism here, which is not isotopic to the identity. So this proposition, we used it here to construct examples starting from an anosoflow and making something which is not isotopic to the identity. So let me explain only one example, which I can maybe say in one minute, which is consider the shales flow in a surface. But I will start changing the surface so that it has a very, very short shalesic in the middle. So the idea is that if you take this tube, very, very large, in curvature minus 1, then if you perform here a dent twist, which turns, you cut this, you turn it, and you glue it again. So the hyperbolic metrics are very close to each other. So let's call the dent twist here a function of rho of the surface. So n is the length of this tube. And so as you do the dent twist, this dent twist is verifies that the distance, the infinity distance, if you want, of the metric rho n, g n, and g n, where g n is the Riemannian metric here, this goes to 0. But then this dent twist induces a conjugacy between the shalesic flows of this metric and this metric. And conjugating by a diffeomorphism sends stable and unstable directions into stable and unstable directions. And so when you apply this conjugacy, you will get this condition. But the function rho is not isotopic to identity. And so you get a new example of partially hyperbolic diffeomorphism, which at this moment we don't understand very well. And so that's it for the morning. Thank you.