 Well, it's the pleasure to be talking here at home, of course, and I'm very glad to have this opportunity. And so I'm going to start by talking about something that says in the title. It says I'm going to talk about robust dynamical properties. So by robust, and in general, every time I'm talking about the topology in which I am looking at the flows, I'm always going to be talking about the C1 topology. Okay, so when I say robust, what I mean is a property that is held by an open set of vector fields or the theomorphisms. And when I say open set, I mean C1. Of course, you can make the definition for CR, but for this talk it's going to be, well, this doesn't work. CR, you can make the definition for CR, but for this talk it's going to be C1. Okay, and so the idea is to try to find ways to understand when a property is robust by looking at only one vector field or the theomorphism. Okay, for example, if you think about one property that's held by a whole open set of systems, like having all the periodic orbits hyperbolic for all the systems in the neighborhood, you would like to be able to detect this by only looking at one system. Also, there is some idea that if a property is persistent by perturbations, by C1 perturbations, there should be something in the structure of the vector fields that have this property that doesn't allow for big bifurcations by perturbations. So that's more or less the idea. And well, when I say that I want to look at dynamical properties, I'm not going to be looking at dynamical properties that are for all the orbits in the whole manifold. I'm just going to be focused on some particular orbits, which are the ones that are important for me, which are the recurrent ones. In this case, we are going to be looking at the chain recurrent classes, which are... So the points that can be... So one point is chain recurrent. If you can go from it back to itself by an epsilon to the orbit, which epsilon to the orbit is going to be to allow to have a mistake every time you make a step in the orbit. So you make one iteration, you can have an epsilon mistake, then you make one iteration and like that. So the points that are chain recurrents are the ones that you can come back to itself by doing this iteration. So one example, and this reminds me a bit of Enrique's talk, because I'm going to be talking about this, which is called dominated splitting, and it's one of my yes toys. So this is an example of what I say when I say I want to look at an open... I want to look at one system and detect a property that's held by a neighborhood of systems. So that picture there, supposed to show a dominated splitting in drawing wise, because I don't want to be too precise, but essentially it's that you would have along an orbit, or you would have a splitting of the tangent space. And this splitting is in a way that if you wait some time, the vectors in one of the spaces are less expanded than the vectors in another space. And you, in the other space. And you can detect this because you can see a cone field around one of the spaces that gets... when you iterate, the cone gets inside itself. And so the space that dominates is the one that is inside all the cones when you iterate. And the vectors that never get inside of one of these cones is the other space. So that will be more or less an idea of what is a dominated splitting. And if you look at this property by looking at the ideas of the cones, you can more or less get the feeling that this actually is a robust property. Because, well, the property of having cones that get inside each other is something that doesn't change much if you change the derivative very little. So this is an example of a property that is robust, but that you can detect only by looking at one system. Okay, so the dominated splitting is a robust property. You can always find, I said this was a splitting of the tangent space, and you can do it in the finest way. You can find the finest possible way of splitting the tangent space into spaces that behave in that way. And, well, so I start with dominated splitting and the relationship with robust properties because if you don't have a dominated splitting, you don't have robustness of properties. So this is a property that you can detect by looking at one system. And if this system has it, okay, then all the systems nearby have this robust property. But if this system doesn't have this property, then the neighbors don't have any shared property that you can say. So the idea now is to ask a little bit more than dominated splitting. Because as a robust property, this is very weak. I say this is an obstruction for getting all the other robust properties. But I would like to understand a little bit what other things I can ask of a vector field that make more precise other robust properties or the other way around which robust properties imply in the vector fields of my neighborhood some structure on the tangent space. So there's these two ways. And so the hyperbolic structures is what I'm going to be looking at. I'm going to be trying to find hyperbolic structures which are my way of trying to understand when a robust property is there or when it's not. And every time I say a hyperbolic structure, I will mean this. I will mean that there is a dominated splitting. And I will mean that this is valid for vector fields or for defiomorphisms. But when I look at the tangent space and I look at this splitting, I will look at the Jacobian of the vector field or the defiomorphism. And so what I want to try to find is when there is a subspace of one of the spaces in my splitting of the dominated splitting, that when I look at the Jacobian restricted to the subspace, I see a contraction or an expansion of vectors or of the volume of something, of the Jacobian. So I look at the Jacobian and I look at the subspaces of my splitting. And I want to see if in any of these cases I can find a contraction or an expansion. This would be a hyperbolic structure. And I'm going to be trying to talk about the relationship of these things with the robust properties. Okay, so in the case where, for example, the subspace that I'm looking at is one dimensional, what you're looking at when you say the expansion or contraction of the Jacobian, you're actually talking about expansion or contraction of vectors. And in this case what you get is hyperbolicity. So what you would have is that for any one dimensional subspace, for example, of the stable space, the Jacobian, which in this case is just the derivative on the vector, is contracted and in the other case expanded. So you can also do other definitions of this kind. For example, partial hyperbolicity, where you see contraction of vectors in the stable space, expansion of vectors in the unstable space, and something in between in the other space. So you're not asking the contraction in all the subspaces of the splitting, but just on the ones in the sides. Well, there's this other which is called volume partial hyperbolicity, which is more, you understand a little bit more why I was talking about the Jacobian of the lab, because in the other case was just contraction and expansion of vectors. In this case, what you're asking is that the extremal bundles, which are the ones that are more in the sides, contract or expand volume. And why am I talking about this? Well, because there is a relationship between this kind of hyperbolicities, hyperbolic structures, and the robustness of properties. One of them was, the one was largely discussed yesterday, which is the C1 stability conjecture, which actually says, okay, hyperbolicity implies that you will keep all your dynamical properties in your recurrent set. And also the other way around, the fact that you keep all the dynamical properties of your vector field or whatever in a neighborhood, this forces you to have some structure in the tangent space. But there are other weaker results like this one, for example, which is, okay, if you have a robust intransitivity or robust transitivity, it depends if you are talking about all of them or an open-ended set, this forces also some structure in the tangent space. So these are examples of the kind of things I'm looking at. Relationship between having something that's persistent and some structure that is forced on the vector fields that have this property and the other way around. Well, these results that I was saying here, they are for different morphisms. The C1 stability conjecture and this one, which is the robust transitivity implies volume partial hyperbolicity. And so I was wondering what happens with flows. And the thing with flows is that some of the results are more or less easy to keep when you don't have singularities. Others just don't. And the development of these kinds of relationships between one thing and the other, it's at least developing slower than in the case of the theomorphisms. So if you try to do a direct generalization of the results of the theomorphisms for flows, you don't get what you want, essentially. So as I said, for the theomorphisms, we have this property that if you don't have dominant splitting, then you don't get robust properties of anything. But this is not true for the theomorphisms. You can have a robustly transitive set and not have dominated splitting of the tangent space. And also, again, related to the C1 stability conjecture, as was also explained yesterday, one step to prove the stability conjecture is to think, okay, I have the theomorphism which is hyperbolic. Then I have a lot of stuff in between, but then I have periodic orbits that are hyperbolic for this theomorphism and for the surrounding the theomorphisms. Okay, but then I'm stable. So there is this middle step that's there which talks about the persistence of the hyperbolicity of the periodic orbits. And this is supposed to imply hyperbolicity and it's supposed to imply stability. But this is not true for flows. There are examples of things that have all the periodic orbits robustly hyperbolic, but that they are not stable because when you do a perturbation in flows there are other ways of killing periodic orbits which is not losing the hyperbolicity. So as a matter, just to make some idea of what I'm talking about, suppose you have a singularity like this of a flow and you have a homoclinic connection, then you can have periodic orbits that are every time more near the homoclinic connection. These periodic orbits can be hyperbolic because, well, whatever behavior is in the periodic orbit is mostly dominated by the time it spends near the singularity and if the singularity is hyperbolic then you can inherit this. Suppose this is pre-dimensional, of course, to the periodic orbit, so the periodic orbit will be hyperbolic, but then you can make a small perturbation and make the periodic orbit converge to the cycle and then you lose the periodic orbit and you never lose the hyperbolicity in this process. So there are new phenomena happening in flows that don't happen in the theomorphisms which make this connection more complicated. In the other case, in the case of robust transitivity and the dominated splitting, the problem is another, the problem is that in your tangent space you have something occupying some space which is the direction of the flow. This doesn't exist in the theomorphisms. So the direction of the flow is playing a role in the splitting of your tangent space and this complicates finding a splitting because if you approach, if you have orbits that are almost tangent to the direction where you're supposed to expand more and then get almost tangent to the direction where you're supposed to contract more then you start mixing the dominated splitting just because there is a problem with the direction of the flow and this doesn't happen in the theomorphisms so you don't have this problem and you can make your theory in peace but you have to deal with this when you are talking about flows. So there's been a lot of work in trying to solve these kind of problems. There are some things about the singularities and there are other things about the dominated splitting of things when there are no singularities. So essentially you have two ways of dealing with these kind of problems. One is when you have singularities and the other one is when you don't have singularities. So when you have singularities one of the problems that I mentioned about the singularities was the possibility of losing periodic orbits by preservation so we still want to try to find some structure in the tangent space that has some relation with the fact that the periodic orbits are all the time robustly hyperbolic. So we want to find some relationship of this sort because there is a really big persistence of something and you want to detect it. Okay, so one attempt of trying to find a structure for these kind of systems is this one which works in lower dimension which is to try to find, okay, it's not hyperbolic when you have periodic orbits of when the periodic orbits are robustly hyperbolic you don't get hyperbolicity but could you maybe find a weaker hyperbolic structure. And well, this works very well in low dimension and the idea is, as I will explain, this is the definition of singular hyperbolicity which appears in Morales Pacifico Pujals work and it works very well in various settings as a way of fixing this problem. So you have the singularity and you don't have a black pencil here. You have the singularity and then you have periodic orbits that approach the singularity. So with green. The splitting of the tangent space of the periodic orbit which we suppose is hyperbolic is of this kind. Yes, times, suppose we are in dimension 3 of course because this works well in low dimensions. Yes, times the direction of the flow times u. And then you have so we call this one the singularity. The tangent space of the singularity in this case as I draw it is two stables and one unstable. So the idea here is that when you have periodic orbits that accumulate on these kind of singularities what's happening is that the direction of the flow at some point becomes almost tangent to the unstable manifold and it becomes almost tangent also when it gets in to the stable space of the singularity. So essentially what you're saying is that these two spaces get mixed because there are orbits that come in and out of here. So what you do is you propose let's get these two spaces together without some partial hyperbolicity. So we are going to ask this space to contract and this space to expand the area. As the definition I gave it is a hyperbolic structure and well, so this does fix the problem that I was talking about here with the periodic orbits accumulating on a singularity and having different kinds of splitting. But when you do this when you try to make this kind of partial hyperbolicity you are saying that the singularities have to be of one kind. So you have this kind of splitting and this kind of splitting and then you put these two together but if you had the reverse time of this singularity then you would have this splitting u times e u u So if you have periodic orbits that accumulate on this kind of singularity here you would have the space where the directions are mixed and then what you would have is that you don't really have any dominated splitting here which is worth looking at. So this works well in dimension 3 and dimension 4 and well, because there is it is not possible to make an example that has two singularities of different indexes like this accumulated by periodic orbits in a robust way. So since you cannot do that you don't have this kind of singularities related in the same class in a robust way in low dimensions in dimension 3 and in dimension 4 this kind of hyperbolicity which is called singular hyperbolicity characterizes in an open and then set the flows that have periodic orbits which are robustly hyperbolic that I will call star flows for the rest of the talk. But in higher dimension what you get is that this characterizes the star flows in an open and then set as long as you make the extra assumption that these two kinds of singularities cannot be in the same class at the same time or else well, you have a problem that you are putting on a priori because of the definition. Okay so there. So what I'm going to try to do now is I'm going to try to explain an example in dimension 3 of two singularities of different indexes that are in the same class in the same chain recurrent class. So I'm going to do that because so there is an example it is possible of two singularities of different indexes that are in the same chain recurrent class in a robust way but in higher dimensions so that is in five dimensions. And telling this example would be complicated and I don't want to get into that so what I'm going to do is I'm going to tell a baby example in dimension 3 which is going to be fragile but in which I can explain how you fix this problem when you have two singularities of different indexes and I want to explain something else by this. I want to explain that it's not so easy to make an example in which you say that a chain recurrent class has these things inside and nothing else. So when you try to build an example like this you have to take care about what happens with all the rest of the points because if you're saying something about a chain recurrent class you need to be sure that that is your chain recurrent class and that the other points don't go there later. So when you do an example in which you're saying something about the chain recurrent class you need to control what happens with all the points on the rest of the manifold. So I wanted to talk about this a little bit. So the idea is the following. I'm going to start with this drawing in a disc but I'm going to ask you to imagine that this drawing is in S2 I couldn't draw S2 in the slides but just imagine that I don't have the pointer either I guess. Let me see. Does this point? Yeah it points. So imagine that this this is a saddle singularity and this is the stable manifold imagine that it connects here by the other side the side that we are not seeing and this which is the strong stable manifold of this sink suppose that it goes to alpha 0 on the other side of S2 that you're not seeing so essentially you have alpha which is a sink no a source sigma 0 which is a sink but it's a particular sink it's a sink that has one strong stable and one weaker stable then you have F0 which is a source and then you have S0 which is a saddle Why do I do this also specifically because this way I'm controlling where the basins that I put are in this drawing I can control where all the orbits are going so what? Alpha 0 is a source so alpha 0 is a source 0 is a source this one is a sink and this one is a saddle ok so now what I'm going to do is I'm still drawing S2 as a disk ok so I'm putting S2 normally hyperbolic in S3 sphere and I'm adding a sink and a source so now I can remove a small open set around the the new sink I added which is this one and I can remove an open set around if you remember so I have this sorry I have this F0 there which is a source and I want to take a neighborhood out of that one so I took a neighborhood out of a source and then I took a neighborhood out of a sink so you can imagine this drawing as this one and the vector field goes in here and out here so now what I'm going to do is I'm going to grab another copy of this with the reverse time of the flow that I defined there and I'm going to paste the so this part with the corresponding part in the other copy and this part with the corresponding one in the other copy but not with the identity so I'm going to do a small rotation in order to paste them in a way that's convenient for me let's go back to this drawing no yeah this drawing so this drawing is the S2 which is in the middle so when you come here and you look at what's going on here in the blue part that I took out you can see that there are a lot of unstable manifolds of all the points in S2 that go to cut the disc that I took out so if you look at the intersection of these unstable manifolds with the disc that I took out you will have this kind of drawing so I'm grabbing all the unstable manifolds of the things that are there in the in the S2 sphere and then I cut them with the with the blue neighborhood that I took out and I get this kind of a drawing where this here depicts the basing of the of the source I took out of zero do you remember there was a source of zero I took it out and this here depicts that and this point here it's the unstable manifold of sigma zero cutting the the blue area I took out so this one is important because sigma zero has a strong a very strong stable manifold a weaker stable manifold now when I put it normally hyperbolic now it has an unstable manifold so this singularity if I choose the unstable correctly I can make it to be Lorentz like Lorentz like means that the unstable manifold has a stronger expansion than the stable manifold which is what helps me here to get like the expansion of area okay so that will make me one connection because I'm cutting the unstable manifold of this of this sigma zero that was Lorentz like with the stable manifold of the other sigma zero which I will call sigma one that comes from the reverse time which is exactly the same but with the reverse time and I can do this in a transversal way because I have to really do things badly in order not to get transversality there I have a lot of ways of pasting one this with the other and making these two things get transversal as in the drawing and I can make it also in a way that all the points that are here don't coincide with the points that are here because these points have the chance of being recurrent if they coincide but if they don't coincide they are going to sink their sources so this way I make sure that the only point that has any chance of being recurrent is the one in the unstable manifold of sigma zero and the stable manifold of sigma one now I need to make the other gluing and so I'm making this drawing here because so recall that this one is the one that I'm talking about now the one that is pink and the one that is pink when I took it out what I see in the boundary is well the intersection of the ah sorry there, the intersection of the stable manifold of sigma zero with that with the neighborhood I took out it's a curve because of the shape of the basin that I draw at the beginning of F zero so essentially what I need is that the two curves that correspond to taking this out I need them to cut transversally again that's very easy to do and in one side of the one side of the curve things are in the basin of a source and in the other side of the curves things are in the basin of a sink so essentially all I have to do is that these curves that have the chance of having recurrent points I need them just to cut transversally in two points and these curves are in the stable manifold of sigma zero and so they are in the unstable manifold of sigma one and so this way I get not only a cycle between sigma zero and sigma one which are Lorentz like but also I control that all that is not in the intersection of these two curves went to the basin of a sink and to the basin of a source and this is the way I control things for not getting other recurrent points besides the ones that I want okay so what is this that I have until now I will so I need these things I have this kind of singularity here I will erase the periodic orbit because we don't have periodic orbits as I said the only recurrent points are the singularities and the cycle between them so there are no periodic points then I have another singularity here wow that was awful and it's of the other kind so I will have unstable here strong unstable here and then stable here okay and I said that the stable of this one had two cuts with the unstable of this one so essentially what you have is something like this this is okay and then the other one had only one intersection so this goes here and the rest goes to sinks and sources so the drawing there is the drawing of the only orbits that survive and that are chain recurrent the rest is all going to sinks and sources so this is the drawing that I have and I have two singularities of different indexes okay so I managed to put two singularities of different indexes in one chain recurrent class but wasn't I doing this to say something about periodic orbits what which one which one why it's not transversal ah wait I didn't look so this goes here and this well the drawing is awful yeah no it's okay it's okay this is yeah there is only one that it's transversal the other one is not possible for it to be transversal and it will break it's a fragile example but when I was in university I was thinking about the normal space no it's not possible for it to be transversal if not it would be robust and it is not in dimension three it's not possible so you get this drawing and now you want to know something about the periodic orbits of the perturbations of this drawing it's kind of strange I mean I have no periodic orbits here so this is what I was trying so the main point I'm trying to make is that hyperbolic structures are nice because they allow you to say things about the neighborhood of a system only knowing one system so what I'm going to try to do in order to say whether or not the perturbations of this thing or the hyperbolic periodic orbits is I'm going to try to put a hyperbolic structure in it that's the idea but which one clearly not singular hyperbolicity because I made this example for it not being singular hyperbolic so what to do what to do so I mentioned another problem of flows I said that there are other situations that are different for flows than for defiomorphisms and that they involve the mixing of the direction of the flow with your invariant spaces and I said there was another way of solving this the other way of solving this when you are far from singularities is looking at the linear Poincaré flow so I'm going to try to look at the other kinds of solutions for flows and try to see if I can mix them together so what is the linear Poincaré flow the linear Poincaré flow is a way of forgetting about the direction of the flow what you're going to do is you're going to look at the normal space to the direction of the flow you're going to project the dynamic of the tension there and you're going to correct it when you flow it because when you flow the normal space to to a vector of your vector field it doesn't have to go normal especially if you're passing through singularities so it can get twisted a little bit but when you project back so the way I understand this happening better is with this sorry, I'm going to go back there later with this diagram that I did there which is okay in this case it's more general but maybe I will do the diagram there ignore the slide so you have the tangent space here you have a regular pointy machine here you have the derivative of the flow here you have your point that you flow it then you take the direction of the flow and you take quotient of everything that is parallel to it so you quotient here by the direction of the flow and you get the normal space to x and then here you do the same but with the direction of the flow you get this point so you get like this and you have here the normal space to this point okay, why I do this this way because this way I'm not depending on the metric that I have in the tangent space but usually people what they do is that they take the normal vectors to the direction of the flow then they flow it and then they project to the new normal space the metric of anything okay, so let's go back to where I was before okay so the linear point of flow is a way of solving the mixing of the direction of the vector filled with my splitting in the case I'm trying to find a dominate splitting but as I defined it here it's not defined on the singularity so I'm going to have to define it on the singularity I could for example add all the directions on the singularity because I just need directions to define a normal space so here I have the direction of the flow in every regular point I could have the direction of the flow and in the singularity I could put all the set of directions the projective space of the singularity but that will give me very little information because asking for something there to be to have some hyperbolic structure in that set it's not really going to make me have the results I want you're asking for too much and it's not really important in the end because there are some parts of the directions of the flow that are not playing any part in the chain recurrent class for example in this case the strong stable and the strong and stable of these two singularities they don't participate in the dynamic so why would you add them to your normal spaces where you want to consider your flow you don't, that's the point so things about this the first one the stable manifold tangent to the strong once it's outside of the neighborhood of the class the fact that this is outside of the neighborhood of the class is robust so I say ok this one is not playing in this drawing and it's not going to be playing in any other drawing that I could make of any of the perturbations of this system so it's out and the same for the unstable so in fact I only have to consider this plane here which is the center stable or center unstable and the same here in some sense this makes sense with the definition of singular hyperbolicity that we saw before so these two don't play and they won't play for the perturbations and these two will play for the perturbations when you perturb if the singularities don't get isolated if there are other recurrent orbits that appear they have only hopes of appearing almost tangent to the center spaces so we will call these the center spaces and these are the directions that we are going to add to to my set of directions to which I will consider the normal spaces so I consider the set of directions the projective space there and I consider here all the directions generated by the regular orbits and in the singularities I can find some stable space that is not playing and some center space that is or the contrary so these are the two singularities that are here there are not any singularities that are here and one has a stable space that doesn't play the other one has there is some mistake there this one should be unstable so this one should be unstable there but one has a stable space that doesn't play and the other one an unstable space that doesn't play I won't add these directions I will only add these ones and these ones so you have the projective space of this center space in one singularity and to that you add all the directions of the regular orbits that's your set in which you are going to take normal in that set you will consider the normal spaces to those directions okay and the good thing is that this varies apersemit continuously why do you want this to vary apersemit continuously because if not the hyperbolic structure you are working so hard to get won't tell you anything about the neighborhood of your vector field you need to have some good property of how this sets by perturbation if you hope to make a structure over them that varies well by perturbation so without base you don't get good structure and this is a good base okay so now we can define extended linear Poincaré flow in a similar way as we do here we take normal to every direction of the sets that we define and then the fact that there exists the linear flow here force you to have a linear flow in between the normal spaces okay so this gives me my candidate to find a good hyperbolic structure because this one the tangent space wasn't working and it wasn't working in a very severe way it wasn't a minor issue that I was having here with my with my tangent space because if you see for the tangent space here I will look at this orbit so I look at the tangent space of this orbit I get the tangent space of X here it's going to be if it's very near if I take it to be very near this singularity I can see a contracting space an expanding space and the direction of the flow is more expanding than the other direction so what you get here is ss times yes times X of X or something like this so this is more or less what you're seeing in the linear neighborhood and when you get to the other linear neighborhood you get like the opposite situation so XT of X when you are here XT of X you get the direction of the flow and then two unstable directions so this splitting you cannot make a splitting that is respecting this situation in the neighborhoods of the singularities and that is invariant because you are understanding the vectors that are in the on one side to the complete other side of the splitting from here to here so you get everything, every vector that's the opposite to a dominated splitting so you don't get a dominated splitting of the tangent space ok so you made a lot of this and you got another space in which you want to look whether or not you have a dominated splitting which is the normal space and the normal space it makes more sense let's look at the same orbit, this one ok this orbit in the normal space when you are in the linear neighborhood what you will see is the normal to X is going to be well some stable direction and some other stable direction ok so this one is stronger than this one and then when you flow it if you did it correctly you didn't but if you did it correctly what you should be seeing is an unstable and a strong unstable but that is not a problem for the dominated splitting because this one is dominated by this one and this one is dominated by this one so as long as you send these vectors to here and these vectors to here when you flow you get the invariant splitting and forgetting this is that the example we built before was having some care on whether or not the stable and unstable manifolds were intersecting transversely is to get this property that the vectors from here get sent to here and the vectors from here get sent to here ok so you can get the dominated splitting for this in this way but what about the stronger hyperbolic structures because I would like to say something about the periodic orbits when I perturb this if whether or not these periodic orbits are going to be hyperbolic or not and how do I do that so ok I said that the direction of the flow was bothering me but now I need it again and that kind of girl I guess so yeah the direction of the flow bothers me but not the information that the direction of the flow carries so let's come here if you had a periodic orbit that appeared here it's true now it has a you can make a dominated splitting that will extend and will be there when you perturb but you would want the periodic orbit to be hyperbolic and so when the periodic orbits are near the singularity of the periodic orbit around this point it's going to be very similar to the one around here which is seeing a contraction in the normal space a full contraction and here a full expansion so what are the hopes of me getting something hyperbolic for this periodic orbits well the point is I'm not going to try for the linear Poincaré flow to be hyperbolic I'm not going to try that because I know that there are things that have linear Poincaré flow which is not hyperbolic and that have all the periodic orbits robustly hyperbolic there is an example the Lorentz attractor for example so well there are many examples but as long as well yeah every time so my point is that's hopeless so I'm going to try to do something else somehow this I mentioned three so we can think about the singular hyperbolicity a little bit and remember that they were asking this to be area contracting or expanding in this case expanding in this case contracting so we need to recover this property somehow to our new flow that it's in the normal space and how do I recover this property the area expansion and contraction well what I need to do is to look at what happens in the direction of the flow compared to what happens in the normal space but in the normal space of the direction of the center because this one is not playing again so I don't need it so what I want is to translate the expansion of area of this center space to the normal and how do I do that well I grab a particular parallelogram which is this which is the direction of the flow and this which is the projection of the center space to the normal space of let's make another so here is the normal space here is the projection of the center space to the normal space and here is the direction of the flow so here is easy projected to the normal space and here is the direction of the flow OK, and I'm going to see the product of the norm of these vectors and see what happens, which is exactly as looking at the expansion of area. So I want this to be expanded because it's a way of translating the expansion of area. And I don't want anything, anything, anything with the other unstable manifolds. So what I'm going to do is I'm going to grab a function which has a support only on a neighborhood of the singularity, and that it's the support norm. It's going to be one outside of a neighborhood of a singularity, and it's going to be something interesting inside of the neighborhood of a singularity, which what it does, it multiplies my linear Poincaré flow with the direction of the flow. But it doesn't do it in all the linear Poincaré flow, but only in this center space. So it's going to be like this. This is the linear Poincaré flow. OK, and so this one, I can, suppose XB, I can write it in coordinates of the dominant splitting. So you can think that this one is at ESS, XB, and this one is at the center space. Well, this changes when I move from one place to the other. So I will call E and F. And I can write the linear Poincaré flow in coordinates of the dominant splitting. Well, I do that, and then I multiply, for example, when I'm near to this singularity, I grab a function H, and I want this function H to be. H is going to be the expansion along the direction of the flow. So it's XT of X, X of X. I want H to be this inside of the neighborhood of the singularity, and I want it to be one outside. OK, so I have that, and I multiply this not everywhere, but just on the space which needs to recover the expansion of area, which is this one, and here it corresponds to this one. OK, so then I get this one, and let's put some gnocchi like this, which is here it's the same, because this one wasn't playing on that singularity, and here is this one times H. I can define another H, let's call it G, because calling this thing the same is a problem. G here, that does exactly the same, but only in this neighborhood. It's like H, it does the same thing of H, only that here it's one, and here it's the expansion along the direction of the orbits. So I can come here and multiply G here, so this one recovers the idea of expansion of area in the center and stable space and a contraction of area in the center stable space, this one. And the whole thing is that this one, if this one is hyperbolic, then all the perturbations of these have all the periodic orbits hyperbolic. OK, that's cool, I can now detect whether or not this is a star flow or not, but even more. This can be done in higher dimensions, the idea is the same, you only multiply the expansion of area in the spaces where the directions are mixed, and you can make this in general in higher dimension. And so, first of all, I forgot to say, this that I made here is not any function with no sense at all, this is a co-cycle over the projective space, and it varies well. So if this is hyperbolic for one vector field, then it's an open property for the vector fields. This is important because if not, it's not a hyperbolic structure, it's not what I'm looking for. And so I made this definition and I took all this care with all of these because this kind of definition helps fix the problem of the dimension of the singularities, and now you can recover all the proofs that were done in the cases where the singularities were of the same index, because you just recover the same proofs, it's not like you're doing anything else, you're just putting the new definition, grabbing the machinery that was already there, and making things work, and so they work. With this definition, you get in an open and dense set of star flows, a characterization via a hyperbolic structure of them. Okay, so not only that, what I'm trying to say is that this kind of way of understanding what happens around the singularity, so looking at this kind of co-cycle, and looking at the, now not only hyperbolicity, but any hyperbolic structure, looking at it instead of in the tangent space in this, in the normal space with this co-cycle, you can now translate the way you were making the definition of hyperbolic structure to this setting, and when you do, you can define singular volume partial hyperbolicity in this kind of way, and you recover the other results of defiomorphism, the other one I talked about today, which is, okay, if you have something that is robustly transitive, then it's volume partial hyperbolic, or if you want a robustly chain transitive, then you get in an open and dense set. And so, still there are a lot of things to do and to understand better, a lot of them I just put some of them, but, well, this is not singular hyperbolic, as I said, but it has this other kind of hyperbolicity that I talked about. So there are things that have this new kind of hyperbolicity and that are not singular hyperbolic, even in dimension three. In dimension three I said that singular hyperbolicity characterizes open and densely the star flows, but there are things in the complement of these open and dense that are hyperbolic for this kind of definition and not for singular hyperbolic, so another question would be, is it necessary to use open and dense in dimension three? Also, the proofs are highly dependable, they depend highly on the fact that the periodic orbits have all the same index, and this in dimension three you more or less have it for free, so maybe you could say in dimension three that all the star flows have this kind of hyperbolicity, maybe, but I don't know. Another thing is that, okay, you can say that the star flows have this kind of hyperbolicity, you can detect a star flow only by looking at this co-cycle, fine, but can I say something about the number of chain classes a star flow has? If it was a stardifio, I can say a lot, I can say the classes are finite, I can say everything that happens inside the class, I can say they are omega stable, I can say everything, but in flows, when you have singularities, they are not omega stable, these star flows, okay, but what can I say about the dynamics? Other than the periodic orbits are hyperbolic, can I say something about the amount of chain recurrent classes? And here I only know partial things, so not the full answer I would like to have. A question related to that, the star property talks about periodic orbits, so do I always have periodic orbits in something that is a star flow? Evidently not, but are these kind of things always fragile? If I perturb something that is so-called a star flow, do I always get a periodic orbit in the class? I have no idea, and this is the main problem with the question just above, so that. That is all. Can you give us a question about robust transitivity? Do you have examples beyond the one we know for the pharmacisms or other factors? So examples that are not star flows, but that are robustly transitive, maybe combining singularities with different indices, do you play with that? No, I didn't, but I'm sure there are, but robustly transitive things. So I stated it with. Do you have an example about robust transitivity, so what are the examples? In fact, I'm always thinking about robust chain transitivity. So this is a problem for me. I never think about anything else. And well, it's just there is these conditions you are doing all the time for getting the expansion. Maybe if you don't ask, for example, the expansion of area there, you still get something that seems to, I don't know anyone, any example now, but there must be. So there is this example by Christian, and I don't remember the precise citation, but it's Christian with other people, with some Chinese guys that I don't remember the names of, which is an example of something that is robustly chain transitivity that has periodic orbits of different indexes. So it wouldn't be singular hyperbolic or it wouldn't be any of that. It has singularities of different indexes, and that one with this is going to be, but I don't know the precise, the names of the other guys, sorry. Leaning. Yeah, that. That, yeah, and there are surely others, but I don't know. What? There is no splitting on the creation of three, the full-dentioned space. No. There is a splitting of three point A to five. Yeah. For negative and negative. Could happen, I have imagined, that there's no splitting at all. At all? What do you mean? The full-dentioned space, you have the dimension five, or is that right? There's no splitting, but it would have a star flow and some kind of a mid-splitting on the way. Yeah, so the example, we will. The tangent, or the non-linear? I guess of the tangent. So I'm not completely sure, but I think the example on the dimension five can be done to not respect any splitting. What? Being a star flow, yes.