 Thank you Dede. So I would like to thank the organizer for the invitation, especially I would like to thank Stefano for his hard work. So single backfield, I mean backfield with singularity, in fact I will concentrate on the singularity. Okay here is that this is a joint work with my student, Lu Songzhen. And this is a framework. So we can see the compact manifold without boundary and can see the differentiable backfields. Later on we were restricted to r equal to 1. And we'll use this small file to deload the flow generated by the backfield. And the compact file to deload the derivative of the backfield. We always use this symbol to adjust for, this is traditional. Okay, so we use this symbol to deload the backfield, that zero also the backfield. We call it the singularity of the backfield. Maybe somewhat called VIX point or REST point, but we will call it the singularity of the backfield. So we distinguish the periodic orbits and the singularity. That means, I assume, we don't think singularity as a periodic orbit. So it's a periodic orbit where we will request that it's not singular. Okay, so we can see the two most dynamics, two kinds of most. The first one is the stronger one. We can see the open and then subset of the whole dynamics. The second one is the weak, it's called generic. We can see the countable intersection of open and then subset. So with their properties is generic. That means it's satisfied for system in the countable intersection of open and then subset. So this is to a large part of the dynamic world. We will only consider R equal to 1 just because for R, for C1 case we have many powerful perturbation techniques. The first one is Frank's Lama and the closing Lama and the connecting Lama. So we will unconsider R equal to 1. If R is a little bigger than 1, it's stricter, larger than 1, then all these Lama maybe, at least in the strongest version of Frank's Lama is no longer satisfied. Even for the, for example, R equal to 1 plus epsilon, it will not satisfy. So it's a big problem for a larger than 1. But anyway, at the end of the talk, I have to borrow some idea from Patent Theory, so I have to assume R larger than 1, but in a very, very special case. Okay, so let me talk about what the system far away from host show, what I was talking about. So this is a revelation of most smell system. It's very, very simple. It's the first kind of system which is proved to be stretch stable. So the law is that the law on the set of the system consists of only finite, many critical, critical orbits. That critical orbits, that means finite, many singularity and the finite, many of periodic orbits, of course, is harmonic. And plus the transverse, transversality. That means a stable manifold of any critical points is transverse to any unstable manifold, to unstable of other. So we can see that this is a picture of a different model, not a picture of a, of a, of a theoretical orbit of laws, just for a fixed point of, of a different model. Since the transverse of, transverse orbit of a theoretical orbit, I mean the diagram is also easy to, to draw. So I just draw a very simple picture for the different morphism case. We assume it's a fixed point, a harmonic fixed point. So it's a harmonic point of, of the, of the fixed point if it's a transverse intersection of the, of the stable manifold of P and unstable manifold P. So we all know that back of Simeon, Simeon said that transverse harmonic orbit one needs to, one needs to horseshoe the existence of horseshoe. So what the so-called density, density conjecture, a kind of density conjecture punish, we call it the weak density conjecture. It claims that most system, and the system with horseshoe forms open dense subset of the whole, of the set of whole dynamics, whole, whole world. So, so this part is very, very simple. It's just the consist, so for the, the knowledge that just consists of finite many orbit. And this part we know since there the horseshoe is very, very complicated. I like this kind of statement very much. It's very simple and of course 1 plus 1 is equal to, it's also very simple. But this is not, it's the proof of, of such a, the statement is not far too far from trivial. So since, as I just said, the back of Simeon system, the back of Simeon theorem, the transverse harmonic orbit will imply the horseshoe. So in, in this talk they are always, they play the same role in, in the, in the following work. So there's a parallel, parallel density conjecture which needs a lot of need, much stronger than this one. So this is, so I, I was comparing with this, I call it strong density conjecture of palace. That, that is, that is a generic system far from homo, biification is hyperbolic. But I was not taught too much about these conjectures. I will not go on this. I just, I will come this, the best one, the weak density conjecture. But from the statement, homecoming biification means that the tangency or heterodimension cycle. Since, since both of these kinds of biification will lead to the existence of horseshoe. So this, the weak is, weak conjecture is real weak than the strong conjecture. So what about the progress of this density conjecture? Oh, let me, I will see more about this. I think palace state this conjecture for different morphism, canary in, in, in, in, in the paper in his paper. But I think that he did not state this conjecture for a fact field. But since it's a very, very close at least in the spirit of his density conjecture. So I also call this density conjecture even for fact fields. It's due to palace. But what the precise statement. And for this statement, this is also if you for palace state a similar conjecture for fact fields for the density conjecture. But in his statement, he, he can, he can see the homecoming biification of singularity. But now we un-consider the homecoming biification of the periodic orbit. This is a difference. I think this is an essential difference from his original, original statement. So in some, I, I strongly that the conjecture, this statement is stronger than his original one. Okay. So for, this is a program for different morphism, of course, just for recognize for different, for C1 dynamics. That means just for C1 dynamics. So this, in, in, in this famous paper, Pujols and Stambrano proved the strong density conjecture of palace. So as a consequence, of course, they proved the weak density conjecture. And with Boratia, when we proved the three-dimensional case for weak density conjecture, but for different morphism. At last, Kravizia finished this story for different morphism case. But how about for the flow case? For the, for the non-singular vectorials, I think they proved Arroyo and Vettric health. Proved the strong density conjecture. Oh, well, although they mentioned the singularity in their paper, but it's essentially for non-singular case in, in my opinion. And very, very recently I found a paper in the archive show and then gave a, gave a, in the paper, they, they follow the idea of Kravizia, use the central model path to the Pankalei, Pankalei map along the section of, of the flow case. So this is almost a parallel tool. Just use the same idea of for Kravizia. It's the idea of central model. So this, how about the singular case? For singular vectorials, previously, just one year, maybe two years, I'm not so sure. I go with Dawei Yang or we prove the weak density conjecture for three-dimensional flow. Three-dimensional is, in some sense, it's very similar to the case of two-dimensional morphism. So, and Dawei Yang, Kravizia with Dawei Yang, they launched the three-dimensional strong density conjecture for three-dimensional vectorials. But how about the high-dimensional case? So this is what I would like to talk about. But unfortunately, I did not, I did not, I could not get a very, very complete answer to the high-dimensional, just a partial result. So I have to introduce some technical things to, for the statement. So first, this is a so-called chain equivalence. So two-point chain equivalent means that for an epsilon, you can find an epsilon chain from X to Y and from Y to X. So we denote, see, are the set of chain equivalent set. That means X is equivalent to itself. And this race relation is a closed equivalence relation of the chain equivalent set. And the chain equivalent, the equivalent class under this equivalence relation is called, I will, I will call chain equivalent class or chain class or just simple class. So what I mean class means the chain equivalent class. So this is so, okay. The, the user way for, to, for proofing the weak density conjectures as follows. So we use this symbol to denote the set of MS, to denote the set of MOS, SMELL system and HHS, to denote the set of system with HOSCHU. So if, if a, so this is approved by contradiction. So I assume that weak density conjecture is not true. If, if the weak density is not true, then we can take a system outside the, the closure of a MOS SMELL, SMELL system and the system of HHSCHU. We can take a generic system. That means C1 generic system. And, and then we, we can find a long trivial chain class, sorry. This is repeated, sorry. We can find a long trivial chain class. Long trivial means it's not reduced to a periodic orbit. Long trivial. And then we can, there exists a long, long trivial, of course. Minimal set which is, sorry, this is contained in C. Where is partial hyperbolic splitting? And central is one dimension. And then to analyze the dynamics of center leaves, I'd like to get a contradiction. This is what the, the central model for, for, for, for Gravezial tells us. So now for my, for today my result is just for the partial answer to the step one. Not, it's a, nothing with step two. But for the, this is a, for the morphic case. But now we have a singularity. So the long trivial case for fact fields has a singularity. So what does a partial hyperbolic splitting look like? So let's have a picture. This is a famous Lorenzo tractor. But this is too difficult to analyze. So we will use the geometric model of Lorenzo, Lorenzo tractor. So this is, I think everybody, almost everybody know this picture. But I should emphasize something and I will come back to this picture later. So this is a tractor. A tractor looks like a boat. And this is like a, okay. So this is the main result. This is a rough statement, not so precise. So we take a C1 generative vector which is far from, which is far from most male system and hostile. So in something we assume that weak density conjecture is not, is not true. So we take, we can take such a kind of for a fact field. And then for any such kind of fact field, this fact field has a singularity such that the chain class contains, contains sigma is long trivial. The conclusion further, every singularity, then this chain class has a same index. The index here will mean the dimension of the stable bundle of, of sigma. So we always, we can always assume sigma is hyperbolic. That means, oh, okay. And every singularity is Lorenzo like, I will give it a definition later. And the chain class admits a partial hyperbolic split. So it's in the spirit of the step one. We put some partial hyperbolic split. I will, I will give a precise statement later for the partial hyperbolic split. Okay. First I will explain the so-called Lorenzo like. So give a single, a hyperbolic singularity when least the level of exponents of at sigma is lambda s to lambda t with multiplicity. So the set value of sigma is defined as the nearest to eigen value which is the, the, the largest negative neopelow explore pollens and the smallest positive neopelow explore. We take the sum of these two neopelow explore. So there's a symbol of the, the sign, the sign of the set menu is very important in the following. So we see the singularity is Lorenzo like if the set menu, set menu is not equal to zero. For example, assume is greater than zero. So the stable bundle of sigma can be split into the direct sum of two neopelow sub bundle such that the central bundle is, is one dimensional and of course this co-dimensional one with respect to the, to the stable bundle. And another condition is the way the strong, the strong statement of sigma intersect the chain class in at only one point, at only one point. This, let me have a look back to the, to the, to the geometric Lorenzo attractor. So this is a strong stable manifold. They will intersect attract only at this singularity. So we can like these two condition as the, of course a singularity satisfying these two condition Lorenzo like. It's like the singularity in the Lorenzo attractor. So, okay, more, this is more precise for the statement. The first one is, the first two are the same as previous. So if the set menu, this is for the passion habit splitting. So if the set menu is equal to the zero, then we get such kind of splitting. And then the bundle is, this is the strong stable bundle is one dimension less than the stable bundle of singularity. And this is a similarly if, if the set menu less than zero. So if, if the chain class contains a larger singularity which whose set value is, is the sign of the, the set value is different from the sign of the set menu of sigma, then we will get such kind of passion habit splitting. This, the center has two dimension, has two dimension. And I think this is the best at this, at this kind of passion habit splitting is the best we can, we can get. Since we have only one possible long, only, only one possible, possible zero depth of experience. I mean module as a flow direction. So this is the best passion habit splitting week. So this is what we can get. But it's not, it's still not so, so good. Since in, in, if we assume that it has two such kind of singularity, we can get such a kind of splitting. But if we consider only one singularity, what can we see? We, we don't know in, in general case, but in four dimensional case, we can, we can get more detailed statement for the result. But anyway the, the, the next result is not corollary of the theorem one. So I state the theorem two. The first two are the same. So we can see the, the index of the singularity. If the index, this is a four dimensional, four dimensional. So now, from now on we will concentrate on four dimensional case. I'll never talk about higher than four dimensional case. So if the singularity, the index of singularity is three, then it's, of course it's, it's well now that it's the upper of stable. The upper of stable, since the upper of stable is very important concept in the volume. So I forgot to give the, the definition. So we gave a compact set. It's a compact, I mean, very set. It's called the upper of stable. The upper of stable. If for any neighborhood U of lambda, there exists, there exists neighborhood V of lambda, V of lambda such that the positive orbit in V is contained in U for any T greater than zero. That means this is, this is an event set for any neighborhood U. You can find using, it's a small one. V such that every orbit in the, in the small neighborhood cannot go outside the, the big neighborhood. So always contained in this neighborhood. This is so-called the upper of stable. And this condition for C1 general case is, is equivalent to so-called quasi-attractive. It's equipped for C1 generic. Equivalent to quasi-attractive. That means you can find a, often, you can find a nest trapping region such that, for example, five, okay, nest trapping region. It's small and small and the intersect is lambda and five T, any say T greater than one. So it's a trapping region for the flow and it's smaller and smaller. So in C1 general case, they are equivalent. Okay. So let's come back. If the index sigma, the index of sigma is three, then that means the, the dimension of unstable manifold, unstable bound of sigma just one. So the unstable manifold is contained in the, in the chain class. Since the chain class is on trivial. So it's not stable. This is, and in this case, we get such kind of, such kind of splitting. And now you, you know, this is two-dimensional. This is also two-dimensional. So this is the best, best partial publicity we, we can expect. So for the index one case, it's almost equivalent to the best one since you can see the, you just can see the back, the back, the, the reverse flow direction. And the set one is more interesting, later we'll, we'll, we'll spend some time on this case. For index two, index to assume the set minute greater than zero. Then according to the, the previous result, we'll get, we get such kind of splitting. But this is one-dimensional. The strong stable is one-dimensional. This is three-dimensional. But now we can prove it's volume expanding. Volume expanding means it has at least one, around zero, near proper exponents. So this, this is very important. And this is very interesting. This, this chain class could not be the upper stable. That means, of course, in this circumstance, this is not, there's such kind of, with unstable dimension, great, there's a so-called Lorentz-Nike tractor with unstable dimension, greater than one. But for all case, this chain class could not be the upper stable. So this is the same. So made, so I talk about, give a color. This is a real color, just a color of the proof. So for C1 generic field, if the back field has a singularity index two, if the chain class is the upper stable, that means the set value of sigma is greater than zero. And single-hybonic. Then, then the chain class contains theoretical orbits with complex eigenvalues. Should be complex. Of course, complex eigenvalues in the unstable bundle. So what's a single-hybonic? So following more like specific, we called a compact invariance set of the effect field, single-hybonic, if we can find invariant partial-hybonic splitting for the tangent flow, such that this one is uniform contracting, and this one is so-called sectional expanding. So for four dimension is just area expanding. If this is just two-dimension, that means it's area expanding. But if the dimension is higher than two, then it's, the condition is so-called sectional expanding. That means for any two-dimension subspace, the tangent flow restricts to the two-dimension space is uniformly area expanding. Okay. This is what I mean. The, the polarity of one ring, and then we are not constructed a tractor, a tractor, such kind of a tractor. So we, according to the current, the, the, the theoretical way, there exists a theoretical orbit in the tractor such that the, the, the unstable, unstable eigenvalues are complex. So let me say more about this kind of a tractor. So this is the, the three-dimension of Lorenzo tractor. It's very easy to construct a four-dimension Lorenzo tractor, just put a very, very strong contraction. Then, so you can increase, increase the dimension of the stable direction very, it's obvious, but it's not so easy to increase the, the, the the dimension of the unstable direction. This is what the contribution of this paper. They construct such kind of, kind of a tractor, but according to our corollary, there's some real restriction on the, on the, on the, on the periodic orbit. So, and also they, in fact, it's almost at the same time, more or less, and put yours, also write a lot of paper on, on the, such kind of a Lorenzo tractor. This expected if all this kind of for, attract, have homotonic tangency. Of course, the example tells, tells that they may, they may have low tangency, but, so, so we have four-dimension of case, we have such kind of a partially hyperbolic splitting. This is one-dimensional, this is three-dimensional. So we have low tangency with, with, of course, with respect to the, with respect to the, the periodic orbits. After some more perturbation, we have tangency with singularity. So we just talk about the tangency of, of, of the periodic orbit. But this is the finest dominated splitting. This is the finest dominated splitting. I mean, for the singular tractor. So it could not be split into fine, for example, C plus in your, your, it could not be split again. This is like, what the color tells us. Okay, let's, let's continue. So, let's, let's, let's, let's continue. So there's a related conjecture is, this is a very, very strong conjecture. The conjecture claims that every long trivial singular chain class contains periodic orbits. So it's a homoconing class. So if this, this is a very strong conjecture. If the, if this conjecture to the, it will imply the, the weight density conjecture for, for, for, weight density conjecture panacea. Of course, with the help of a central mode of convivial issue. So it's, since it will imply, if the chain class naturally contains periodic orbit, then it's a homoconing class, of course. And it will imply the existence of hot shoe and then, then imply the weight density conjecture. So this is a very, very strong conjecture. Of course it's open, even for three dimensional case. But in our, when we solve the, the weight density conjecture for three dimensional case, we solve the, we solve this problem under the, this kind of splitting, this kind of partial hyperbolic splitting. Here is one dimension, here is two dimensional. But for general case we don't know. This is for, for three dimensional. But for singular, I mean, this is a, for the polarity conjecture. So for singular hyperbolic case, if class, for gen, this is for, for any, if the class is the upper of stable, then the answer is yes. I mean, for the polarity conjecture. So for three dimensional, I think it's a result of, for, for less than the specific. And for general case, it's a result of, for Vienna and Jagangyang. It's, I learned this from a lecture at the IMPAR Internet, the web of IMPAR. And of course, Jagang told me. And they reproved this result very, almost at the same time, I think. And I will mention the method in the paper later. I think, okay, let me go fast. So this conjecture, of course, is even for the singular hyperbolic case, it's open for the set of class. For set of, set of means it's not as near plus stable. It's also not near plus stable for minus x. We don't know. So if, if you have such kind of a good splitting, for example, if you have a very, very good splitting, this is a one dimensional, one dimensional, two dimensional, and, of course, of a chain class. Even if you assume it's, for example, it's a very expanding, we don't know the answer. So it's, even if you, it sounds a single hyperbolic, it's, it's very, very close to hyperbolic. But in this case, we found that it's very different. So, okay, let's talk about, more about the proof. So this is, we'll use the so-called linear power flow. This is a tangent flow. So this is a normal bound of the, of the vector field. That means it's, the normal bound at x consists of vector field which is orthogonal to the vector, to the flow directing, to the vector, to the vector at x. So the linear power flow is that we can see any vector field which is orthogonal to the flow direction and can see the image of, its image on the tangent flow. Of course, the, the image usually is not orthogonal to the flow direction here and then we take the orthogonal projection to the normal bound. This is a so-called linear power flow. And, but, okay, this is a basic property for the linear, for the linear power flow. This theorem, this theorem has proved for different morphisms. But, but, I think it's, the idea is almost the same according to the Frank's number of flows. But the Frank's number of flows usually is not so easy to stay. To, to stay, of course, it's not so easy to perform. Anyway, we have such kind of Frank's number. So this is a, I translated from the different morphism case. You can, if here, h t means the set of vector field with a tangent associated to a periodic orbit, just associated to a periodic orbit. So I assume the vector field is far from tangency. Then, there is a neighborhood of the vector field and the uniform t greater than zero, such that for any hyperbolic periodic orbit of the, the small perturbation such that the period is greater than t, then we have such kind of domination over the periodic orbit. This is uniformly for all periodic orbit of the system and its small perturbation. So it's very strong, strong domination of a periodic orbit. Here it is, we can say that the hyperbolic splitting for the normal bundle, okay? We want to use this kind of domination. But now we have singularity. You know that the normal, the linear punga flow is not defined over on singularity. So if the, if the, if the sequence tends to the singularity, how to use this kind of information? How to use this kind of information? This, this is what we do for the extent, this is the aim of the extended linear punga flow. So it's not, it's defined, the linear punga flow is not defined on a long compact set, but it's had as a natural compactification. We call this compactification extended linear punga flow. So we, what's the, what's the compactification? So we can see that the projected bundle, that means that it consists of one dimensional subspace of the tangent bundle. And this is a grass, it's kind of a grass manifold. So we, you know, this is the projection, the boundary projection. So, and then we can see the pullback of the tangent bundle. So tangent bundle is over m. So we can pull back to a bundle over the projective bundle, the projective bundle. So the pullback is a subset of for the product space. So we can see that the, this is a one dimensional space L. And V is a vector field in, in the, in the tangent bundle. And the pullback means that they have the same base point. So we are seeing, this is the, the non-bundle of this pullback. We, we can see the sub-bundle of the pullback, such that the, the the vector field is a, is orthogonal to the base point. So this, and then you can, you can easily define the extended linear power flow over this large, large bundle. So just if you can see the flow direction at the, if the vector field, the non-zero vector will generalize a one-dimension subspace. So it can, can be naturally embedded in this flow. This is what we call extended linear power flow. And now if we discuss dominated around a singularity, we will always use the extended linear power flow. This is very, I think I'm too slow. So first we will approach so-called Lorentz like this way. This is a very simple observation. This is a three dimensional diagram. But from this diagram you, you, you can find, this is the, the compact vacation at a singularity. So this is at, at the tangent bundle, at the tangent space at singularity. And then I think I have no time to explain on this. Okay. Just escape it. The second one, you can see the homogeneity of singularity. We just, if, if you have two kind, if you have two singularity with different, with different index, this is the index two and this is index one, then you're just connected to a loop, to a heteroclinic loop by, by connecting them. It's possible anyway. And then you choose two, two orbit arc here and here. And then you, after four small perturbations, you, you transfer this to a periodic orbit. And then by choice, by a proper choice of the, the length of the orbit, this orbit arc and this orbit arc, you can, you can give the contradiction. Of course it's not so easy to, I will escape it. Okay. And last, we want, we just gave some idea to prove that if the index is, this index of sigma is two, it's not Leopold stable. So I assume on the count, on the count you assume it's Leopold stable, we'll get contradiction. So we should, we'll prove this is impossible. So I assume it's Leopold stable, then every singularity is also indexed, since we already prove it's the homogenous. And the sedimentary is greater than zero. Since the stable, unstable manifold of, it's contained in the chain class, so it should be greater than zero. If less than zero, we'll get contradiction immediately. The local star property is, that means you can find a neighborhood of the vector field such that any periodic orbit contain the small neighborhood of the chain class. And for any small perturbation, it's harmonic. So I try to prove this. This, if you all have proved this, then you can use some previous result to show it's a single harmonica. And since it's a Leopold stable, single harmonica means the existence of a periodic orbit. So we'll get contradiction. Okay. I have no time to mention the result, the favorite result of the Latian tunnels. Okay. Nice. So if it's not, if the star property is not satisfied, and then you always have a 2-1 domination, of course, first you'll have a 1-2 domination. But if it's not, if the local star property is not satisfied, you'll find another domination. And then you give a splitting for the unstable bundle of the singularity, of course, contained. You can find such a kind of set. This set, according to this property, it's not the whole set. It's not the whole chain class. And then you'll get such a kind of splitting. You can prove that every round-travel environment supported and the chain class should be supported on the small set. And then you'll get the splitting over this whole chain class, such kind of the CO boundary is bottoming spending. And then, of course, we have some, something like machine and entropy. I just give you a half a look at it, and I think I'll stop here. Thank you very much.