 which was a 211 map. Today we are going to try to introduce some tools that are good to study some perturbation of this linear system. Like if you have a nonlinear system very close to this cut map, how can you prove some topological properties? Like for instance, I'm going to prove that if you have a perturbation of the 211, you still have the property of being topologically mixing. So as usual, we always have our favorite 211 map. And it induces a map on the two torus, as we have been seeing all the time. And as it was said repeatedly, if you look at it on the square, at every point we have two eigen directions that are orthogonal. One we denote by the eigen space, which is EU, which is associated to the eigen value, which is bigger than one. And the other one, which is ES. And it has this property that this is the whole space. This is like R2. If you take the subspace generated by the stable and unstable eigen space, they have a direct sum. It's always transverse. They are in particular orthogonal for this map. And it also has this nice property. Why do we call it like a hyperbolic system? We have two eigenvalues, lambda, which was one over lambda R, eigenvalues of A. And we have this property that if we apply dfa at a point x to a vector vsx, it is one over lambda times vs at fax. And dfa x applied to a vector. After I will tell you what these vectors are, because lambda times vu at fax. So where vu and vx is like a choice of vectors that lies on ES and EU, like you choose an orientation that works globally for all the points. In that case, you have a family of vectors, like a vector field that vs defines, v sigma defines a unit vector field in e sigma for sigma equals SU. So this is a very important property in hyperbolic system which is saying that this is an invariant splitting. This is invariant by the dynamics, like invariant splitting. Meaning if I take dfa x applied to e sigma x, I get e sigma at f of x, which is for sigma equals SU. And it has another property which is called hyperbolicity, which is that if I take, so I use here the Euclidean norm that you have here on the square that comes just from the Euclidean from R2. dfa restricted to ES is equals to 1 over lambda, which is less than 1, and 1 is greater than lambda, which is dfa restricted to EU. So these are two properties that are very important in general in our study, which is like invariant splitting and the hyperbolicity of the splitting, like the, yeah. OK. So for me, at least when I see this property for the first time, you see it for the cut map. You can see it easily. But to see that this property is a kind of robust under perturbation, I could not see it just by this definition. At least for me, I could not see like if I perturb fA a little bit, like if I take fA plus some perturbation like Amy did in the class by adding epsilon science or something. So I could not see exactly why I should still preserve, like have invariant splitting for the new dynamics and have hyperbolicity for the new dynamics. Now there is a very interesting tool which I think it helps a lot to see that this is a very robust condition. It is preserved under perturbation. And that leads us to introduce the notion of cone fields. So today we will use tools from invariant cone fields. So let's stick to this map, to the cut map, because we are going to study perturbation of the cut map. So we just define invariant cone fields for the cut map. You will see the definition is very general. So given alpha positive and x in T2, you define the stable cone fields, stable cone at x. So stable cone at x by csx of angle alpha, which is defined to be the set of vector v, which any vector v here, you know that you can write it as a direct sum of vector here and here. So you have vs plus vu. Now you have the condition that it should be in the cone, meaning the vu is less than alpha vs. So you see that this defines a cone of angle alpha. If you take a vector, you multiply by a real number. The vector you get is also in the cone. Similarly, you have unstable. So here I am writing this. I'm just meaning the usual Euclidean norm on R2. It's defined by v vs plus vu. Just the reverse. It will be a cone around the unstable direction alpha vu. So this gives me, for every alpha, I have a family of cone. At every point, I have a cone. Like here, you will have this is a cone, c, u, at a point x of angle alpha. So around es, I have the same. This is csx alpha. I mean, it's clear that you can see this is a cone around the stable, because if vu is 0, this is trivial satisfied. One very important thing about this cone field is invariance by the dynamics, which I said here. So there is this fact. I don't know if I should write it as a lemma or, but it's very simple to see that for every x and t2 and alpha positive, we have what do we have? We have that dfa applied to cux alpha is exactly equal. This has to do with the fact that the map is linear, but I will tell you what we have in the general case. It's cu at fax1 over lambda squared alpha, which is a subset, which is a cone inside the cone. Like it's obvious cu at fax alpha, because this lambda is bigger than 1. Similarly, for the unstable cone, we have the invariance, but in the backward applying backward, it reads the s at f minus 1a x1 over lambda squared alpha, which is inside the cone of angle alpha. Strictly. This is a very important property of this cone, saying that stable cone is invariant in backward and unstable cone is invariant forward. Let me tell you what the picture you have. Let's say here we have x and here we have fax here. I have the unstable direction, which is here. And this is a cone I have of angle alpha. This is alpha. Here also I have the unstable direction, which is here. And here I have the same angle alpha. So the invariance is telling you that if you take this, the image of this by the derivative by the differential of fa, which is just a linear map, it is mapped strictly inside. So this will be mapped to an angle lambda minus 2 times alpha. So this is strictly mapped inside here. So maybe is it clear to everyone that this is true? One could just try to, one could just write, you take a vector v in, OK, proof. You take a vector v in cu x alpha. You write v as vs plus vu using the splitting. And you see that dfa v equals 1 over lambda times vs. I should, I should, I should, I should, OK, I should use the, because I defined this as being the unit vector. So I should use some coefficients here, eta s and eta u. So what you will have is it will be 1 over lambda eta s times vs plus lambda eta u times vu. When you vector, you can do this. And you see clearly that the ratio between this and this, you will have 1 over lambda squared times the original factor. And the other way also is clearly satisfied using the fact that the inverse has the eigenvalues that are just inverse of each other. Is it clear that for the 2 1 1 map, I have invariance of the cones. Given like this here, I have an equality actually because of the linearity. Yes? You apply the Jacobian. Yes? Yes. It does not open it. So it squeeze it around stable and stretch it along unstable. You squeeze around stable. This is squeezing here. And this is stretching. So that's what you see here. This cone will get this and this. That's this hyperbolicity. You squeeze and you stretch. This is a very, very, very important. To lies it important now. So now I can study the perturbation of the cut map. Let's consider like let f epsilon is that the norm of f epsilon minus f, c1 norm, is less than epsilon. c1 norm meaning the norm of the c0 norm and the norm of the derivative. Like the maps are closing the c1 norm. So the claim is that I know I have invariant cone field for the cut map. I want to say that the same cone fields are invariant by perturbation of the map. So given alpha positive, there exists epsilon such that there exists epsilon and lambda bar, which is probably not 1 over lambda, such that df epsilon at, yes, yes, yes. So how should I call it? Yeah, but it's not exactly the same. Kappa, OK. df epsilon of c u x alpha is in alpha, f of x, f epsilon x. And can you see here? And df epsilon minus 1 at c s x alpha is in c s f minus 1 x kappa alpha. So I have the cut map. I have the 211 map. I have invariant splitting. I build a cone around this invariant splitting. And I'm claiming that the same cone is invariant by applying the differential of a perturbation of the cut map. But you see this is easy. It just follows from the continuity, right? Because you see what you have. Let me write the picture again. So I'm here at x. I have the cone of angle alpha. I'm here at fa x. I know that this guy is mapped strictly inside. There is a space here. So this is like lambda minus 2 times alpha. So if I take a perturbation of the map, so x will not be mapped exactly to fa x. It will be mapped to somewhere very close. So let's say x is mapped to somewhere f. You can make it as close as you want by choosing epsilon small, in which case, if you draw here also the curve, the cone that you have here, the family of cone is continuous. You can choose them so that it is continuous. So the angle, the cone changes continuously. So what you have, if this guy is mapped strictly inside here, the image of this guy also by, so this was by dfa. And by df epsilon, it will be also strictly mapped inside here. I don't know. My picture is very messy. Guys, forgive me. This follows just because this is continuous. If you look, this is the same. It is a continuous family of cone. And the differential, I change it just a little bit. So I will preserve the invariance of the cone. Is that more or less clear? OK, now what is good now about this, about the invariance of cone? Actually, one can see that it is exactly equivalent to hyperbolicity, what I define here. I will do that. I plan to give that as an exercise in the afternoon. Move that this property here in the claim, let's denote it by star. Star implies that there exists and not, such that for all x in T2, we have Tx T2, the tangent space of the torus, splits into two new bundles. So I will denote it by tilde just because it has to do with the perturbation. I should put an epsilon. And it is invariant, df epsilon applied to e sigma tilde x is exactly e f epsilon x sigma for sigma equals SU. And 2i, if you wait long enough, you will see the same hyperbolicity happening. df and not restricted to e s tilde is strictly less than 1, which is strictly less than df and not restricted to e u. So here we see what I was saying about the fact that the hyperbolicity is robust. Because I have the cut map. I have cone fields. I put up the cut map a little bit. The cone field will remain invariant. And here the exercise is telling you that invariance of cone field like that implies the existence of a splitting, which is invariant and which is hyperbolic. OK. That's good. So this, actually, yes, yes, yes, yes. If you have this for some alpha, just for some alpha, you don't need it for any alpha. You have these two properties. Now you can, yeah? Yes. Can you say louder? Depends on? Yes. Yes. The change is continuous. So this df a minus df epsilon is less than epsilon, the C0 norm. C0 norm. Is that what you're saying? Yes, f epsilon, yes, thank you, yes. Yes, yes. So actually, let me just look at my note to do not mix the step of what I want to say. So actually, this family of map that have this splitting and the hyperbolicity, it belong to a bigger family, what we call the uniform hyperbolic systems or the Anosov systems as well. So invariant splitting plus hyperbolicity, this defines what we call uniform hyperbolic maps or Anosov diffuse. So this what I just sketched for the card map, it tells you that you have two facts. Like the fact that the being Anosov is an open condition is C1 open. That's a very important fact, just using cone fields. Like if I have an Anosov diffimorphism, what I do, you can do it for anyone. Like if you have Anosov diffimorphism, what you just think of, you have the splitting and you have the hyperbolicity, you can build your cone around this invariant splitting and you have the invariance of the cone field. So you will have the invariance of cone field for the perturbation of the map, C1 perturbation, because you see, I use the fact that the differential is closed, like they are closed in the C1 topology. And another property that I want to say, I don't want to forget anything. Yeah, I said, so having invariant cone fields is equivalent to being hyperbolic, like being Anosov. So we come back to the perturbation. Now what we want to prove is that the perturbation is topologically mixing, epsilon, for the cut map. The proof I will give here is, I mean, it has to do with the fact that I am in this square and I am perturbing very specifically. It has to do with the geometry of the domain, but there are also proof using other techniques, just to tell you. So this is our next goal. Yes, yes, yes, yes, yes, yes, yes, yes. Yes, yes, yes, yes. Yeah, yeah, that's a very important point. Thank you for bringing that up. So the thing is, the fact that you have cone fields, just invariant of cone fields doesn't see what is happening inside. It doesn't tell you whether inside you have expansion of contraction, but it just tells you the relative, like, the domination. Like, the vectors in EU, their norm, dominates the vectors in ES. So it doesn't tell you which this one is contracting or expanding, but it tells you that you have what we call dominated splitting. Like, it just tells you this, for instance, without the one. Yeah, for two dimension, it just tells you this, without the one. That's very important. But in our case, since we are perturbing the cut map, we use the fact that it's a C1 perturbation, so you still have the contraction and expansion. Thank you. So our goal is to prove that the perturbation epsilon is topologically mixing. I mean, since I want to save time, I will just tell you, remind you, what topologically mixing was. That was the definition of topologically mixing. Any two open sets you take, there is an iterate after which these two open sets, they always intersect after that iterate. Like, if you iterate one. Yeah. So how do we prove this? To prove this, there is some technicality, some technicality that has to do with the geometry of the space that we're considering that we have to use here. What is the geometric property of the cut map? So I write this fact from the cut map, and after we can discuss how it follows. So there exists a positive such that any piece of stable, unstable manifold FA of the cut map intersect at least, any piece of length at least intersect. Is that any? What am I saying here is that if I consider the cut map, so I have eigenspaces that is doing this and this. This is the unstable eigenspace, and there is the unstable eigenspace, which is also orthogonal to it. I'm saying that you can fix the specific lengths for the piece. I mean, I'm considering these two. It's in the torus. So this is just one piece of unstable manifold. So any two piece that I consider from here, so you see very, very if I go unstable like this, it will intersect. So what I'm basically saying is that there exists a length. It just has to do with the angle, with the angle of the unstable direction, with the angle that it makes here, which is irrational. And yeah, can we believe this? So if you have this fact, now, I'm very slow. So there is also this observation. Observation is that if you take limit when alpha goes to 0 of c s c sigma x alpha, you get exactly e sigma x. This is trivial, right? This is trivial. Like this cone is around for sigma equals SU. So what I derived from this property is that now I can fix an alpha such that any piece of curve that is tangent to the cone, I need two pieces of curve that are tangent to stable and unstable cone respectively of a certain length will intersect. Gamma s and gamma u tangent to stable, tangent to c s. So I should say there exists alpha, tangent to c alpha. And you, respectively, length of gamma s, length of gamma u bigger than l, gamma s intersect gamma u. It's not empty. Because what is this basically saying? So if I make alpha very small, what I get is a curve that is very close to being the stable manifold. For instance, if I take a curve in c s alpha, where alpha is very small, I get a curve very close to the c one to the stable manifold. So from this fact, I can choose alpha small enough so that any curve of a certain length will intersect. Any two curves of certain length will intersect. Now having this, I can now come to the perturbation. Because this is a very useful fact for the cones. Because I know the cone holds, like the invariance of cone holds for the perturbation. So what do we have here is that now, so what do I need to prove is that if I take two open sets on my torus, actually there is a picture for this. I should use it. Nice picture. I take two open sets. I want to find one iterate for which this guy, for instance, keeps intersecting the other one. So what do I do? I take a curve. So remember, I want to prove f epsilon. By choosing epsilon small enough, depending on how alpha is, I can guarantee that the vectors, I can guarantee that a curve that is staying inside the stable cone will get stretched like this. Why is that? Because you can just look at what is f minus epsilon, minus of minus 1 epsilon gamma, this curve. Let's say you consider this, you call it gamma 1. You take the gamma inside the stable cone, the stable cone given by alpha. So you can see that the length of this curve is growing. And as the length is growing, the angle, the cone is getting narrower and narrower. So you can still apply this fact. So what will happen is that length of gamma n, there exists some m such that length of gamma m will be bigger than gamma m. Yeah, thank you. So I do this. Gamma is a piece of curve that I consider in the stable cone of angle alpha. So gamma is a starting curve which the tangent vector to the curve is in the stable cone. It's inside of you. Yes, yes, yes, yes. Thank you, yes. So I take a piece of curve inside u because u might be very small. Maybe this curve gamma that I consider doesn't have the length that I want to intersect. So what you do now, you iterate this curve backward and you know the vectors in the stable cone they expand when you go backward. So the curve is growing in length. Like you have the limit of length of gamma n is going to infinity as n goes to prancing. So you see the curve is growing in length because it is in the stable cone when you trade backward. So at some point it will achieve my good length that any other unstable curve in the unstable cone of that length would intersect this one. So here you are lucky. Just here you can see intersection. But you can really go further to see that the other one also will achieve the certain length so that you cannot avoid intersection. And this proves mixing. It proves that whenever you do a perturbation, having this geometry of my space, you remain mixing using cone fields. And yeah, so I have about seven minutes and I wanted to give some kind of motivation. Like why do we care about this mass? Why do we care about having splitting? Why do we care about having cone fields? So for that I would like to just play a video that I like very much that comes from the Lorentz equation. And this famous Lorentz equation that you all know. When we look at the movement of the atmosphere, we quickly realize that it is infinitely more complex than that of the solar system. The atmosphere is a fluid that has at each altitude above each point of the Earth's surface a speed, a density, a pressure, a temperature, and so on. All of this data varies over time. It is, of course, impossible to understand this practically infinite amount of data. It is almost as if we were in a space with an infinite number of dimensions. To understand something about it, we must make approximations. In 1963, Edward Lorentz simplified, then simplified, and simplified the problem again. He simplified it to such a degree that there is no guarantee that his equation has anything to do with reality. His model of the atmosphere was reduced to just three numbers, x, y, and z. The evolution of the atmosphere was reduced to a simple equation. Each point, x, y, z, in space, represents a state of the atmosphere. The evolution follows a vector field. For example, and this is only an example, the first coordinate could represent the temperature, the second, the wind speed, and the third, the humidity. Over here, it is cold, breezy, and rainy. Here, the opposite holds. When we follow a trajectory of the field, we are following the evolution of the weather. The weatherman just needs to solve a differential equation. This is what Lorentz saw when he studied his model. Does this have anything to do with real weather? That is far from clear. This is what physicists often call a toy model, used to try and understand the broad outlines of some complex behavior. In fact, Lorentz only had these sorts of graphs to look at because his computer in 1963 was quite primitive. Let's look at two atmospheres, represented by the centers of these two balls that are extremely close together. So close, they are almost identical. Let's observe what happens to them. At first, the two evolutions are almost indistinguishable, but then, they split up significantly. The two atmospheres become completely different. This, then, is chaos, sensitive dependence on initial conditions. So I wanted to show you this to tell you that this system that we're considering, like the uniform hyperbolic system, they belong to a bigger system, which has a chaotic system, which has this notion of sensitive dependence on initial conditions. Like, if you look at, let's say, two points on the same unstable manifolds, you see that their feature looks very different. They do very different things, even though to start with, they were very, very close. So this system that we're considering, the uniform hyperbolic system, they have the sensitive dependence on initial condition. And actually, for this system, the Lorentz system, maybe time will not allow me because I want to talk about something else very soon. You could, like, what we do in our fields is we introduce what is called the geometric model. And the geometric model is a vector field that looks exactly like the floor near the origin. And you can study what is called the Poincare section. Poincare section will look something like this. So I'm in my three-dimensional space. The geometric model of the Lorentz floor is something like this. And what does it do? It takes, you have here this line, it takes this and maps it to something like this. So this is really, I'm giving a very rough idea of what's going on here to see the invariance of these bundles for the Poincare section. What happened here is that if you take a, here you have the family of, for some parameters, you have a family of these vertical lines. What happened is that these vertical lines, they get maps to vertical lines again. Like, this guy will be mapped to vertical line here for some parameters. And this is really showing you that you have some, some like invariant splitting. And you have also the contraction because also for the parameter for this exact equation, the map is strongly dissipative. You have this area contracting. So you see a lot of hyperbolicity. But nevertheless, there is something more important. Should I say more or should I? Yeah, so there is a lot to say about this, but I think the time will not allow me to elaborate. So I will stop here. If there are questions, okay.