 Maintenant, c'est la dernière lecture de cette introduction à la dynamique hyperboli. Dans cette dernière lecture, j'aimerais présenter plus de travail en progrès et des perspectives. Pour concentrer plus sur les attracteurs, comme Raphaël a déjà fait le matin, donc, une dynamique de partenaires hyperboli. Alors, nous allons summariser ce que nous avons dit ici. Nous considérons un déformorphisme qui est partenaire hyperboli. Cela signifie qu'il y a une filtration. C'est une séquence de trappage. Une séquence d'esthétique. Ils sont trappés. Et puis, le set maximum en variant entre les régions de trappage consécutifs. Donc, le set maximum en variant en Ui plus 1 minus Ui est partie hyperboli. Donc, il n'y a qu'une sainte ou une source, donc juste un point périodique. Ou il y a une sainte dominante avec des bundes non-triviales, contractées et d'expandes. Et une partie centrale qui explique des bundes en 1 dimension. Donc, cette notion de partie hyperboli, c'est ce qui se passe quand f est loin d'homoclinique. Donc, c'est la dynamique que nous espérons ne pas être too wild, et que nous voulons décrire. Donc, le problème que nous avons adressé, que nous avons discuté, c'est celui que j'ai discuté ce matin. Donc, up to consider subclass, but which is dense. Donc, j'aimerais savoir s'il y a un set open, but dense contained in the set of partially hyperbolic system, such that for this system. So, the dynamics in general splits into classes, chain record classes. And here we want to say that the chain recurrent classes are finite, finitely many one. And then maybe it's possible to describe more the dynamics inside the classes. So, properties of these classes, let's consider such a class. So, what we discussed, what I've discussed yesterday was about the transitivity, but one may address many questions. Here we are more qualitative, so a question that we want to know is, for instance, if we have a periodic point, but then you may do a ergodic theory. And so, today, this afternoon, to conclude, I'd like to say more about this property, what we expect about this one. Just to say that among these classes, there are several cases. So, basically, three cases. So, the first case occurs when the class is a whole manifold. It's a chain transitive case. So, this is exactly what Raphael discussed yesterday. And we have the case C is an intersection of trapping region. So, it is a quasi-attractor. You may think it is a, okay, C is a quasi-attractor. Among them you have attractors. And the last case, so the other case, let's say that C is a saddle type that we understand the less. So, I'd like mainly to discuss this case. So, this morning Raphael told you that. So, there is a better class, which is a class. The center here only has one bundle. And so, let's restrict to this class, in which case Raphael said that the number of quasi-attractors is finite. So, we'd like to say more about this attractor and discuss in particular the transitivity in this case. Yesterday, it was discussed the transitivity. In the case, there is only one quasi-attractor, which is a whole manifold. Now, we would like to see what happened when there are several ones. Okay, so, as I said, it's more a work in progress with Raphael. So, let me start with something more classical. So, but what we would like to say. So, about the dynamics inside an attractor. We didn't discuss up to here so much about the periodic points. So, now, let's do it. So, about the periodic structure on the class. So, we have a notion of, that is called homoclinic class. This is defined as follows. So, you take two periodic points and you assume they are hyperbolic. And you say that they are homoclinically related. If there are orbits which go from P to Q and from Q to P. So, if the unstable manifold of the orbit of P has a transverse intersection with the stable manifold of the orbit of Q and the conversal. So, the homoclinic class of P is the closure of the set of periodic points homoclinically related to P. So, there are many nice properties. I won't detail them. So, just to mention. This relation is an equivalence relation. So, if P and Q are homoclinically related, both homoclinic classes are the same. And H of P can also be obtained as a closure of the transverse intersection between its stable and unstable manifold transverse. And from that, we can deduce that the dynamics on the class is transitive. So, all of these properties are nice consequences of hyperbolic properties of hyperbolic sets. So, for instance, to show the equivalence relation, you have P, you have Q. The unstable manifold of P intersects the stable manifold of Q and Q is related to another one, R. And then there is a property called lambda lemma, which tells you that when you look to the global unstable manifold of P, it accumulates to the unstable manifold of Q and then will intersect transversally the stable manifold of R. So, you may prove that. And just with this lemma, you will get most of this. OK. So, one thing I would like to do is to compare a chain record class with the homoclinic classes, which in principle are smaller because they are transitive. Does this occur in general? And does it occur frequently, robustly? So, there are cases where it's OK. The answer is yes. So, we discussed about Manier's example. And I will show you that for Manier's example, so non hyperbolic dynamics, which is robustly transitive, in fact, the whole manifold is a homoclinic class. So, for Manier's example, remember, you start with a three torres. You have hyperbolic automorphism acting on the three torres with three real eigenvalues. And you deform it, keeping the splitting. And so, Raphaël explained that it's possible to get something robustly transitive. So, what I would like to add is the fact that under the property he told you, f, so for f, the whole manifold is a homoclinic class. So, the proof is very similar to the proof of the robust transditivity. So, let me say it shortly. So, we have a fixed point. So, with one stable manifold and a two-dimensional unstable manifold, we use a strong direction. And now, to show that, we have to look to the unstable manifold, the stable manifold, and to try to find a transverse intersection near any point. So, you pick z here, and you would like to find a transverse intersection near z. So, what you can do is to look to... So, you take a ball, a neighborhood of z, and then you take the stable manifold of z, the strong stable manifold of z, and you iterate backwards so it expands, it gets... So, it attain a macroscopic size, et éventuellement, it will intersect this stable manifold. So, then you iterate backwards, forward, again, and so you bring this unstable manifold of p now close to z, because this stable manifold gets shrink. So, this is part of the unstable manifold of p. And now, you iterate forward this disk, and you use a property called sh property. So, Raphael explained that yesterday, which gives you some expansion at some point in the future. So, this is fL of d. And once it's large enough, you will meet the stable manifold of p. So, then you iterate backwards again, and this point comes here. So, you find a transverse intersection between stable and stable manifold of the orbit of p. And here, I used only properties that are robust. So, now, I'd like to say a bit more about the classes. So, not only we may have a good structure to fill it with periodic points that are linked together, but sometimes one can recover a weak hyperbolicity. Because remember, when you are partially hyperbolic in the center, you don't know anything. So, you want to say something which is maybe more topological. So, let me introduce the notion of chain hyperbolic class. Ok, so, now, we consider a homoclinic class. So, the setting is a bit more general with a domination. And we assume that there are two plagues family that are tangent to E and F. Usually, I say local invariant, but here it's better, we assume that they are trapped. So, the first one is trapped in the future. So, the image of a plague is sent into the plague at the image strictly. So, let's assume the thing you have disk, open disk. And the second one is trapped in the past. So, you don't know about the size of this plague. It's given, so maybe it's large and too large compared to what happens on the class. So, you want to link this plague to the class. So, what you assume furthermore is that for P, the plague along E for P is contained in a stable manifold of P and the plague for F at P is contained in the unstable manifold of P. So, here, the dimension of fibers of E correspond to the stable dimension of P. And under this property, one gets some of the usual good hyperbolic properties, qualitative properties. So, you may imagine that the plagues are like local stable manifold and local unstable manifold, but in a weaker sense. So, what you get is that for any point X, the E of X is in the, you like the stable set of the class. Here you have the chain stable set. This means that for any Y in the plague and for any epsilon, there is an epsilon pseudo orbit between Y and some point in the class, so P, you want, or X. And in the same way, the other plague is in the chain stable set for F minus 1. So, let's say, chain unstable set of the class. So, in particular, for any two point closed, you have a local product structure. So, the two plagues intersect transversely at a unique point. And so, from that, you deduce that the point Z is in the chain record class of P, but you have, in fact, better. P belongs to the homoclinic class. And also, you have, so this is a local product. From a local product, you're close to have a shadowing lemma. You want to detail it. Just, you don't have shadowing lemma at arbitrarily scale. Small scale, you have it at the scale given by the plate. And the last property is that you have some robustness. So, there are some conditions. You start with a differmorphism with a chain hyperbolic class, or a moclic class, but you have to assume, at first it is also a chain recurrence class, and when you perturb, you keep the chain hyperbolicity. So, what you have to imagine is that chain hyperbolic classes are like hyperbolic sets when you look at large scale. You see some contraction and some expansion. If you zoom, then points may separate. So, it's exactly what happened in Manier's example. You started with something hyperbolic, and you deform in a very small neighborhood of a fixed point. So, remember, you had a fixed point P0, and you deform in the center to create now. So, this is supposed to be unstable, and you deform to create a new periodic point and so to change an unstable dimension. But this is at a small scale. So, if you look at a larger scale, you don't see this. The remix is still semi-conjugated to the initial one. And so here, you're chain hyperbolic. The plagues have to be taken at a large enough scale. So, this is a first tool I wanted to introduce. Then, now, still the difference with the uniformly hyperbolic case is the center. So, we would like to understand more what happens in the center and to discuss the different possibilities. We would like to classify the dynamics along the center. So, for the moment, let's forget about homoclinique and chain record classes. And let's discuss this locally. So, you have k with a partial hyperbolicity. And then stable, the center has dimension 1. So, a local description. What you do, you don't know that maybe the tangent dynamics doesn't say anything about the center. You may have neutral points. What you would like is something motorological. So, you introduce plagues, plagues-families, a plagues-families, tangent to EC. We know it exists, which is locally invariant. So, you have a point, you have the center and you have a plagues here. So, while you have a point in a plagues close enough to Z, you may iterate. It remains close to the orbit of Z. You remain in a plague. And you reduce to a dynamics, which is like fiber. And so, there are several cases. So, first, you introduce a plague-familie. And second, you discuss the cases. So, this is what I call center models. So, it's very similar to what happened in the center for a fixed point. So, what you can see is this, either for all the plagues, you see a trapping in the future for all the plagues at the same time, or you see a trapping in the past. Or you may, if the center is orientable, you may see a mixed case, which is like a parabolic. And there is a last case, which tells you that there is a segment of an interval in the plague, which is contained in the set. So, you can show that there are the only possibilities. This occurs only for one plague, not all. But here, it's for all the plagues at the same time. So, you see what you can say locally, and sometimes you may deduce something, we will see. But then, if you want a global description, you would need a global place, and then it becomes more complicated. Global description. So, to say something, you will need a periodic point. So, this time k is not any set. We assume k is a homoclinic class, so there is a done set of periodic points. And then you may use, so the existence of large plagues, kind of plagues. So, it's provided on the periodic point to an argument by Federico Herz and Raoulourez. So, in this case, any periodic point belongs to, how did I say that? Ok, so to a curve, gamma, tangent to ici. So, gamma is tangent to ici, not only at the point, but at anywhere. So, you need that ici is defined everywhere. So, if it's the case, fine. Otherwise, you can replace by your confin. So, at any point of gamma. And this curve is invariant globally, so it is, which is periodic. And it's infinite on both directions. So, both branches, when you take gamma minus p, are infinite, have infinite lengths. So, now if you have a periodic point and such a curve, you may look to the dynamics on the curve. So, you know for one-dimensional dynamics, invertible dynamics, any point goes to a periodic point and the period is more or less all the period you see on a curve, there is a unique one. Or, well, you may have a fixed point and point of period two. So, let's assume there is a unique one to simplify. So, you may have a collection of of periodic point, of fixed point for instance. In between this fixed point, you have some contraction or expansion. And then, under a very soft assumption, you may assume that all periodic points of a given period you have only finitely many of them. So, in a plate, in a such a curve, you have only finite number of periodic point because they have the same period. And so, after some, so if you go far away, enough, then you don't see any periodic point and so you see either an expansion or a contraction. So then, what are the possibilities? Either you see something at large scale which is trapped again for f minus one or for f or you have a parabolic case and something else may happen when you built this curve Well, I didn't precisely say curve but maybe what you have in fact is a circle. So, these are the possibilities and so, you may wonder which cases are possible. So, for each of these cases you may build an example but you may ask if it's possible to have several of them at the same time. So, yes, this could happen. There are examples built by Christian Bonatti André Gogoleff and Raphaël Potry. Well, they wanted something else but on their example they show that you may have at the same time circles and lines but on this example they don't know which is a case here that appear maybe several. So, this is quite open now. I like to give an example quickly where we could have the parabolic case because for the so this one appears like a skew product with a circle fiber this one appear among hyperbolic system. So, what about the parabolic case? It's not a degenerate case. This is something that can happen for an open set of system. So, Raphaël explain it's possible to build example of partially hyperbolic set that are skew product that are similar to iterated function system. So, what you can do is this you want it's a saddle it's a saddle class so not an attractor the dynamics is of the form Z T goes to so it's fiber locally F of Z GZ of T and here you have something hyperbolic you have a horseshoe so on a on a surface this means that you have a rectangle some horizontal smaller rectangles that are mapped by F0 to vertical ones and now you define G and G is locally constant on each of these smaller rectangles so you you are only interested by point that are here so I only have to say I fix 16 maps here maybe I don't need no 4 maps the maps are constant here sorry this is the image so here you have G1, G2 up to G4 so I have to draw the dynamics of these 4 maps so on the interval so they start with a contracting point and they end with a repelling one so each time you have a 2 fix point and you have to see how to put them so so there are 2 pairs of maps so for the 2 first I am interested by the contracting part so you have weak contraction so that if you imagine you are on the line and you are allowed to apply one of these 2 maps you can go to the right or to the left to the left or to the right and since the contraction is weak you may choose the sequence to adjust an orbit to come to any place in the interval ok so this one I will correct then I have another pair of map that do the same but they are repelling weakly repelling and then since I want a parabolic behavior I have to complete here with a repelling point but I choose it in such a way that when I look to all this fix point the last one is repelling and here I have to complete a contracting point and I want also that the first one is attracting so if I look at a large scale I see contraction expansion so we are parabolic here this region is linked together because of this weak contraction it's possible to travel and to go to any place here it's linked together for the same reason by weak expansion and these two regions are linked together because there is an overlap here so everything is linked there is a single class and this is robust and so we are parabolic and this example is interesting for another reason so remember we are wondering if the classes are finite but also if they are transitive so basically they are transitive but we would like to have also robust transitivity and we hope that each time we have generic transitivity we also have robust transitivity and this is what occurred in the case the whole manifold is transitive here in the saddle case it's no more true so to say there are special cases where it's possible to get a homoclinic orbit between the point here and here and this homoclinic orbit will be ejected from the homoclinic class you have built here so it's belong to the chain record class but not to the homoclinic class oh well I have no time to to describe this ok so here it's saddle so what happens when we have quasi attractor so this time so we have a class see and it is a quasi attractor and it is so partially hyperbolic with one dimensional center and so there is using the tools I introduced you can show the following so remember we first try to have properties et then try to extend them to an open region of deformorphism so F is generic and then you can show that either C is a chain hyperbolic class homoclinic class or C is saturated by so it's a lamination by center unstable disc and when you have such a lamination transversely to the leaves you have the strong stable foliation so if you pick a leaf this leaf is in the class and now any point in a neighborhood is in the strong stable manifold of a point of some leaf of the class and so you attract any point in a neighborhood C in this case is an attractor there is no other class there are no other class in a neighborhood in this case we don't know but we have some good structure we have the chain hyperbolicity so this is a robust property and this is also a robust property because when a class is isolated for generic morphism it is robustly isolated just by abstract bear argument so how to show that so first you use oops the local idea of the proof so first you use a local argument so you introduce center plates so remember there were several cases in this case you're trapped so you're chain hyperbolic well you have to build a periodic point but you're chain hyperbolic then the other cases either on one on one side you see something repelling or you see which cover the trapping for f-1 and the parabolic case or you have an interval but you're a quasi attractor so from a quasi attractor you cannot escape you remain in the quasi attractor so here this is in the class and this you have points in the class and then you may saturate this interval this center interval by a strong unstable manifold which are also contained in the class you have a center part you saturate by the unstable leaves until this way a center unstable disc in the class so now generically you will have periodic point close so a periodic point close has some strong stable manifold which hit this disc which mean that p belongs to the class and now you're generic so when you're generic one periodic point in the class you are homoclinic class by genericity and now you have many periodic points so you may use a global argument the global place leaves, center leaves at the periodic point so center leaves at periodic point and you discuss the possibilities either for all you have so you have periodic point maybe several one but when you compare the two last these lengths here is uniformly bounded and then you have some contraction then you fill the class by such curve and so then you may extend to the closure so to the class center place but you see they are huge maybe and they are trapped so you're chain hyperbolic so it's the first case or there are you may find two periodic point in the same leaf that are arbitrary far so they are arbitrarily large segment so periodic segment periodic center segment intervals in the class and then you may saturate by the strong and stable and so you're building a center and stable leaf which is arbitrarily large so you take such a sequence you saturate by the dynamics c contains center and stable lamination inside but here I explained that such a lamination should be an attractor so the class has to coincide so it is a center and stable lamination and so you got the result so this is fine we got something generic we have to say more to have something more robust one word before you're discussing the non hyperbolic case because in the hyperbolic case you have a hyperbolic attractor which is fine so there is a result I obtain with Henry Kay which else to address palace conjecture but which imply this so if c is non hyperbolic quasi attractor and partially hyperbolic with center of dimension 1 then c contains periodic point with different stable dimension and again this one is robust it's true also for perturbation so not only in the center you are not hyperbolic you are not contracting nor repelling but this lack of contraction of expansion can be seen because you have periodic point with different center behavior in the center ok so now how to say something robust conclude last step then you need more geometry I don't know 4 or 3 to use unstable lamination laminations so c is a quasi attractor so as a quasi attractor is an unstable lamination and there is a question we would like to know that if f is generic if you have a chain record class which is an unstable lamination is this lamination minimal or dynamically minimal dynamically minimal which means that there is no invariant compact lamination strictly contained in c if yes then you may use a kind of argument we said yesterday et conclude c is robustly transitive but we don't know this question seems difficult and you may ask even in the uniformly hyperbolic case a related question cause to show that generically you are dynamically minimal it's enough to show that nothing from any dynamics you need to perturb to get something minimal so we found the following problem consider an anosoph on t3 so conjugate to a linear one which is partially hyperbolic so as an anosoph you have an unstable but it's too dimensional and I assume so there is a strong unstable and this one is unstable so we know that the stable foliation and the unstable one each of them is minimal but what about the strong unstable ok or dynamically minimal ok so anyway we we try to find another argument and so what happens when this lamination is not minimal then you decompose and you consider the minimal lamination inside and as yesterday you can show that the minimal unstable lamination in c are have the property s s h more less as what Raphael proved yesterday and then another property is that I won't explain this now but it uses the fact that in the class you have both contraction and expansion at some point then if the minimal if the lamination unstable one is not minimal the strong stable one has to be the strong stable leaves of c have to be dense in in c generically so this was known for in the case the whole class is partially hyperbolic by well maybe several words but Bonati Diazures and for attractor there is a work by nobiles to show that so now you have s h on one on the unstable lamination minimality on the other direction you have transitivity generically and then you get robust transitivity and so the conclusion if I have a few minutes I don't know the conclusion is that so in progress is that so for you open and dense so the quasi attractor there are many of them but any quasi attractor is robustly transitive and robustly a homoclinic class we would like them to be isolated so to be an attractor so either it is cu lamination and it is an attractor it is chain hyperbolic so it is like at a large enough scale it is similar to her hyperbolic attractor so you may try to find example et so here what you can do is to take so on a surface an attractor hyperbolic attractor on a surface one dimensional attractor and to to take to do a skew example with circle that you have to link so you have something two dimensional the circle along the center and here you may find a topological semi conjugacy a hyperbolic model so there is a classification of hyperbolic attractors in dimension 3 by by born classified hyperbolic attractors in dimension 3 et you may use his classification to build example in this class so what are his example you may have the whole so an automorphism on t3 and from this one you may open some leaves do some derive from another construction so et from that built example like this they are construction by so here you have many example but using a da you have a car value an example by car value so you have open in a stable leaf with a strong stable you have open a bubble which is outside the attractor we have a picture in the notes but this is the first case the second case the the dynamics so here it's 3 dimensional the attractor now it's 2 dimensional so you have a 2 torres inside in your manifold so but to make so this 2 torres either is normally contracted in this case you may not perturb it so it's only a hyperbolic attractor so what you can do is to take it tangent to the strong stable and the strong unstable and transversally you put a weak contraction if you perturb that you still keep a hyperbolic set but so it's homomorphic to the torres but not C1 and then you may deform it like in Manier example to open in the center so in the center you create some line so it's semi conjugate to another in dimension 2 but you have open in the center and then the last one is a solenoid so you have the solenoid solenoidal attractor and you may deform it to create a non hyperbolic attractor ok so this and the picture of this partial hyperbolicity with one dimensional center so it was a nice week I have to leave tomorrow but I wish you to next wonderful week