 So, today is the last day of our course with Sylvain and this is my last lecture, so I wanted to profit to thank first of all Sylvain for inviting me to share this course with him, which is an honor, and also to thank Stefano, who's not here, but for he was an excellent host and an excellent organizer. And so, now I will start, so in the afternoon, I think Sylvain will make kind of a summary of what we are doing, but today, in the morning, I will try to explain something which is related to what we did yesterday, but it's kind of independent also, so you can, hopefully you can follow without having followed yesterday. So, the question we posed today, again in the context of trying to understand how to decompose the dynamics, so yesterday, we tried to look at this dichotomy, either there exists a trapping set, or said F is chain recurrent, okay, and the work we did yesterday was to try to see under which conditions we could improve this recurrent property, okay. So, today, we will focus mainly on this other case, which is the rest of the case, and it's not at all well understood, so now, we know we have a certain trapping region, U, which is below, and now we would like to understand how can we decompose the dynamics, so essentially the question is how many chain recurrence classes are there. So, in the first lecture, Sylvain explained that there are some phenomena that nowadays is known as Newhouse phenomena, which implies that under certain conditions, one has infinitely many chain recurrence classes, in particular, infinitely many things, this is one of the important cases. So, now we are trying to understand if we can put some conditions that will forbid this phenomena to appear, and the condition we will try to test is partial hyperbolicity, okay. So, before I enter exactly into this, let me remember, this was also an exercise in the first tutorial lesson, is that if F is hyperbolic, then it has finitely many chain recurrence classes. This is sometimes called the spectral decomposition, and let me try to very, very briefly explain the idea of this, because in a certain sense, it is quite very related to the argument we did yesterday to show in that transitive and also diffeomorphisms are robustly transitive. So, the idea is that if you have a hyperbolic diffeomorphism, meaning if you don't remember that each chain recurrence class is hyperbolic, then you have to show that there are finitely many chain recurrence classes, and so how will we do this? You will take a sequence, you suppose this is not the case, and so you take a sequence of chain recurrence classes, which is converging to a certain set, so let's call these classes, and so these classes are compact set, they will converge to a certain compact set lambda, which will be contained in a chain recurrence class, but so this is a hyperbolic set, so we have lambda here, and so we have, it's a hyperbolic set, so it has a uniform sized stable manifold, and a uniform sized unstable manifold, which covers the whole dimension, and so hyperbolicity is something that holds in a neighborhood, and so as classes CN start to accumulate here, you get that the stable manifold of the class will intersect the unstable manifold of lambda, and the stable manifold of the class, unstable manifold of the class will intersect the stable manifold of lambda, and if you look this closely, this means that you can go from pseudo-orbit from here to here, and from here to here, and so they are in the same class, showing that this is not possible, okay, so again the key point is that when you are hyperbolic, the stable and the unstable direction, which is where we have some control on the dynamics, cover the whole dimension of the manifold, and so the idea now is try to extend this to the context where you are partially hyperbolic with one dimensional center, so before I state the main problem, let me first mention that this phenomena is very associated with homoclinic tangencies, which was explained by Sylvain, and the result by Sylvain, Enrique Puschels, Martin Sambarino, and the way young implies that far from tangencies, one has partial hyperbolicity, and the other with low dimensional centers, I won't explain precisely how Sylvain did the first day, so essentially what we expect is that, far from tangencies, we will be able to show that this phenomena cannot occur, and so the conjecture is by Bonati, it has many forms, but I will state a simple version, is that for F partial hyperbolicity, it's partially hyperbolic diffeomorphism, center dimension one, or at least open and dense subset of partially hyperbolic diffeomorphisms with center dimension one, F has finitely many chain recurrence classes, and so essentially you can change this, this would be globally partially hyperbolic diffeomorphisms with one dimensional center, but as we will be working in neighborhoods of chain recurrence classes, it's no problem to assume that you are partially hyperbolic in this weaker setting that chain recurrence classes are partially hyperbolic, but for the purposes of today, I won't matter about the distinction, and you can assume that you are globally partially hyperbolic with one dimensional center, so the difficulties are the same, and so only let me mention that this conjecture is also related to what we know as palace conjectures, in particular there's a conjecture by palace that tries to describe dynamics far from homo-clinic tangences and another kind of bifurcation which are called heterodimensional cycles, and in that case palace claims that the dynamics should be hyperbolic, okay, and so essentially what I will tell you today is in a certain sense a continuation of a big work that Sylvain did with Enrique Puschels, who studied the case of dynamics far from homo-clinic tangences and heterodimensional cycles, so now we will focus on this setting, so let me mention which is the result I would like to explain a little bit about the theorem, showing with Sylvain and Martin Sambarino, which says essentially the following, that there is an open, C1, open and dense subset of partially hyperbolic difumorphisms with one dimensional center, such that every F in, there's an open dense O, every F in O has finitely many quasi-attractors, okay, so essentially we are not able to show that there are finitely many chain recurrence classes, as I explained the other day, they are some chain recurrence classes which stand out from the rest because they capture the dynamics of most of the points, okay, and so we are able to show that there are finitely many such classes, okay, the other ones we don't know, but at least these ones, which are the attractors that capture most of the dynamics, we know that they cannot accumulate. Before I continue, let me say that this is related to a work by Bonati, Gan, and Lee, and the way Yang, which studies, in which they show that far away from homoclinic tangences, quasi-attractors cannot accumulate in a quasi-attractor, okay, so it's not so different, but here what we are showing is that you have only finitely many quasi-attractors, and of course what we will try to do is try to reproduce this argument, we will take a sequence of quasi-attractors, conversion to something, and try to show that this is impossible, okay, what Bonati, Gan, Lee, and Yang did in a more general context, in the general context of far from homoclinic tangences, they showed that if these quasi-attractors converge somewhere, this cannot be a quasi-attractor, okay, so I will explain a little bit better, okay, so anyway, so let me recall what a quasi-attractor is, Q inside them is a quasi-attractor, if two things happen, Q is a chain recurrence class, and there exists a basis of neighborhoods that can be trapped in sets, this meant that they are compactly sent into the interior, such that Q equals the intersection of these open sets, so in the rest of the lecture I will try to explain the ideas behind this result, and so the first thing I want to do is try to relate this result to what we were trying to do yesterday, okay, so how can we relate these two things is because essentially quasi-attractors are saturated by unstable manifolds, so quasi-attractors, and the important thing about quasi-attractors is that they contain minimal and stable laminations, okay, so let me explain that, this is a proposition, let F be a partially hyperbolic defymorphism with one-dimensional center, and let Q be a quasi-attractor, then for every X in Q we have that the strong and stable manifold effects is contained in Q, okay, so when Sylvan explained the other day that when you have a partially hyperbolic defymorphism, every point has an unstable manifold, a manifold which is tangent to the unstable distribution and which is contracted in the past, okay, so let me explain this result, and let me explain it in the context of an example I did the other day which was the example of solenoids to remember what we did the other day, and so essentially if you have a point X in Q, so you have a sequence of open sets which are trapped inside themselves, so and take UN containing Q such that F of UN is mapped inside UN, so for every K, okay, this is an open set containing a compact set, there is a uniform distance to the boundary, so at every point in Q we have a uniform local and stable manifold which is contained in this open set, okay, so for every Y in Q there exists, we have that the local and stable manifold of Y is contained in UN, so if you recall Sylvan's lecture the other day, you know that you can recover the whole unstable manifold by iterating this manifold forward, so the unstable manifold of X which is equal to the Nth iterate of the local and stable manifold of F minus N of X will also be contained in UN, right, because we have this local and stable manifold is contained in UN, but UN is sent inside UN, so all forward iterates are contained in UN so this belongs to UN and therefore since all UN's determine the quasi-attractor we get that Q is saturated by unstable manifold, okay, why did this, I did this drawing because the other day in a certain sense when we explained that here you had sort of a quasi-attractor the way we did it was like iterating forward and seeing that these sets were like spiraling around these torus, essentially now that we know about unstable manifolds what we were trying to say is that this set is saturated by unstable manifolds and unstable manifolds have to intersect one of each one of these disks, okay, and that's what we did the other day without having the words to say, so what's the trick here is that you have, what would happen if we have infinitely many quasi-attractors, so suppose one has infinitely many quasi-attractors Qn, so these are compact invariant sets and if they are infinitely many they should accumulate somewhere, so there should be some accumulation Qn converges to lambda as in the hyperbolic case and what is the important information we have about quasi-attractors is that they are saturated by unstable manifolds and since unstable manifolds vary continuously in compact parts this information extends to the limit, okay, so important point lambda may not be a quasi-attractor but at least it is saturated by unstable manifolds and now we are more or less in the setting we were yesterday of trying to understand sets which are saturated by unstable leaves and minimal such sets, so let me make two remarks which are the main points here, one we have that, so let me recall what was a minimal unstable lamination a minimal WUU lamination was a compact set say lambda used before, so let's call gamma saturated compact eff invariant saturated by WUU leaves and minimal for these requirements, so the two important remarks to go to point to this direction is that each quasi-attractor contains a minimal WUU lamination and this is trivial because once you have a compact set which is saturated by unstable manifolds, so as we explained yesterday, it's like a Sorn-Lehman argument tells you that it has a minimal one, at least one, it may have many but one it has and the second remark is that if gamma one and gamma two are minimal WUU lamination, so lamination and the stable manifold of lambda one of a point X and there exists a point X in lambda one such that the stable manifold of X intersects lambda two, then the two laminations coincide, like this, they are compact invariant sets, if they have a point, if they share a stable manifold, this means that their distance has to be zero, they are invariant compact sets, they have a positive distance, so if you have a stable manifold that intersect both, they have to be the same. And so with these two remarks, we will try to perform the same argument in hyperbolic setting to try to show that if you have a sequence of quasi-attractors accumulating somewhere, you will have an intersection of stable manifolds. So let me first show the difficulty and then try to explain how we deal with this problem. So what is the possible problem that we may have is the following picture, which I draw the other day. So we have here a sequence of points, like this, so we could have something like this, right? All blue guys are attractors and they are accumulating to some other quasi-attractors and here you may have infinitely many. So in principle, the fact that we have a center direction, where we don't have a control on the size of the stable and unstable manifolds may provide a difficulty to show that each quasi-attractor occupies a uniform amount of space. So what are we going to do? We are going to try and perform a perturbation, as we did yesterday, to get some geometric condition that forces each quasi-attractor to occupy some uniform space in the manifold. So today, I won't expect to give a definition and so I will only make a drawing. So we have the lamination, so let me put here transversality or non-short integrability. What we show is that it's the same perturbation result we used yesterday, we use it in this context and it says essentially the following. It says that for an open and dense of partially hyperbolic diffeomorphisms with one-dimensional center, each time you have a minimal unstable saturated set, you have the following property. Each time the minimal set has an unstable leaf, has two unstable leaves which are connected by some stable manifold. So let me show here blue is unstable, is stable. We have the property that there is another point in the unstable leaf which does not intersect here. So we have some space in the center direction. So essentially to do this more or less correctly what we prove exactly is that in a minimal unstable lamination of a partially hyperbolic diffeomorphism which belongs to a certain open and dense subset of this, what you have is if you have two unstable leaves that share a stable manifold and you can quantify this by making this is more or less a certain distance one and so if you move here a distance also more or less one then you have a space here of size more or less delta. This is more or less, you can quantify this, it's done in the notes. And so the point is that now as you approach this limit then you will get some intersection of stable manifolds because we recall here that each one of the points in this unstable manifold has a stable manifold and this makes, let me make another drawing so I don't mess up with this one. So I have one stable manifold here, one unstable manifold here is connected to another unstable manifold here and what we know is that if I saturate this one by unstable manifolds I get like a roof like here and if I saturate this one by stable manifolds I get something like this. Can you see the picture? I have like two co-dimension one surfaces, remember that the center dimension is one which are blocked by the stable manifolds of our lamination. So what do we know now is that there cannot be a point here which belongs to another minimal and stable lamination because of this easy remark. That's the key point. So let me explain a little bit how to conclude from this geometric property. Yeah, this is, as I said yesterday I tried to explain quite unsuccessfully this perturbation argument and what I said is that this is very related to accessibility. Accessibility means that you go by stable and stable, stable and stable and you can reach everywhere, okay. But accessibility you can go in the unstable wherever you want. When you have a minimal lamination you don't have much choice because only the stable manifolds that connect points in the unstable are useful. So in a certain sense which I said yesterday our perturbation result is an improvement of Dolgopiat Wilkinson's result of accessibility but the key difficulty that we have to control the continuations of these sets and make a breaking of integrability everywhere, okay. So a point can be here but the key point is these two blue lines belong to one unstable lamination, say lambda one. What I say is that if you have a point here which belongs to another unstable lamination so its unstable manifold has to cross either this roof or this other roof, I don't know, I don't know the word. And so this means that the unstable manifold of this point will intersect the stable manifold of one point of the lamination and so we can apply this argument to say that they are the same lamination. So each lamination is occupying some uniform amount of space. But there's a difficulty that I had here which is the thing I would like to tell you because it's the easiest part of the argument which is we know we are able to prove that whenever two points belong to the same stable they have this non-joint integrability, they make this drawing. But what happens if your lamination does not have a pair of points connected by a stable manifold? Okay. So essentially what I showed here is that there cannot be infinitely many WU saturated, minimal saturated sets, gamma n, sharing such that they have point x different from y in the same WSS leaf. Okay. If this happens, if you have infinitely many laminations sharing a stable leaf, you can always iterate so that the distance in the stable is more or less one and you get some uniform space that's occupied by this quasi-attractor. So only finitely many can exist, can appear in the manifold. However it could be in principle that there are, let me say it like this, but what if there exists a sequence gamma n of minimal WU saturated sets such that for every x in gamma n, okay, the intersection of the strong stable manifold of x with gamma n is just the point x. Okay. This is really a case we have to deal with. But fortunately, we are able to prove that we can use the theorem that Sylvain explained the other day saying that this property of being intersected only once by a strong manifold forces the set to be contained in a surface tangent to the CU direction. Okay. So let me recall that statement. This is done. So I recall the theorem by Bonati and Krovisie, which says, it's not the one I said yesterday, that says if gamma is partially hyperbolic and the strong stable manifold of x intersected with gamma equals x for every x in gamma, then it's not the one I said yesterday. There exists sigma surface, surface, it's not surface. The dimension EU plus one or plus the center, tangent to EC, locally invariant, tangent to EC plus EU such that sigma gamma is contained in sigma. The drawing should be something like this. You have the lamination and you know that each time you throw a stable manifold, it intersects your set only once and what they are able to prove is that this allows you to construct a surface here, sigma which contains the lamination. So the dynamics of this lamination lives in a sub manifold of dimension EU plus EC. Okay. So now the problem is you may have an accumulation of these things, of these surfaces which are getting together, the stable manifolds are missing each other and so they might not intersect. Okay. We could have a sequence of surfaces converging to a surface so that the stable manifolds start to miss each other and they don't intersect. So what we are going to do here is to do a kind of weird argument an argument which is different from what we've been doing to show that if this is the case then there are at least some points in the lamination which have very big stable manifold in the center direction. Okay. And this will imply that as you throw a stable manifold here, it will have to intersect the stable manifold of another one and you can apply again this argument. So to finish, let me explain this argument. So what I want to show is that proposition given F in partially hyperbolic with center dimension 1, there exists delta bigger than zero such that if say gamma is a minimal WU, U-saturated set, then there exists a point Y in gamma whose stable manifold has size delta in the center dimension of the center. So essentially what it says is that we can apply the argument of hyperbolic set because now if we have the strong stable direction where the manifold has a uniform size and if in the center direction we have points where the stable manifold is big, now we complete the whole dimension and we don't have the problem of not knowing the dynamics along the center. And so how do we prove this? This is proving two steps. The first step is to show that any, there exists say epsilon bigger than zero such that or let's call it H, a uniform constant H such that any WU-saturated set has entropy larger than H. This argument does not use the fact that lambda is contained in a surface. It just uses that the set is saturated by unstable manifolds. So the proof of this is very simple it's related with the inequalities one knows about entropy and what are called Lyapunov exponents. So proof is the following. So in the unstable direction we are expanding by a certain amount by a certain factor say 2. Each time we iterate an unstable manifold we get the double of size. But the set is saturated by unstable manifold. So if we take a small arc in the unstable manifold we iterate once and we can choose here two points which get separated in one iterate. But now inside each one of these we iterate again and we have four of such arcs. And so we have here four points which are epsilon N such that separated like in the definition of entropy. And as we continue to do this you might say oh okay but this unstable manifold is getting back together. But before it comes back together it has to separate somewhere and so we get an exponential growth, a uniform exponential growth of different orbits giving this estimate on the entropy. And now the final argument is to use the fact that we are contained in a surface to try to do the inverse argument. This is called Ruel's inequality. I think that's called Ruel's inequality which says that for a given invariant measure I haven't spoke so much about this but the entropy of the measure is smaller or equal than the integral of the unstable Lyapunov exponent or something like that. In a certain sense what this says is that if you have a certain amount of entropy you need at least the same amount of expansion. But now lambda is contained in a surface tangent to EU plus EC. Okay so the expansion we are seeing it here. We used the expansion here to see that lambda had some uniform entropy. But now if we use that lambda is contained in the surface and we apply the same argument, this argument to the inverse of F, okay. So for F minus one one has to see expansion in the EC direction. You apply this to F minus one. You have that gamma is contained in a surface here so the expansion cannot appear in the unstable manifold because for the inverse it's contracting and so it has to appear here in the center direction. And this is exactly what we want. We want some points which are contracting for F in the center direction. And so this was what I wanted to say. Thank you.