 Okay, welcome to this third lecture on my course on market petitions and young towers in dynamical system. The first thing I would like to tell you that the notes change at every class so I have a new link for today, which is this one. You simply have to change the two to three in the comparison to the previous one. And in the next day there will be a new set of slides. So we have in the first two classes we have seen this first two sections in this table of contents. So we have seen made a brief introduction to physical measures and SRB measures and the key of correlations. And then we have seen this gives mark of maps, which are useful when we induce, given a dynamical system, still for for the endomorphism for for non invertible systems, we induce in some region of the face space. And if the induced map has some good properties and properties are those that define gives mark of maps. Then we have some good measures there with invariant measures, and they can, they give rise to good measures for the original dynamics. In particular, when we could see the smooth dynamical systems and the reference measure for the gives mark of map being the back measure, then they give rise to two SRB measures or absolutely continuous measures with respect to the back. Then we use tower extension tower extension to do just to care of correlations. And we apply that to the family of intermediate maps into situations, one in the interval map the other one in the circle map. Essentially, the results are the same techniques are similar. The only difference is that in secret in second case we need to induce twice to get the gives mark of map, but essentially, it's the same thing. The good thing of the second one is that it allows us to introduce this solenoid with intermittency today. I'm sorry. I click there. It's not very wise to click, because it's a clickable link. So, today we are going to see this solenoid with intermittency as an application of the results we are going to see. So today, the situation is essentially the same of the second section. So now for for maps with contracting directions for the film often vertical maps with contracting directions for that I'm going to start introducing special sets, and with some good structure and I'll call these structures I'll call it young structures. And then they have associated to these young structures there are return map or recurrence time and induced dynamics with risk associated to the original dynamics. And the idea for those structures is that when you so they are made of stable and unstable discs. And when you collapse through the stable discs, you obtain for the induced map you and you obtain an induced map of the gives mark of type. So, we will also consider associated to it a tower extension, and a quotient of that tower and using the results from the, and the morphism case will be able to deduce the care of correlations in the diffior morphism case. Let me start introducing this young structures. So the setting is the following, we have a remaining manifold that I'll call M. So highlighting NF a diffior morphism onto the image. We say that the compact set landing and has a product structure. If the set can can be obtained as the intersection of two families of discs, a family of stable discs, and a family of unstable discs. The set is precisely so you consider the unions of those stable and unstable discs. And the set is precisely the points that are in the intersection. So this is the set having a product so so you have here picture. And there are some conditions for it well the dimensions. The dimensions is the dimension of the manifold. And each stable and unstable this they meet in exactly in one point. So, this is kind of product structure for hyperbolic attractors that you know the product structure exists. So given a point we will use gamma s to of that point x to denote the stable this containing it and gamma you for the unstable this containing it. We will use holonomies that's what I introduced here is this theta transformations. So given to unstable discs that's what we have here gamma and gamma prime. We can slide along the stable affiliation from one to the other. So this is what I call theta gamma gamma prime. And we can slide from any point in the set with the product structure to a fixed leaf. Of course, the points intersecting lambda in the same way so you consider the unstable leaf to that point. And you consider the previous transformation from the unstable leaf to the stable. We say that the set with the product structure I call it hyperbolic I think well here appears the word hyperbolic well you can remove it's not so important to put it here is measurable if the if the maps are measurable for all leaves gamma and gamma prime. An important concept is the this notion that I introduced now that the s subsets and you subsets. We say that the subset of lambda is an S subset. If it that's the first picture here. If that subset can be obtained by intersection of the of all leaves all unstable discs that define lambda and a subset of stable leaves that's what we have so here in blue. We have all leaves that define lambda. And in red we have a subset of the stable discs that define lambda so lambda zero is the intersection points here of the reds and blues. And similarly for you so so this is an S subset, a user set is defined similarly simply considering a subset of the unstable discs, and the all stable discs. So let us now define what I mean by a young structure. So assume that we have a set lambda with a measurable product structure, and we say that it has a young structure if the following five conditions, I'm going to introduce now. So this is the, the whole for this set plan. This conditions appear that originally, not necessarily in the way I'm going to present them now. Now it's simplified. So they are simpler than the original ones in the work of the end from 98 the paper in honor. The improvements in these conditions carried out by well in a paper of mine with being able, we, we need to make some, make some simplifications having in mind some example we would like to apply. And recently, Koropanov, Koslov and Melbourne. They also made some some simplifications. I will tell you what are the same. Well, some simplifications are merely in terms of the statements. The state, the conditions of young imply some other conditions and those that are implied in some sense are simpler than those that young stated as her condition so I think it's, it's better to put the, the simpler conditions that are implied by, by the original ones. But there's a simplification which is actually a strong simplification, which deserves to be included here and I will tell you in a minute what it is. So my first condition is what I call the mark of condition. And it's close to the mark of condition in hyperbolic dynamics. So we say that this mark of property holds, if there are pairwise the joint as subsets of the set lambda. So the subsets are made, I obtained intersecting as subsets of the stable disks and intersecting with the unstable disks. That's the picture here. So such that these two properties hold. Well, when you consider the union of all subsets, that's what I asked subsets, they essentially give everything in the unstable disks. I say essentially because this is in terms of the labor, labor measure and gamma. I don't know if I said that I forgot to say that but m gamma is the labor measure, the back measure on on the unstable disk gamma. And so this says that when you consider the union of all stable of all as subsets. This gives a new intersect with any unstable disk. This gives essentially all the points in in lamp. And also, it's an important point is that the measure of lambda intersecting gamma is positive. So this is another important property. The second one is is is, I would say the usual mark of property, but not in every rate of the dynamics improper iterates of the dynamics. So for each set. We choose. There is some iterate, such that the image under that iterate of that you set. So I'm taking as subsets and their images are you subsets so they they in the unstable direction they feel it all so they feel stable unstable news. So that's that's the tendency when when you iterate the stable has tendency to to contract. That's what happens here to the reds, and then stable has a tendency to expand. I say the tendency because he is not uniformly hyperbolic than and so non uniform in the applications. So, in the future, this we've eventually will occur. And so this is the standard condition of mark of property, but the only difference here is that we have these only in the return times. Okay, so the stable stable if so this happens in a strong sense. The image of an asset set is a user set in a strong sense, in the sense that the stable leaf goes inside a stable leaf, and an unstable leaf contains the respective unstable. And the recurrence time and the return map so this allows us to introduce a return map. Since these goals. These goals from from lambda to lambda, every every lambda I sent into a subset of lambda. And so we can introduce a return map from lambda to lambda, simply considering on each lambda I the respective power, and also the recurrence time. Similarly to what we have done for in the case of gives mark of maps, we can introduce a separation time. So it's the meaning in the smallest and for which they lie in distinct elements of this. This is a mode zero on on leaves mode zero partition so I will refer to this as a partition. Okay, so this is the standard definition that we use before. So we can use separation terms as before when you consider a power beta some number between zero and one and beta to s why can as of xy can be interpreted as the distance of the two points. Okay, and so this is the first condition let's go to the other four conditions. The first one is is where the big difference to do young presentation of her structures is where the big difference appears. Young consider the condition why to for well she didn't call it why but I call it why condition similar to why to is that the stable disks. They when it's rated forward by the original dynamics, they are contracting uniform in the future. In terms of some applications that we have, this is a very strong condition. So, and thinking of what is done it would be reasonable to consider a weaker form of contraction for the unstable discs for the stable disk I'm sorry. So that's what I'm talking about now. And the condition is this one. So we have contraction see that I'm taking this constant see and beta our uniform constants, we take from the in the very beginning. And so, and the condition is that when we return, we have the good contraction so this is contracting by uniform cost. And in the middle. This is put here essentially because of systems where we can have a single irate is and so, in general, this is, this is not a big problem so this is the condition that in the middle. We don't expand too much so the Tennessee unstable leaps is to contract so this is not much of a restriction, but we have to put this condition, if we want to consider very general systems that this needs to be the needs to appear as an assumption. Okay, so this is the contraction on stable disks, and we have similar properties for the unstable disks, and we have. So this is the expansion on the stable disks. So the expansion is stated as usual, in terms of when we go, we go to the image and when we look back, we have contraction so that's this condition here. And also a similar to the second part of the previous property, we don't. It's not very wild in the in the middle of the iterates to the return. Okay, so these are the expansion and contraction on stable and unstable disks. So this is, let me remark that this, this is only the contraction and expansions is only at the return times. The third property here in this in this page is absolute absolute continuity of the stable foliation that here we have those allotment maps we can state it very easily. So, saying that given any two unstable disks, we have this allotment map that I have introduced for sliding along stable disks from one unstable disk to the other. And this condition is saying that the push forward of the low back measure above is well above or below depends on where you represent the disks gamma gamma prime but the push forward of the low back measure on gamma is the loop is absolutely continuous with respect to low back measure on gamma prime. So this is the usual absolute continuity property, but we need some regularity of this, how long it is this is an important property that we need to impose. And the regularity is this one so since this push forward is absolutely continuous with respect to the back measure on the second disk, we can consider the density of this of the push forward with respect to the measure of the measure on that leaf. And the density needs to have these good properties. So the first one is well it's bound uniformly bounded from above and below. And the second one is some regularity of the logarithm of the density. And this is a kind of as before in the whole, the gives mark of situation, when we consider the Jacobians are related to these densities. If you think of these as being the distance the beta to the separation time so we are imposing some type of holder condition to the Jacobi here to the density of the push forward with respect to the measure on the leaf. So this is the the continuity property. And yeah. Payman asked the question is he wants to know if the distance that you consider is this distance on the leaf. Yes, it's always the distance on the leaf. Yes. Okay, thank you. Even though in application doesn't make much difference, because curvature of those disks is very regular and so it's essentially the, the, I'm sorry, here. No, here the distance is the distance in the manifold. But as I said, in terms of applications, these disks are very regular in terms of a curvature. And so it's not important if it's the distance in the manifold or in the disks. Actually you make no assumption on the on the curvature of this stable and unstable discs. Yes, there is no assumption here. Okay. So when we build them we have, we need some some control on curvatures for otherwise we don't have these properties. So, in principle, I don't need this but in practice then I will update situations where it's in the disk is the same. Okay. And the third one is the is a regularity of the Jacobian so this is the Jacobian of the return. So the return map was from one unstable leaf to the other unstable leaf. So here we are considering is from the very beginning we are considering smooth maps. So this, this Jacobian exists. And so we, so TX here is the is the tangent space that unstable disk. So comparing the logarithm of the Jacobian at two points, we have this property. So again, it's a kind of a holder condition type. Okay. So this is the regularity we need for the Jacobian of the return map. Actually, as we had before in the Gibbs Markov situation so this is parallel. The only thing that we didn't have there was the contrast of why to it's make it makes no sense in that case, and why for also makes no sense because in the Gibbs Markov situation, we don't have stable the other properties are similar. Okay, so we say that the end structure has integral recurrence times. If the integral of the recurrence time in. Well, it's for some and by the by these absolute continuity property then necessarily for all, we have this integral integrability condition. This is this is not asking in the definition. In many cases, the unstable discs are totally contained in lambda. This is the case for instance when we apply to partially hyperbolic attractors. This is not the case in the construction of young and benefits and young for hand on attractive. They've even obtained the property of the discs contained in in in the, in the, in the young structure lamb. These are contained in the handle attractor but the structure is a subset of the handle attractive. In such case, this is just for one of the results I want to talk for for an application I have in mind. I will refer to this as a full young structure so full young structure means the discs that define the young structure, the unstable discs, they are contained in the end structure that in the set with the young structure. Can I make another question. Yeah, of course. So is this why five property a sort of bounded distortion, just for the return times. Yes, it's bounded distortion. Yeah. Okay. Well, actually, I call it gives, well, first because gives properties close to this for the special, this special observable logo of the Jacobian. I, I, I usually come bounded distortion for iterates, when you try it and in the foreign iterates, and you usually have that property. So when I consider it rates by the return time and so I will refer to that as bounded distortion but of course this is a bounded distortion property in that sense. Thanks. Okay, well, since the first case for an amorphous I called it gives mark of. I would like to stress something related to gives and what is related to gives is the bounded distortion property. So, now I will state several results which are parallel to results we obtained in the endomorphism situation. So the first one is for the return map for for the return map if we have a set with the young structure and we have the corresponding return map, then it has a unique ergodic SRB measure. The SRB measure here I mean simply a measure so the return map is for for the structure we have the unstable disks, the stable disk that geometric very regular picture. And so, SRB measure means a measure that leaves on lambda, and the, the, the conditionals on the unstable disks are absolutely continuous with respect to the conditionals. Okay, so it makes sense. Formerly introduced SRB measure for smooth dynamical system. This FR is not smooth but there is a smooth structure behind it to define it. And so we can talk about the continuity, the absolute continuity of the conditionals in those unstable leaves. Moreover, the densities of the conditionals of these SRB measure, they are, of course, by definition they are they are absolutely continuous, but more than that the densities are bounded from above and below by constant. Well, the situation is, as I said, similar to the situation we had before without stable directions. And the fact is that for the induced map, we have uniform estimates of expansion in this case. And so, this leads to very regular properties of the densities. And the proof is similar to theorem 2.5 is the theorem that says that when we have an induced Gibbs Markov then the Gibbs Markov map has an absolutely continuous invariant probability measure. And remember in that case I, well I didn't give you the details but I said you can see the Lebesgue measure you can see the push forwards by the Gibbs Markov map, and you can see that the time averages and have a sequence of measures, and then you can see the weak star accumulation points and you control the densities here is the same situation. The only difference is that you start with Lebesgue measure on an unstable disk. And then you consider the push forwards by the return times, by the return map, and you consider you also can see the return map is uniformly expanding. You can control the densities of these measures and you obtain uniform bounds from above and below. And so a weak star accumulation point will be the SRB measure. And again, if we have, since we have a unique ergodic SRB measure for the induced return map, then we can push it to, we can push it to the original dynamic. So if F as a young structure with integrable recurrence time and here the interability as before is important, then F as a unique SRB measure, of course, again, we cannot say, cannot stop here in the part of highlighted, is the unique that intersects that young structure. Of course, the system can have other region which does not interact with it and so there's the possibility of having other ergodic SRB measures. Giving positive weight to the young structure, that the set with the young structure is unique as before. And again, it's obtained with this formula. So you can see the push forwards, the measure for the induced system restricted to the set of points which return after time J and then you can see this sum. So it's exactly the same formula. So if a fully young structure appears here, because of applications I'm going to give you to on the next lecture, if, if the set has a fully and structure, and the fully is contained in some compact transitive set. Then the support of new coincides with this compact transitive set. Now the idea is simple because the measure is contains disks in if it's a fully and structure, the measure contains disks in the support, and then you iterate unstable disks, and then you iterate these unstable disks and by transitivity, it gives that the support of the measure necessarily is the whole set omega. These omegas in practice will be attractors for for the dynamics, transitive attractors. Okay, so this is the only point today I need this fully and structure so I recall that fully and structure means the unstable disks that define lambda they are contained in them. So you can see proofs well, they are similar of course, technically somewhat more involving them in the previous situation of gives Markov map but the ideas are the same. Okay, so the key of correlation so inducing schemes are important to obtain SRV measures. It was in the case of endomorphism, and now in the case of defumorphism, and merely using the inducing scheme we don't need that any tower tower construction are useful for other purposes for obtaining the SRV measures. It's important only the only important thing is to have this induced map with the good properties. So, the key of correlation so this is what I would like to tell you today is to give you an idea of the proof of this first theorem here. So what is the content of it so assume that we have F as a young structure with integrable recurrence time. Well if this holds then it has a unique ergodic SRV measure intersecting the set. So this is the contents of the last theorem. And if the greatest common divisor of the recurrence times is one, then we obtain the same conclusions that we have in the endomorphism case so the decay of correlations is polynomial with respect to that measure. So the recurrence times decay polynomially fast, and the decay of correlation is stretch or exponential stretch exponential or exponential if the recurrence times decay stretch exponential or exponential. Okay, so this is exactly as in the previous situation. As in the previous situation we also have a result for the case that the greatest common divisor is not one, and as before is the proof is very similar. As before, we can decompose the unique, so mu is the unique ergodic SRV measure given positive mass to the set lambda. And so we can decompose into a finite number of exact measures, and mu is precisely the mean of those measures. One measure is the push for there's a cycle one measure is the push forward the other one. And we have the decay of correlations for the respective power. And as before the number of measures can be strictly smaller than the greatest common divisor. Well, as before the here to obtain this young theorem on the cuff correlations. Again, we will use a tower construction. And again, we will have a semi conjugacy from the tower to the original tower system to the original system. And that's me conjugacy is not necessarily one to one and so it's not conjugacy. And so it may happen that two measures in the tower, give rise to only one in the original system for that reason. That's the reason why we can have less measures than than the greatest common device. So in the conclusions are seen in so for those measures we have the decay of correlations, and with the same rates point normal stretch explanation. Okay, so the idea now is to prove a theorem to give you an idea proving I would need more than four lectures, but to give you an idea of of the proof of this theorem. Okay, so, as I said, it uses a tower construction and the tower is introduced exactly in the same way as before. Since we have the induced map and the recurrence time are you can introduce so this is the same expect well I'm using I'm using different notation so here I'm using hats. This is because I'm going to, to consider caution so see that lambda the points in lambda through the points in lambda they are stable and unstable. And so there are points in lambda belonging to the stable or unstable this off another point in lamb, in particular, there are points in lamb belong to a stable this of another point in lamb. And we'll make a quotient through stable this and for that quotient that quotient map for the return map has some good properties, it will be a gives mark of gives mark of map. And for that, I will, I will consider a tower, which is of the type introduced in the previous lecture. And for that tower, I won't use the hat so essentially here the novelty is the notation is defined exactly in the same way. We introduced the tower goes up to one unit before the recurrence time and then the dynamics in the tower the levels. And we have a natural partition in the zero lab because the zero level is identified with lambda in this case. And we have there the natural partition of the lambda eyes. And so we can, can push that partition in the zero level to two partitions in the higher levels. And we obtain a global partition of the, the whole tower. I will call this global partition q q hat. And again we define the same exactly the same projection from the tower to the ambient manifold. We define the same map or defined in the same way as before. And again for the same reasons we have that it's a semi conjugacy between the this tower and the original dynamics. And this tower also has some good properties. The unique ergodic SRV measure, new hat, and this ergodic SRV measure the projection or the push forward under this pie here is precisely the unique ergodic SRV measure for the original system, for which such that gets positive, positive weight to the set length. As I said we have with the recurrent the return map, we can construct the SRV measure of the original system with the tower we can also construct an SRV measure and they coincide. So the idea is that the projection gives the measure that we constructed before. And this is, this is one of the good things of this power is that it's the SRV measure is a projection. So we have a semi conjugacy of the system. So it's not only this this property but also the projection of the good measure the tower is the good is a good measure in the original dynamics. Okay, this is again parallel to what we had before. In the endomorphism case. So the novelty appears here. So this allows. So we have the return map, and we can introduce a quotient map of the return map. So recall that the return map is defined for the points in intersections of unstable disks with the stable disks. So I'm going to define a quotient associated to a fixed unstable disk so I fix an unstable disk. And I take a point in the intersection of that unstable disk with London, and it has an image. And that image is belongs to some other unstable disk. So this image is FR of that point. And then we slide through the stable leaf through this point, FR of X, and we obtain the point in the original leaf gamma zero. Okay, so this is what I'm going to call F. So this is the quotient of the map FR. Okay, so recall that theta gamma zero is projecting to the leaf. And gamma zero, or this comes in. Well, and this map. Having in mind all properties that we have put for for the, the, the set with the young structure is not difficult to believe that this map is a Gibbs Mark of map with respect to the reference measure is naturally the on the disk and the partition. So for a Gibbs Mark of map we have a reference measure and a natural partition for that map. And with respect to the good partition what is the good partition is intersecting gamma zero with the elements of the partition associated to the young structure lamp. Okay, so it's not difficult to prove this result. It follows from the properties that we have imposed to the young structure. And this projection is actually, well, this is essentially because FR sends stable disks into stable disks by the first property, the Markov property in the young structure. And we have that this that the transformation theta zero so theta gamma zero is is the projection to the unstable this gamma zero is semi conjugacy between this new portion return map and the return map that we have. This is a good very important property. And it happens that if new is the SRV measure for FR, then the project so the push forward of new through, through this, through this projection map, which is that the whole of me is the invariant probability measures such see that since F is gives Markov map by the results we have seen in the last days, it has a unique invariant probability measure which is absolutely continuous with respect to the reference measure which is the back here. And precisely the push forward under this projection of the good SRV measure for the return map. So we have this very good relation between the measures that we construct using different methods. So, we have this gives Markov map we can associate to it as I said we can associate to it a tower as the one constructed in the second day in the second lecture. So, consider the quotient map and consider the tower map with the recurrence times which are the recurrence times associated to the return map. Notice that the recurrence times for us for this map are precisely the restrictions of the recurrence times that we had in beginning and I called it our eyes. And notice that the levels of this tower are contained in the levels in the respective levels of the tower with the stable stable directions embedded in the structure. This is because we made a quotient of stable it so what remains is an unstable it and it's contained in the respective unstable it and so, and the tower is constructed in a similar way so this happens. And it also happens that he is the restriction of T hat to, to doubt. And moreover, we also have a natural projection. So, is given by the projection from one tower to the other is given by the projection that we had before in the from the set lambda to the lambda intersection with gamma zero. And so, see the towers are essentially in high levels copies of lambda. And so what we are doing here is in the second coordinate we don't do anything second coordinate is the L here. And it's kind of simply collapse to the, the leaf gamazine, and that's it. And it's very natural to see it's very natural to obtain this, this semi conjugacy property. And moreover, one more relation which is important. If new hat is the ergodic we have seen that new hat. T hat. So the tower T hat has an ergodic SRV measure which I call a new hat. Then, the projection through the push forward under this theta transformation is the unique ergodic team very probability measure which is absolutely continuous with respect to M gamma zero. This is the one that we obtained in in our second lecture. Okay, so there's, as I said there's all the possible good relations they, they hold so this projections always project the good measures in one system, we have a several dynamical system we have the we have FR then we have the quotient, and we have the two towers associated to the FR and associated to the quotient which I call capital F here. And then we have natural projections, the Thetas and the pie, and all the good measures the SRV measures they project under the projections to the good SRV match. This is key feature that will help us. So, as I said, I'm trying to give you an idea of the proof of the decay of correlations of young theorem in this situation. So we have so let me recall some of the things that we have, making the point of the situation. So we have this semi controversy between the tower and the original dynamics here with stable directions. We have this semi controversy between the two towers. And given age stands for the set of holder continuous maps defined in a manifold. So we can consider. Maybe we can can look at this picture here so given a five here, we can consider the composition of five with pie and so we have an observable in this tower head. So that's what I call a five hat side hat and similar for five hat. Okay, so we have a unique SRV measure for T hat, we have a unique SRV measure for the original dynamics intersecting the set with the young structure, and we have a unique SRV measure or absolutely continuous with respect to the black measure of that tower, the tower there the question tower. Well by the results that we have just told you we have that this relation here. Well by one of the exercises. So we have this situation, then the decay of whenever we have exercises whenever we have a semi conjugacy. So the exercise was stated in in the endomorphic case but there's nothing specific of the endomorphic so whenever we have a semi conjugacy, and such that the measures are exactly so the projection of this measure hat is the other one, then we have the correlation term is exactly the same. So see that five hat and side hat are precisely the composition with the semi conjugacy. So this is exactly the exercise. And so we want to obtain estimates for these decay the decay of correlations with respect of f with respect to the measure. And so we pass the problem to this tower tower head so it's not the question to this one, this is the one which is semi conjugated to the original benefit is the one that is an extension, the original dynamics. So, it's enough to obtain estimates for the tower system. The idea here is to reduce to a problem on the question tower. And you could say, well, maybe this is easy because we have also a semi conjugacy between the tower and the question tower. We use what we already know for the question tower previous lecture, and then use it for the new tower and so for the original system. The problem look at this picture the problem in this picture well is a big mess here. I guess the picture is that the arrows here this arrow tops I cannot. This has some, some problems this is one. So the, the, the, this semi conjugacy here goes from the tower hat to the tower without head so these goals in the wrong direction. And then we could deduce the decay of correlation of T, knowing the decay of great. So, this is to say that T hat is an extension of T of the dynamics is an extension of T with the measure new hat is an extension of the T hat with new hat is an extension of T with new. In the wrong direction. Nevertheless, we can manage to bring the information from T to T hat. Okay, so this is the idea of what comes next. So it's not. It's not trivial, but still doable. So let's, that's my next goal is to give you an idea of the proof of this. So we are going to reduce the problem that we have here. So we have we now want to compute this, this, this correlation term, which is highlighted here. And so I'm going to reduce it in a certain way to a problem in the portion town. Well, first of all, I'm going to take some an arbitrary and and is the end that appears there and I'm going to take K of order. And I'm going to consider the refinements. So we have this tower T hat. We have the partition in the tower, the natural partition, and I'm going to consider the dynamical refinements of this partition. So that's what I call until the iterate K. So that's what I call Q hat K. And I'm going to consider a discretization. So we have this. This observable fee hats. And also I will also consider for side hats. So I define I just tell you how to do for a generic fee hat. And so it will be also be defined for side hat. Let me recall that phi inside here. They don't have the exactly the same roles. So it's not, it's not symmetric the roles. But for this definition. It's no problem. So what, how do we define so we consider an element in this refinement of order to K, see that I'm taking to K, the refinement of order to K. And then I define restricted to to the element in that refinement to K. I consider the infimum. So you consider, we consider the case images of points in Q. So we evaluate fee hat on that. And then we take the infimum overall points when X ranges in Q. So this fee hat fee, this map, Phi K, this map is constant by definition in the elements of Q. So this belongs to the two K refinement of the regional partition. So, see that the elements, the partitions is dynamical partitions that they are dynamical partitions obtained by the dynamics in the tower, the tower the partition the tower makes no partition in stable it so we can think of this function also as being defined in the quotient in the quotient in the domain of the quotient power, because the infimum is is constantly does not depend on the leaf so well, so let's start the reduction of the problem. Well, this is a straightforward calculation takes one or two pages, but there's no, it has no significant ideas behind it, and just straightforward calculation, we can prove the following. The correlation with respect to the tower hat is what we want to estimate. And I can, we can prove that it's bounded by the correlation with respect to the quotient tower. And the good measure there. Well, we pay a price is not anymore this, the same observables is this Phi K and Psi K. Okay, but they are discretization, see that this domains when we we iterate so when you consider the refinements of the partition. They are smaller and smaller so the discretization is close to the observable itself so it's not a big difference. And then it appears here to, well appears the zero knobs, since we are taking Phi and Psi. So both holder continuous, then the zero norm is finite. So this is the soup norm, and we obtain two L one norm terms so L one norm here is with respect to the probability measure in the tower. So, and essentially is, is, is Psi hat evaluated on the case image minus the Psi hat. So this is this this will be ideas that is that this will be small, and it's not hard to believe that since look at the definition of Phi hat or Psi hat. When you iterate a lot then see that we are taking iterates much higher than the iterate we much high well is double than the iterates we take here so it's still there's there's some refinement there. And so we can we can control this L one norm term so we have three terms that we need to control now. Correlation term and the L one norm term so that's the way I will refer to them. So we first consider the correlation term. And this is the point where we are going to reduce the problem to a problem in the, the tower the quotient. Suppose that Phi K is not zero well if Phi K is zero then the correlation term there's nothing to be done because the correlation term is zero so see that by definition of the correlation term whenever the observable Phi is zero then the correlation term is zero. So we can assume that Phi K is not zero, not equally zero, and we can we consider this Phi K star. What was Phi K star this has been introduced in in the previous day. So this is the way is a step towards passing observables in the original dynamics is the non invertible case to observables in a certain family of the, the tower associated to that system, essentially is adding a constant to this to Phi K so how do we observe it, how do we construct it, we add a constant in such a way that Phi K becomes strictly positive, and then we divide by the integral with respect to the measure new, in order that it has integral one. Okay, so that's what I'm going to I'm not going to show you what is in 11, because then I will take a lot of time to get to this point here. But if you go there is essentially what we did. And then we consider the probability measure I call it lambda K probability measure whose density with respect to the good measure in the in the quotient tower is precisely this function here. Well, it follows. Well, we introduced this star, this new function star precisely to obtain this result. That's the content of this lambda 220. So we have seen that we can reduce the correlation problem in that case but the objects are exactly the same to a problem on the way the push forwards of these initial density is are approaching on the speed at which they are approaching the measure the measure the invariant measure for the tower. So this this term here is precisely the the total variation of this sign measure which is the difference of the two. Okay, well, you could tell me well now simply show that so simply apply that what we had because we made some estimates on these terms, there's a theorem saying that if this term, I might have a highlighting here, the case is much faster than the decay of correlations is polynomial especially fast exponential up. The problem is that this function, the density which I call of lambda K star is not sufficiently regular so the density is precisely five K star is not sufficiently regular. So we have to make it more regular. And for that, we are going to iterate under the tower system to K times. The idea is that is as for uniformly hyperbolic dynamics, iterating densities transfer operators and things like that, densities provided a line the good spaces, they become more regular. And here is the same. So, I consider the that measure whose density if I try to estimate estimate is not sufficiently regular and sufficiently regular is to lie in the space associated to the this limit to 20 and the space I'm going to highlight it now is the good space for the observables in the quotient power. So consider a sufficiently a sufficiently good number of iterations or push forwards of this measure, and then consider the density of this new measure so see that there was a star before now I get rid of the star iterating under the two case top. Now, and five is the density so see that I have this by definition and this is this and this is this, where row is the density of the internet measure with respect to the reference match. It's easy to see from what I've defined before I'm not going to check in details but we can easily see that this is true. And so this, well, the important thing now is to the important thing is now is to see that let me just put this here. So the important thing is that with this we reduce the problem above on correlation to this problem here. And now, as I said, these lambda k has a density which is good. So I'm see such that 5k 5k is precisely the density. So that's what I call the density I call it 5k 5k lies in the good space. And there's a uniform see that there's a uniform control, as I said, in the result where I've given you the. So now it's not here anymore so now it's here. So there is a result saying that if the recurrence times the case polynomially fast, then these the case problem and we fast. If the recurrence times the case financially fast, then these the case exponentially fast. And I observed in the end of that result that the the constants in the conclusion they only depend on on bounds for the constant associated to the observables. And I observed that well if you go there to the theorem, the observation is that it only depends on these constants and in fact on upper bounds for for this constant. And since we have can prove that we have this uniform upper bound, then we can prove now notice that k is of order I've chosen it of order and over four so and minus 2k is of order and over two so is of order and okay. So we can apply that result and obtain the estimate the good estimates for this quantity here. So the drama could be in the result they are calling the conclusions they are constants, which depend on the constants depend on the observable. The problem could be the constants for which we bound this total variation here, the constants could depend on k that would be a big problem, but they do not depend so that that's the message because we have this uniform bound for for this. The constant associated to the fun functions in this space okay that the notation I used. And so this is the way we control the correlation term in in in this proposition. Let me now give you a brief idea of on how to control the L one on terms for the so let me recall what. So these are the L one on terms this one and this one. So they are the same time so the functions find side they are. They are holder continuous functions and hold continuity plays an important role at this level. And so we are going to control those L one terms. So, given a point in that in the tower. So let's be K of that point be the number of times that up to time k, the point returns to the base level to the zero level. Okay, so BK is the number of times it goes down to the zero level. And going down is good because we have, when you go down we go down through hyperbolic map where we have contraction in the stable uniform contraction the stable leaf leaves uniform expansion in the stable is for the return to the base so we are going to put some estimates in terms, depending on the number of visits to the zero level. Well, see that, if we take any two points in an element of the second to K refinement or dynamical condition. Then we have by definition because the points have the same itineraries up to to K, after to K we don't know but since we refine to K times. And consider points in that refinement. If you consider until iterate K, they always visit the same elements in the partition. And so we have this, and in particular this is what matters in partition BK of y is equal to BK events. And so, if we take a point in an element of the partition. Given you have given two points. So there. So the idea is that there is a point so we have this product structure. So given a point in that element. There is a point such that the intersection of the stable lead disk of one point with the unstable disk of the other is another point in that partition. So this is according to the way we have defined this object. So this is the local product structure. We can control the distance this distance, as you will see this distance will be natural. Why is it natural so let's look. So we want to control these terms here down. Because integrating this with respect to new hat, we obtain precisely the L1 norm we want to estimate and see that. So this is a five hat. Well, five hat is precisely five composed with pi. And so we have here is five composed with pi composed with this tea hat cake. And so we use the fact that five is holder continuous. And here you say well but here you have five, five K. According to the, the, how we have defined it is still something of this type, because we have defined precisely as the minimum overall points X in Q, according to what we have here. So it's so five K is of X is again five hat tea hat K of some Y. Okay. And so using the fact that five is holder we pass this to estimate this distance here above. Okay, so essentially only using the definitions of these objects five K and five hat. And so this distance is bounded by the triangle inequality. So we can compare the distance of these two points. So it's the same type of distance but now with X and Z and X and Z they belong in the same unstable disk and the idea is to control these difference here using the fact that we know that they have any two points so BK, Y and BK, X, they are equal. And so we can control. So every time so the idea is that every time we come to the ground level we have expansion in the state in the stable direction and contraction and just is the other way around. We have expansion in the unstable direction in contracts contraction in the stable direction. And so we can control this by this quantity here is essentially so Sigma is the rate of essentially is the rate of expansion and contraction. Okay, so we can control that essentially estimated the number of visits. And then we have that, well, since BK is constant and stable on stable disks. We can easily see that integrating so we want to estimate the L1 norm we want to estimate to integrate these with respect to new hats. Or now using this estimate integrating this with respect to the new hat. Well, we have here, why do we have to hear this TK term is because of the unstable direction because the contraction is put in terms of the of the images under. We need to take TK iterates. And so is the contraction is stated in terms of the images and so that's why this appears here. But this is not the problem because we are going to integrate with respect to a measure new hat, which is invariant with T hat. So integrating comp. So we divide this the integral into two integrals and they are equal because we are integrated with with respect to composition with the dynamics for which the measure is invariant. And this, this is to, to say that integrating with respect to new hat is the same of integrating with respect to new, because this BK according to the way we defined. We are constant on stable disks and so since the measure is one measure is the projected through the autonomy on stable disks to the other one we obtained this. And the last last estimate is obtained. So for that I need to introduce these so see that this is the sequence of of recurrence times. So see that F here is is the quotient dynamics so I'm considered the first recurrence time and then I consider the image through the quotient dynamics so through the return map, and then I can see the next one so this is the sequence of consecutive for a point of consecutive consecutive return times. And we can. So, as I told you, we need to know we need to estimate this integral, and we can estimate this integral in terms of these measures here, and it takes in particular into account this air RL here. And the point now so this is somewhat involving calculation. And we can. So we obtained this. And so we have passed the problem on the other one or two to this thing here. And it's not difficult to see that in polynomial. In the polynomial case, we obtain a polynomial estimate, losing one unit, even though we have here RL it's no problem, we obtain a polynomial estimate, you losing one unit in the in the exponent of the polynomial rate. And this exponential exponential case we don't, we don't obtain. We don't lose anything. This is clear in the first term here. In the first term of course, if it's exponential then the tail of the series is still exponential. For the other one is not so clear but it's, it's still still can be proved. So this is the way we obtain the control on the L1 norm terms. So, now I for for this lecture, I finish with an example of application of these results. And in the next day, I plan to obtain some more applications to partially hyperbolic system. So, here I will give you a simple example where we can see some of the ingredients that appear in the general situation of partially hyperbolic systems. But here is easier to handle because the dynamics is very well known. And what is this, this, so the dynamics is what I call a solenoid with intermittency. So this, this is due to my, this appeared in a paper by myself and being able in 2008 where, what is the dynamics, we essentially go to the solenoid map, which I introduced in the very beginning of these lectures. And in the solenoid map I describe it as, as a skew product in S1 times disk, and in the S1 direction I took 2x mode one. Well, we can take any, any expanding map in the S1 direction, and in the disk direction is a uniform contract. So essentially we obtain the, the usual solenoid map is a different morphine from, from the solid torus into the image. And this new system is essentially you go to the solenoid map, and you replace. The idea is to replace in the, the dynamics in the base so we can think of the solid torus as a product and the solenoid as a skew product in this product system, and we replace the expanding map by one of those maps that I consider the intermediate circle map. So it's, it's, it's, for instance, go to 2x mode one and in the fixed point, you, you, you make it tangent to the diagonal. Okay, so this is the intermittent circle map, and we obtain some results for this intermittent circle. So this is what I call solenoid with intermittency. So the intermittency appears in the base dynamics, thinking of the solenoid map as a skew product. See that when, so for that map, intermittent circle map, we have seen that it is, so I cannot highlight this part here, I don't know why but no problem. It appears so actually it's a family of maps that we introduced with some exponent alpha, and we have seen that if alpha is between zero and one, then the map, the little f and then the capital F now, the map is only C1 plus alpha, if alpha is smaller than one. It's not, it's not C2. If alpha is greater or equal to one than the map is C2. So, the result, the result is exactly the same as for the base dynamic, but before the result, let me say that this solenoid maps, again as an attractor that I call omega, and it's obtained precisely, computing, so M here is the solid porous. So, again, is obtaining intersection of the forward images of the solid porous, and so the geometrically is the same as pictures of the solenoid, so when you cut you obtain a counter set of lines passing through the points. Well, it's easy to see that we have a fixed point for the base dynamics corresponding to the zero. And so since we have contracts, so the fixed point gives rise to a fixed disk in the solid porous, and it's a contraction so there is a fixed point there and in fact it is unique. So, let me consider this point, I call it P0, we can compute this, this fixed point, and well this is for 2x mode one. We can see that, well, due to the direction in S1, the little f as derivative one, so it has an eigenvalue one, so this omega is not a hyperbolic set. Okay, so this is just to be sure that we are not reproducing results that we already know in the hyperbolic set. Well, and the conclusions are exactly the same that we had for the intermittent map in the circle. So if alpha is greater or equal to one, then the direct measure now at this point here is a physical measure and its base in covers almost all the solid porous. And here is the Lebesgue measure in the solid porous. So that's the situation for which the physical measure is very simple is the direct measure. So this is for alpha greater or equal to one. For alpha smaller than one, again as in the previous situation of the circle map, the dynamics has a unique ergodic SRB measure mu. Moreover, it is exact. The support coincides with the whole attractor and the basin of this SRB measure, which is a physical measure, contains the solid porous almost almost every point in the solid porous. And moreover, the decay of correlations is of this order. So this is the same estimates of the same type of conclusion that we had in the circle. The idea is to again to induce now obtaining a set with a young structure, but the recurrence times will have exactly the same estimates that of the base dynamics, little f. And so you have the same estimates that we had before. And so that's why it, it yields this decay of correlation. Okay. And that's the idea of what comes next. So to induce and deduce. Here we have a semi-conjuricy between the circle dynamics and the torus dynamics of the circle and as little f and capital F, which is the projection of the first coordinate. It's very easy to see that. Well, it's a school product. So this happens. And so we have this. Let me give you a brief fight. Well, I'll give you the proof of the second night. The second item is the simple one. So we already, we already know that the DRAC measure is the physical measure for the base dynamics. The basin, as we have seen in the previous situation is almost so M1 here is the Lebesgue measure in the circle map. And so the basin covers almost all S1. So this is what we have seen. Well, it follows that the preimage by the projection of a full measure set in the circle necessarily has full measure now with the Lebesgue measure in the solid torus. And this is because the, the, the projects is the projects along the, the disks which are very regular. And so we obtain this full measure. So given any point here so this is the point such that the projection we know it falls into B and B is a set of is the set of points for which you know that we push forwards. The points are converging to the DRAC measure and Delta zero. So we take a point in the preimage and consider also the set that these push forward measures of the DRAC measure on that point. So we want to prove that these converges in the week start apology to the DRAC measure on P zero. Okay, so this is the definition of physical measure. As I said, so we are proving in the second case, we are proving that P zero the DRAC measure at P zero is the physical measure for the system. So we want to prove that this sequence of measures for any P and this has total a bag measure in the solid torus for any P in this set. This accumulates in the week start apology in the DRAC measure of P zero. So this is given in this simple way. So any week so what we actually are going to prove is that any week star accumulation point is delta P zero. Well if any week star accumulation point of a certain sequence is a certain measure that necessarily it converges to that that measure. See that any sequence here has accumulation points and we start to pull it. So, the proof is very simple so assume that it converges to a certain measure. And since the push forward is continuous we have that the push forward of this sequence converges to the push forward of the accumulating the measure it is accumulated. Yeah, the push forwards are linear and using so 34 is this fact that it by semi conjugates, we can see that. So this is so the measure the sequence is defined here so I'm applying pie star so I can take pie star inside. So I can see that pie star of this is by the semi conjugacy is is pie of the image so this is the semi conjugacy is very easy to see this property, but, and then this is equal. Sorry, here is not the semi conjugacy so here's is a simple feature of push forwards and DRAC measure the semi conjugacy is using the next, the next step. We obtain that this is true. So, the sequence of push forwards is this sequence here. Well, and we know that pie of P belongs in B. And so this sequence is converging to delta zero. And so we obtained that easily that necessarily mu has to be equal to delta of P zero. And here we also use that in the fixed disk, every point converges to the fixed point. And so we obtained this. Okay, so the interesting case is when alpha is between zero and one. Well, in this case, the situation is much more delicate than in the in the circle map. In the circle map we have constructed that natural partition, which appears here. The natural partition was so we induced in that case, we induced in enough for the circle map, we induced first to to use the first in inducing map. And then we use a second which I did not explain how to obtain but we obtained a second induced map. And the second one could be obtained. Well, I told that it could be obtained in J one but actually it can it can be obtained in any union of the J is here is not difficult to see that part. If you go to the slides I improved it a bit to convince you that it can be obtained in any unit. So I'm going to consider a union of two days, and you may ask me why union of two days and not just one and not just J one. Well, union of two is to be sure that when I induce the the greatest common divisor of the inducing times is one. Because the inducing times. Well, according to the way we have created the induced map for for F, which I call the FS. It is related to the returns to to well now to J and zero union J and zero plus one and see that the for that we first consider the iterates for which it escapes from this region and then we consider the returns. And this points escape so in J in J and zero escape one unit less than in J and zero plus one. And so this makes that there will be points with consecutive return times and so the induce time, the greatest common divisor has to be one. So this is the only reason the other reason to obtain. Not in J one and one of these is that I want to induce in a very small region. So if any is very large this will be very small because these J's are accumulating in zero. And so, so we have an induced map FS. So there is a natural partition, I call it Q. And at least the elements is accountable partition associated to the induce maps which is as a Gibbs Mark of map, and I call them omega one omega two and so on. And, and associated to this omega C that the only as they are in S one. So now look at this second picture. So the only as they they are in S one. I will consider the points so this. So associated to, for instance, this little omega one, I will create the, the omega one, what is that is the set of points that are in the attractor the solenoid attractor and project into it so the lines here. See here on them, no, no, no curvature, but they have some curvature, and they are, they are lines here and so I consider the elements in the attractor that project down to the element in the partition. So I'm creating a partition in a certain region of the solenoid. And this, this, this omega guys will be used to construct a set with a young structure here in the union of the guys. Well, we can prove this is this example this is the advantage of this example is is easy to to manipulate and make some calculations, and we can prove that there is an invariant calm. Well, according to the expression I've presented the cone is this one here. Well, it's not so important expression. The only important thing is that for every point in the attractor. We have a cone which is invariant under the dynamics and the angle converges to zero. See this, this is clear and uniform hyperbolic context here, we have a point with derivative one but still we have this conclusion. And moreover, we, I cannot assure that I have expansion uniform expansion, but I can tell you that I have this, at least this bound on derivative for points in the cold. And, also, when I return. Well, when I return. Yes, I have expansion for points in the cold. Okay. So this is the idea that when I return I have a uniformly expanding map now for see that F now is the solenoid map. So using this we can we obtain partially hyperbolic because we still we also have a uniformly contracting direction is the direction direction of the discs. And since we have see that for contraction in the discs, we have this contraction the discs. So that's why I've taken in the discs I've taken somewhat strong contraction to compensate so this is the best we could achieve in these calculations. So this contraction is stronger than any possible contraction in this, in this direction so the fact that we have the cones and the slopes go to zero makes it possible to obtain an invariant splitting. And then in the invariant splitting. This is the estimate that we had in the cone. This view before. And so this contraction is stronger than any possible contraction here. And so this is an invariant splitting dominated the composition and with a strong, strong conclusion, a strong contraction in the other direction. This is what we call a partially hyperbolic set. There's a classical result by shoe, which says that various center unstable discs. Tangent to this center unstable direction. And they are invariant in this sense so at least the part. This is the ball be be absent is the ball. So at least the part of the. So a neighborhood of the disc that's what it says is sent into the disc in the image. And also, and this is why I'm mentioning this result is that these discs that depend continuously on the in the C1 topology, they depend continuously on the point. At some point I need this dependence of certain family of this I'm going to consider and so this is the way I can assure that they depend continuously on the points. So, well, these discs are not necessarily unique. This family of this is not necessarily unique by the result of show. And there's nothing that assure that we have expansion. So we only have expansion in the return so this will be enough to prove that the measure new. This is the measure we have in the dynamics in the in the circle. So the measure new can be lifted to a measure in in. So the measure in the circle can be lifted to a measure in in the solid torus. This is essentially using a bone argument of lifting measures for for extensions. And this measure has the support of that measure coincides with the whole attract. And we can prove that that that measure has a positive level of exponent in the in the in the center and stable direction. If it has a positive so the positive level of exponent is because it projects down to the measure, which is the good measure in the in the circle and in the circle we have a positive level of exponent so we can manage to prove it has a positive level of exponent above. And so with that we have local and stable discs. And we can prove that the local and stable disk is contained in this. This, which are also discs obtained by by by shoe. This this set has said obtained by shoe is essentially to, to, to assure that these most local and stable discs, they also depend continuously. So we have these local and stable discs. And so, using this disc, this discs, I will be able to construct a young structure. So what, how do we do that. Well, the, the family of stable discs will be easy so is the the the discs that we have in the solid torus contracting uniform. We need to construct a structure as as the one I mentioned in the beginning of this lecture. I need to build a family, which I call gamma you off unstable discs, and such that we obtain in the intersection of the unstable discs with the stable disc that we have, we obtain a set with a young structure. First, we take and sufficiently large, such that the projection of this of this unstable is not a center unstable disk by shoe contains this element here. This is an unstable this so I want to prove what I want to prove is that there is a center is there is an unstable disk of passing that projects to this region here. That's what I want. Okay. The idea I'm not going to follow this explanation. The idea is that in in the base dynamics. So this disk contains a center unstable disk. And then it projects down to this interval here. This interval here, necessarily, it expands in the band base dynamic because the base dynamic is strongly expanding. At a certain moment, it becomes everything. So when we reproduce that above, it tells us that the any this guy take so any unstable this guy take gross and gross to a scale of order. I zero so I zero is when I project down. Okay, when I project down is order of I zero. So this is the idea of this life. So this. So this is a disk. So we have a disk in the solenoid, such that the projection by the system vertical projection fine is a good region for the base dynamics. So is the region where we induce in the base dynamics. So I will use this this this will be so, as I said, there will be an inductive construction to build the set of unstable leaves. Inductive construction starts. So there will be a certain sequence of of gamma gamma and the very first is precisely made by this gamma zero. And how do we obtain the assuming we have one construct how do we obtain the next one. Well, let's look at the first step. The first step so I have this gamma zero and recall that I have so gamma zero projects down to I zero in I zero I have the partition in two elements omega I will go one of the two the little omegas and the pi minus one of these. When I intersect with the soul with the solenoid pi minus one of these little omegas give rise to the big omegas. And so I have a partition here on the set which is I zero times the solid well is the part. It's not the solid torus is the part in the attractor that projects down to I zero. I have a natural partition in particular in this gamma zero is the intersection of the capital lambdas omegas with the leaf gamma zero. And I know that down the corresponding part in the corresponding so associated to omega one I have a time as one is the return time. So the corresponding dynamics down since the little omega one to the whole I zero. So it means above that the corresponding part here intersecting gamma zero in S one iterates comes to a disk that projects down to I zero. Okay. And so these will be another element in the family of unstable discs. See this is in some sense a minimal construction because if we want to have the young structure with the same inducing times with some recurrence times that we had before for the base dynamics. The image under S one of this first element here which is this new disk here necessarily needs to be there. This is one of the requirements of the young structure. And so we have this first one. And we need for the other ones, the other image discs. Okay. And so that's the way from gamma zero I obtained gamma one. So considering the partition on the leaf gamma zero. This is the natural partition induced by the partition in the base dynamics, and then considering the respective images of those elements that I obtained gamma zero. And so I obtained the new leaves. And, well, we proceeded inductively and this is the way that we describe the next steps. Consider the union of all these leaves. The problem is that the union cannot may not be a compact set. So, and the union of all these discs is here. Okay, in the elements that we construct in this way. And then we take the closure. It's not that we have discs, but when we pass to the closure. We want the compact, the compact set lambda to be foliated by stable and unstable discs in the young structure through each point that must pass a certain unstable this can unstable this with certain properties. Taking closure can be a problem. Well, it's not a problem. In fact, we can prove that through any accumulation point that's an explanation here but I'm not going to read this through any accumulation point, we have unstable this passing through it and then stable this is still contained in lambda. See that in this way we obtain a full young structure so the unstable discs that define the young structure they are contained in the in the structure. And so that's the way we build this, this set with a young structure. And then we consider the stable family is the obvious one so you consider the discs transverse to a certain point in the circle and these are the stable discs. And so this is the last slide. In this way we can prove that it has a young structure. So, most properties come from what we know already from the base dynamics. And so many of the properties, some of them come from there, the geometric ones at least. And then we can prove that it has a full young structure with recurrence time the same of the base dynamics. And according to the choice of the recurrence time the greatest common divisor will be one. And the estimates on the on, I think here there's one unit less. So it's one over alpha minus one. So, I'm sorry, he is not the decalculation know this is correct. So this is the, the, the tail of the recurrence times to the tail of the recurrence times is this and for the decalculations we lose one unit. Okay, so we this and that's what is here so the care of correlations we lose one unit. So the measure the support coincides with the the attractor by a proposition I presented you before. And so it coincides that the solenoid attractor is still transitive as the classical one, and almost every point belongs in the basin. Well, that's an explanation here I'm not going to read it. Because in the base dynamics almost every point belongs in the basin of the measure it's lifted it so we can use this argument to prove that the almost every point belongs in the base. So that that's all for today. So in the next day I'm going to give you some ideas of applications to non uniform next pending maps and the morphine case and partially hyperbolic systems. Okay, thank you very much. Any questions. Yes, can I ask a question. Yes, please. So, wouldn't it be possible to do this, this argument for a different class of measures instead of fill it back measure. Yeah, that's, well it's related to the question of Lucas last day. Yes, provided you. Well, you need to consider conformal measures, but then you obtain partitions which are related to the measures. In all these, they are in this case will back mode zero partitions. Let me say that we'll don't have some results with, I still haven't seen an adaptation of the results in the other morphine case to the different morphine case, but there are some results where in some sense you obtain some universal and I guess using that we could obtain some something for at least conformal measures, I guess so. I didn't check details actually I didn't try to check the details but I believe that something can be done in that direction. I'm asking because yesterday he mentioned that we in the workshop next week we will talk exactly about these induces systems to prove the uniqueness of the measure of maximum entropy for Vienna match. Maybe he will use those results I haven't talked to you for a long time. We haven't updated the novelties. So maybe that that will be good if he talks about about that. At some point we had the idea of adapting but there's not, there's no time available for ideas we have. Okay. Yes, Lucas. Yes. Sorry. I mentioned you. I was here listening. Okay, so at some point I got confused and I think the confusion. I'm sorry. It's right in the beginning because these leaves that we have over there. Let me let me let me let me go to the beginning so the beginning of today. Yes, of today. This year. You should have told me that in the beginning. Yeah, but I realized I was confused just later. No, my question something something like this. These local structure that you have here. Sorry, this product structure that you have here. It's not clear for me if it's given locally at each point in. Well, it's globally land down. So land is intersections of these points. So there's a family of stable discs and a family of unstable discs with these properties. So this is usually a small region. Okay, so so these leaves. They can be dense in M for example or. What happens if you for instance iterate forward this little unstable discs. They may be happy dancing in and it depends on what is the dynamics out of this region. Yes, it may happen. Okay, but if these are local. These are local stable and local unstable discs. Okay, so they're local. So yeah, I got confused in the picture of the induced map where you your collapse. You take the quotient map. Because the picture is not it can happen that the image is very far away from the the original gamma zero. Yeah, this picture. Yes. It can happen that this image is very far away. I mean, it's still a return. So it's lambda, but it can be far away, even though from gamma zero. Right. So there's no one. Yes, it's one of the of is in one of the blue discs in the previous picture. So FR of X is in one of the blue discs. Yeah, it can be far away. Why not. But there is no autonomy to project back to gamma zero then. No, it's a point still in lambda. So if it's still in lambda, there is a point through stable leaf through it, and then a stable leaf through it. Yes, but the stable leaf through it. It's necessarily passing through gamma zero. Yes. Yes. The picture is exactly this one. So lambda is the intersection points here. It's a full product. So it's global in some sense. It's global in this local structure. Yes. Okay, okay. It's a global product in lambda. Lambda is very small, but it's a global product. Okay, okay, okay. So when you take a point here, it comes to some other point here on which there is a stable disk and unstable disk so you can, you can project down I'm not going to do but imagine you take this gamma zero as this very first one here, and it comes here so you can project down here. Okay. Okay, so. Yeah, now it's clear. Thanks. It's clear now. Yeah. Thanks. And see that. So each, maybe I didn't give a sufficient highlight to this property here so each stable disk intersects an unstable disk and in exactly one point. Okay. This is not so this is in the product structure for for hyperbolic set this doesn't happen globally in the hyperbolic so this is just local product structure. Here, if you think of lambda as our pro hyperbolic set, then this is a global product structure, but in lambda, which is a local set well, just 10. Okay. Yeah, yeah. Yeah, that's it. It answers my question. Okay, so that's why we can make a global question. Otherwise, we'll have problems. Okay, any more questions. Okay, so you have the notes available in this link. Okay, so there are more questions. See you on on Friday. Thank you. Thank you.