 Okay, so Stefano and Zé and Matos, do you think it's a good time to start? Yes. Yeah, go ahead. Okay, very good. So welcome everybody. Sorry for the late start. We are five minutes late, but no problem. So let me first you make an announcement and actually it is related to the organization of this school and workshop. So as you know, ICTP is one of its goals is to foster mathematics in developing countries of which Brazil is part of. So I discussed a lot with Stefano Luzato ways to try to make this development, especially in the region that I come from and I work, which is in the northeast of Brazil. And one of the ways that we found was exactly by making this event. And another way was trying to select people for postdoctoral positions that could work here and also have transit at ICTP. So I would like to advertise a postdoctoral position. So I'm sending here the link that is paid that is funded by Instituto Serrapilheira. And this postdoctoral position is for people in dynamics here in my university. And they will also be allowed to have an extended visit at ICTP. As a matter of fact, the link that I sent you has two jobs. One of them is in dynamics funded by Serrapilheira. And there is another one in differential geometry also funded by Serrapilheira. But I should focus that the extended visit is only intended for those applying for the dynamics position. Okay, so you can find further details in this link. So another above on top of having this postdoctoral position, we also decided to have this school and workshop on the field of non-uniform hyperboleicity, which is the field that we both work and on which dynamics is focused in a lot in the past years. So there are two important techniques in order to understand non-uniform hyperboleicity. And they are Markov partitions and young towers. Markov partitions, they were developed in the dynamical perspective in the late 60s and early 70s of last century by Sinai, Adelaide, Weiss, Ratner, Bowen, and others. And young towers, they were introduced with success by Young in the 90s and then later developed by Chernov and other people. So our idea was exactly to try to present these two techniques and their developments, their very recent developments to allow students, postdoctors, and also professors to learn these two tools and to compare them. So in the past three weeks, we had two mini courses, one given by myself and another given by Jose Alvis, which was, I presented the Markov partitions perspective and Jose Alvis presented the young towers perspective. And everything was recorded so you can find on YouTube all the videos. So this was the part of the school on the event. And now we arrive at the part of the workshop. So this workshop will finish the activities of this event. We will always meet in this week at the same time slot, which in Italy is from 2 p.m. to 4 30 p.m. in Brazil, on which many people are in the audience is from 10 p.m. to 12 30 10 a.m. Sorry, to 12 30 p.m. And we will have two talks every day. And today we will have Carlos Matheus and Yakov passing. So it is a pleasure to to introduce in Carlos Matheus, which works at CNRS and they call it technique. And Matheus will talk about young towers on non positively curved surfaces. Thank you, Matheus. Yeah, thank you, Yuri, Stefano and Zé for organizing the event and inviting me to talk here. So it was a lot of fun following the mini courses on YouTube. And again, I hope you have fun also seeing my talk. So yeah, as Yuri said, I'm going to talk about the aspect related to this course, which is young towers and the key of correlations in the specific case of non positively covered surfaces and everything that I'm going to talk about is trying to work with Yuri Liman, Ian Melbourne. So before entering into the subject, let me quickly show you the plan for the talk. So the plan is to recall you something about what is known about ergodic fury in negative curvature. So it's classical top, but it's always good to refresh your memory. Then I will start talking about non positive curvature for surfaces and what was known prior to our result, which is basically a result of constructing young towers and obtaining statistical laws in particular the key of correlations and central limit theorems for a particular class of non positively covered surface. And then in the final part of the talk, I'm going to explain the proof of the result. Not everything because there are many aspects to this proof, but at least some what we could call the main ideas. I don't know. Yeah, so that's the start. So for me, S will be always a Riemannian manifold. So finite volume always. I'm not considering it's volume here. Yeah, dimension at least two and coming with a Riemannian metric. And I'm doing that to not mix up the manifold with the phase space of the jedezik flow. So GT for me will be the jedezik flow, which is a flow taking place in the unit tangent bundle. Or cotangent bundle depends on your preference. And then just to make things clear, I'm going to denote by me deal with measure. So the lab bag measure on these Riemannian manifold, which is invariant under this flow. And I'm going to call M for the phase space of the jedezik flow. So the unit tangent bundle. So as for me, it'll be the surface. But as everybody knows, the jedezik flows take place really on the unit tangent bundle. So takes place on M. Okay. Now, what people, well, the history of analyzing ergodic properties of jedezik flows has a long history. So of course, we should pay tribute to the work of Hopf in 39. And Hopf in a similar work, he introduced what people call today the Hopf's argument to prove that the jedezik flows on surface of compact surfaces of negative curvature. They are ergodic. And as I told you, this is what people now call Hopf's argument. And the idea is that basically he studied a big of average along invariant manifolds. So he noticed that basically forward big of invariant average are constant along stable manifolds. And similarly for past big of average along unstable manifolds. And then he used a kind of Fubini streak, which was possible because the foliation, the invariant foliations are C1 in this setting. And so by combining these two arguments, he was able to prove ergodic so that functions which are invariant are constant almost everywhere. But this property of having C1 invariant foliations is very strong. And in particular, it was discovered by a Nozov that it almost never works in higher dimensions, in some sense. But in any case, several years later, so in his thesis in 69, a Nozov noticed that actually to run out the second part of the argument, you don't need the full strength of C1 in regularity. You just need to know that the foliations are absolutely continuous. So to apply this Fubini argument, just need this weaker property, which hopefully, which luckily holds in our setting. So thanks to dynamics, the foliations which are usually just holder, they are holder plus this extra absolute continuity property, which is sufficient to conclude. So yeah, thanks to dynamics. We don't have arbitrary holder foliations, but a special kind of foliations. And interestingly enough, another regularity discussion come up in our setting when we discussed our work with Iurian member. But it's not exactly this kind of regularity, but it's interesting that there is always a kind of regularity discussion that we should make when talking about these statistical properties. But this is the context in negative curvature. Next, once we know that the flows are ergodic and actually mixing, it's not hard to see, you can ask about the speed of mixing. So in other words, you look at the correlation functions. So for instance, you could think of these functions here, phi and psi to be characteristic functions. And then this would be just the probability that mu, the probability that the intersection of a and gtb minus the probability of a and probability of b goes to zero. So basically how fast these things correlate when you apply the flow to one of the sets. And these speed well, it goes to zero because of the mixing property. And then you can ask for rates of decay, at least if you pick observables which are smooth. I mean, it's known that if you pick arbitrary observables, then even the best chaotic systems have slow decay for arbitrary observables. We can always cook up very tricky observables making this very slow. But if you look at, say, C1 or C infinity observables, then there is a hope to pick up some speed here. And actually, you can ask, well, why you are interested in these questions of the key of correlation as well. It's a natural mathematical questions once you see mixing. But not only that, it has applications outside dynamics. So in particular, it was used by asking McMullen in the 90s to count integral points in algebraic varieties, special types of algebraic varieties. So in other words, you can use exponential mixing to count things. It was used in 98. So asking McMullen, I think it was from 93. Klein-Bock-Margulies is from 98. They also used that to solve problems of, I mean, diophantine problems, trying to pick up diophantine vectors inside analytic manifolds and the method to pick up these diophantine properties along manifolds was to use exponential mixing and distribution properties. And more recently, in 2012, if I remember correctly, exponential mixing for the frame flow was used to build surface subgroups inside hyperbolic, inside the pi1 of the mental group of three manifolds. And this is the work of Ken and Marcovic. And the list continues. But I'm not going to mention that. So the message here is that rates of mixing are important not only for, say, intrinsic beauty of mathematics, but also has applications through number theory and geometry and so forth. Okay, but let's get back to mixing. I'm not going to mention the application of mixing, but just how to prove some results. And in this direction, I want to say that in 87, Marina Hatner got a very precise exponential mixing result, but in the case of hyperbolic surface, so constant negative curvature. And her idea was to use what people call a Bargman's classification of unit representation of SHR. So basically the idea is that from the spectral gap that we get on the laplacian of the surface, you can convert that into isolation property from the trivial representation. And this basically converts into the key of correlations. But of course, this theory is very algebraic, so uses representation theory of SHR. And so there is little chance that this kind of argument extend to variable negative curvature. That's the point. But much later, Chernov was developing a method which he called the time Markov approximations, which actually was he started developing these methods together with Bonimovic and Sniper billiards, but then he wrote in 98, a paper on geodesic flows. And he introduced this technology of what he called Markov approximations, which was a kind of pre-Yangtaber somehow, if you wish. And then a condition called non-uniform integrability of the foliations to derive a stretched exponential mixing for certain geodesic flows in negative curvature. Okay, so a stretched means that this is the case at not exponential rate, but exponential minus the square root of t, basically. Okay. And then famously, Dogopiat was able to go from stretched exponential to true exponential using what people call today Dogopiat estimates, which you can summarize in one phrase. I mean, of course, this is a kind of disrespectful to Dogopiat, because I'm trying to summarize this beautiful work in one phrase. But if I had to do that, I would say that it's applying Von Degpore put lemma to temporal function. So basically, he wants to pick up oscillations in the Laplace transform of the correlation functions. So you look at this correlation here as a function of time, you apply Laplace transform, and then you try to get cancellations in the kind of Fourier or harder Laplace expansion. And to get this cancellation, you apply this idea of that the non-integrability translates into a beautiful oscillation along temporal functions. And you use a classical lemma, due to Von Degpore put telling when exponential integrals or sums exhibit cancellations. And then, of course, once we got this result for exponential mixing for a class of systems, then everybody got excited and there was a huge literature, what was developed after that. I'm not going to mention everybody, so I'm sorry if I'm not mentioning your work today, but I would like to mention just a few names on this direction, because the literature is huge. But let me mention that Leverani was able to extend Dogopiat's work for another contact, another flows. And more recently, Suzy and Jiang announced a solution to this so-called bow and well conjecture, saying that if you have in three dimensions this time, so you have a compact three-dimensional manifold, you have another flow on it. And then you can improve topological mixing into exponential mixing. So it's kind of funny that topological property implies into a quantitative property. And this is for all equilibrium measures for holder potential. So maximum entropy measures, Liouville measures, yeah, all of them, all interesting measures. Okay, so this is the scenario for negative curvature. And now I'm going to add some zero curvature in my manifold. And so again, the setup is S now for me to be a compact surface with non-negative curvature. M is the phase space of the geodesic flow. Mu is the Liouville measure for me. Gt is the geodesic flow, sorry. And then I'm going to denote, I've been given a vector. I'm going to denote gamma xt, the geodesic generated by this direction. And then I'm going to define the degenerate set to be the set of vectors generating geodesics along which the curvature vanishes identically, okay. And then the regular set will be the complement of that. Now, I want to tell you why a gadget called Jacobi Fields are interesting when talking about geodesic flows. So you probably know this, but let me remind quickly what is the rationale behind these considerations. So the idea is the following. What is a Jacobi Fields? So a Jacobi Fields is you have a geodesic and then you have vectors along this geodesic, which are these vectors in Gt. And then it's a vector field satisfying this equation here. And so it's a little bit bizarre if you present it like that, a better way is to say the following. You start with a geodesic and now you consider one parameter variations of the geodesic by other geodesics. Okay. So now I have two parameters, I have a family which is gamma t comma s. Okay. So t is the parameter along the geodesic and s is another geodesic. And now if you derive the variation at each point of the initial geodesic, you get a vector field and the vector field because of the condition that the variation is by other geodesics satisfies this equation. So that's the point. This equation here characterize the fact that you are doing variations by other geodesics. So in other words, what you are trying to do is understand how geodesics are behaving nearby, given one that you are interested in. In particular, if you use these variations of geodesics to parameterize the tangent space to the unit tangent model, then you can do it. You can make an identification of, I mean, a variation. I mean, by definition, a tangent vector for the geodesic is a variation. And you think there's more variation of the geodesic, which we saw that it's identified with these Jacobi fields. And the advantage is that in these coordinates, the derivative of the geodesic flow is very simple. It's just, it starts at the initial point, zero, and then you just flow for time t. And you look at what fields you got. Okay. So these are coordinates which make the analysis of the geodesic flow very easy. And so let me just remark that, I mean, the manifold was the surface is a remuner manifold. And since it's a remuner manifold, the unit tangent model is also a remuner manifold. It carries a metric, which is called the sacchymetry, which is very simple. And then I'm going to compute what happens to sizes of vectors in a positive derivative with respect to this metric. And if I do that, well, I'm going to do that along two subspaces, which are the stable and stable subspaces, which by definition are the initial points. I mean, this is a second order of the, so to give, to determine a solution, any initial position on the velocity. So this is why I'm putting G and G prime. So given this guy, I have a Jacobi field. And now I'm looking at, so the stable space is the space of Jacobi fields, which never, which stays bounded when I go forward. And the unstable one is the space of Jacobi fields, which stay bounded going backwards. And these spaces, they are nice because you can check by this simple formula that they are invariant on the digitizing flow. I mean, if I'm bounded forward, if I just go a little bit to keep this property. In our situation of non-positive curvature, they are one-dimensional. And they depend continuously on the point. And more importantly, they coincide if and only if, so I see a tangency between these spaces, if and only if I'm sitting on the degenerate set. Otherwise, they are perfect splitting if I add the flow direction. Okay. Moreover, I can quickly compute, as I told you, the derivative of the digitizing flow along these spaces with respect to the Sasaki metric. And then I get, well, after some computation, I get some function here, which I wrote the basic problem. So there are functions u plus minus, which are continuous functions, which vanish only at the degenerate set. And by compactness of my surface, this term here is bounded. So if you forget about this bounded term, what you see is just exponential of the integral of this function along a geodesic trajectory. So if this function is positive, so if you are away from the degenerate set, this function has a lower bound. And in particular, you are integrating exponential of some constant times t, at least. So you see exponential growth along, yes, if you are away from that. Similarly, this function is negative away from that. So you see exponential contraction along the stable direction. So in other words, the simple calculation shows that this space are really deserved to be called stable and stable subspaces. Because if you are away from back, you are really seeing exponential contraction and expansion. The only problem is that when you get close to that, these directions tend to close up. You get consciences. Okay. So this is the situation. And now let me remind you that there is a beautiful application of Passing's theory by Passing himself from 77, which explores these features of these bundles to show the following result that if the measure of the degenerate set is zero, with respect to the Liouville measure, then the degenerate flow is ergodic. Okay. This is basically one of, I mean, an important application of the theory of a non-informal hyperbolic system developed by Passing. Much later, Knieper considered other measures. So in particular, he considered the measure of maximal entropy. And he proved that there is a unique measure of maximal entropy in my context, if I'm rank one. In this context, it just means that the regular set is non-empty. Because I'm talking about surfaces. And more recently, even more recently, there was a series of works by Le Drapier, Lim and Sarig on one hand, and Bern, Bern, Skleeman, Haga, Fischer and Thompson, establishing many results about these types of measures of, I mean, equilibrium states in general. So we start establishing particular uniqueness and Bernolicity, which is kind of the top, the strongest abstract ergodic theoretical property that you can ask for in the chain of ergodistic mixing, K property, and Bernolicity. Bernolicity is kind of the strongest one, and they could establish these results in this context. For certain equilibrium states, I'm not going to give precise statements here, but I recommend reading the papers for precise statements. But in all in all, these results, yeah, so the advance, of course, the knowledge about statistical properties of these flows, but once you get to this state, then you can ask the same question that people ask at the end, another proof of mixing, which is, what about the key of correlations for these flows? And, well, this was not so well understood. And so the idea of the next result, which is, as I told you, I'm trying to work with Yuri Lim and you remember, is to offer some polynomial mixing statement force for a certain class of surfaces with no positive coverage. So the result is the following. So you take S to be the surface that you get by first, you take this function here, so say set R equals to four. So this means that you're taking the profile on the plane, y equals one plus x to the power four in an interval. So you draw the graph, and now you make a revolution about the axis. And now you glue to make the surface boundary less. You glue two negatively curved surfaces on the boundaries to close up the picture. And then this is my surface. So this surface, I'm going to explain to you in a moment that it has negative curvature everywhere, except at the geodesic in the middle, corresponding to the revolution at the parameter x equals to zero. Okay. And so this surface has curvature, which is non-zero except a long one single close at geodesic. And in this context, what we can prove is just to visualize, this is like two balls that are glued in a point. No, I'm going to show a picture later. It's like two tori. Yeah, it's like two tori because I want to have in the end at least genus two. Yeah, because you start getting hyperbolic matrix in the torus. Could you show the picture now so that we understand? Yeah, I can show the picture now. Yeah, before engaging the results. Let me show the picture now. Yeah, so this is the picture. So this is the profile of revolution that I told you. I mean, this is supposed to be the curve one plus x to the power four, and then there is the x here. I'm rotating about it. And then I'm going to things of negative curvature just to cap it and make something with dot boundary. Okay. So this is the kind of surface that we can treat. Yeah, recently we think we can treat more things, but let me just keep to a simple statement. So let's take this example, which is very concrete. And then the result is that in this context, the presence of that single geodesic along which you have zero curvature makes the rate of mixing to drop from exponential to polynomial with a rate which depends on the profile. Okay, so it's this number. So the precise relation between the exponent in the profile and the rate of mixing is given by this function r plus two over r minus two. Okay. But in particular, since I'm right between four and infinity, I'm getting something between one and three. Okay, so I get cubic pk at r equals to four. And if r increases, my profile gets flatter and flatter somehow. And this speed gets closer and closer to linear. That's the morality here. And there is a work in progress by brewing Melbourne and tertiary saying that this rate should be almost optimal in the sense that in the same context, there should be, it should be possible to construct observables whose correlations are not very far from this rate up to some logarithmic term. But this work, as far as I know, it's not on archive yet. But yeah, I heard from Ian that this work is in progress and exists. And so presumably, the optimal, I mean, the exact rate should be about this thing here, maybe with some logs, some little powers, I don't know, but about that. Okay, so this is the result. Another result that we can get from the same, from the same arguments is a central limit theorem. So I'm just recalling what it means is conversing distribution to a normal distribution of this kind of average when I normalize by the square root. We also can get weak invariance principles, etc. Okay, so I'm not going to talk about all these results, but the idea that the same machinery gives everything. And of course, if you followed the course by Z, you know that all these statistical laws are going to follow from a nice young tower once we get the hands on a nice young tower. And so that's what we are going to discuss now. It's how to get young towers for this geodesic flow in this context. And you can believe me, once we have that, it's not hard to get polynomial mixing, central limit theorem, etc. Okay. Okay, so are there any questions? Okay, so if not, let me advance. So let me try to explain what we are going to do to get this young tower. So before that, let me compare with the situation for billiards. So actually it's funny because there is a paper from 2005 by Chernoff and Zang where they kind of consider almost the same scenario, but for billiards. So the idea is the following. You start with a dispersant billiard, but then you modify one of the obstacles to have a profile, which is exactly one plus X to the power R. And then you put two of them, one in front of the other. And so this makes a period of period two, where you have this kind of neutral behavior because the curvature is going to zero. And they could prove things for the map. But as far as I understand, you can have results for the semi flow, but not for the flow. Okay, so the analogous results for billiards are open as far as I know. I mean, at least that's what was said in a paper from by Melbourne in 2018. And I think the situation has not changed since then. But yeah, it's funny because for the same geometry for billiards, we know some things, but not everything. Yeah, but for the geodesic flows, we could do something. Okay, so let me get to the test shows. This is the picture of the surface. And as I told you, it's not hard to see that. I mean, this geodesic here for this profile has the Gaussian curvature is zero. So actually, I have formulas for these things. So you can check, for instance, these formulas in the book, it's a very nice book on differential geometry. And so we can look at the chapter on surface of revolution. And so what the car most says is that if you have a surface, which is the evolution of some profile function psi. And then you put cosine theta sine theta to do the revolution. Then the curvature is going to be is going to be given by this formula here. So it has some second derivatives in the numerator and then something which is not zero in the denominator. And in particular, you see that, well, the function one plus x to the power of four has the derivative zero at x equals to zero. And so the curvature is exactly there, but not nearby. Also, another thing that we're going to use is Clairro's coordinates. So there is a very nice remark by Clairro when people was trying to study geodesics on Earth, which is a remark that on surface of revolutions, you have this meridians, which are the circles. And then the remark by Clairro is that when you look at the geodesics moving on the surface and you measure the angles with respect to the meridians, then the profile times the cosine of the angle is a constant of motion. So it's an integral of motion. And so this is particularly useful for us because it allows to reduce easily a second order differential equation, which is the equation for geodesics, into a pair of first order equations, which are easier to estimate. And this is why we chose these examples first because at some point we need very precise estimates. I'm going to tell where. But the idea is that the analysis of the Clairro's equations, we can decompose well, we can first check that the dag set is just that geodesic in the middle, the curvature venison only there. And also that when you are trying to understand geodesics, which enter this kind of neck, which is the region that where I have this evolution profile, I'm going to call it neck. When you enter the neck there, you can do three things. Either you can aspire and converge to the geodesic in the middle. So I'm going to call this profile asymptotic. So I'm in the stable or unstable manifold of this closed geodesic. Or I can do what I call what we call bouncing movement, which is just you enter the neck, you aspire around, but then change your mind and get out by the same part of the neck that you came from. Or you can cross, you spiral across the middle geodesic and go out by the other side. So I'm going to show you. And these three behaviors correspond to a certain value of the Clairro constant. Okay. So these are the pictures. So we can either come here and spiral around if the Clairro constant is one, if it's bigger than one, then you change your mind at some point and go back and get out by the side, or you cross the geodesic effectively and go out here. Okay. And so now what I'm going to do is the following. I'm not going to construct myself the young tower. I'm going to take advantage of the fact that the turnoff and co-authors of Markarian and Jeng notably working on a set of geometric conditions, which I'm going to call today turnoff axioms, which is a set of geometric axioms that allows you to ensure the existence of young towers with exponential tails. And then once we get that, we can take advantage of the theory of young towers and conclude this statistical law that I announced. So the key idea here is really to stick to turnoffs axioms, which are a kind of geometric gadget allowing you to ensure the existence of young towers. So he worked hard to show that these axioms implies the existence of young towers. So I'm not going to reinvent the wheel. I'm just going to use these results. And so what are these axioms? These axioms are basically, yeah, so, yes? Sorry, Matheus. Can I ask you another question? So at which stage do you go from the energetic properties of the young tower to the flow? I'm going in the very end because actually the idea is that I'm going to construct a nice section to be able to check for the map first. And then I have to analyze what happens to the return time to the section. And it's then that I'm going to get a polynomial mixing. It's because of the returns. So the moment everything you're talking about is you're going to, so when you say channels axioms, these axioms for the flow, for the map, for the map. They are axioms for a map. No, they're axioms too that will give the existence of the map, but the axioms on the flow or the axioms? Well, they are axioms on a given map. I mean, you have a map from a surface and then it's a list of eight axioms. I'm going to show some of them. And then if you check those axioms for a surface map, I mean, it's not to be a surface map. Then this guy has a young tower with exponential tails. And then there is the problem of passing this information to the suspension. And then we have to discuss, of course, return times to the section. And in our case, this part of the return time is not very difficult because we are dealing with contact flows. So the kind of only condition that appears in dogopiat's work and turnoff's work is automatic here. So this is why I'm not talking too much about the return time function because we are contact flow. So because we are geodesic, so yeah, the kind of uniform non-integrability affiliations are kind of automatic in our setting. Yeah, but I'm going to mention these two things in the end, I hope. Yeah, but the idea that we are going first to construct a very nice map and then try to check these axioms and then understand the return times. So what do we do? Yeah, so as I was telling you, we don't need to check too much. We don't need to care too much about the suspension function because we are in a flow, in a setting where the flow is very nice as content. Okay, so what are these axioms? So these axioms are a list of usual properties in dynamics which is uniform hyperbolecy, curvature bounds on invariant manifolds, distortion bounds, absolute continuity, and the growth of unstable manifolds. And then, I mean, this method was applied in many papers, especially in the case of Peter's with lots of success. And what we are going to do is kind of use the same scheme with a little tweak. And why we are going to tweak it, it's because in my list of properties here, there is this kind of uniform curvature bounds, which is kind of annoying because when you talk about curvature, you need objects to be situational. In particular, the invariant manifolds, if I'm going to talk about curvature, they need to be situational. And the point is that they are not situational in our setting. So this was noticed by Bauman, Brie and Burns in 87. So in particular for the profile that I wrote, 1 plus x to the power 4, glued with something that could prove that they are not situational at the closed geodesic. So you say, okay, you don't have the axioms for of Chernoff. But fortunately, in 99, Gerber noticed that it's not situ, but it's uniformity one plus leap sheets. And this is the right regularity to make the Chernoff scheme work. Because what he asked when he was talking about uniform curvature bounds, basically what he needs is a slow variation of the touch of its planes in order to compute measures of sets coming from projections of the tangent space to the unstable manifolds. And to that, you don't need the curvature bound, but you just need to know a notion of upper bound of the curvature. So in other words, that the tangent planes are not far into fast. And this is given by leap sheets. So in other words, even though we can't strictly apply Chernoff axioms as written in the literature, it's not it's not difficult to tweak them to include this slide generalization. And this is precisely the condition that we have for our examples. So we are saved by this work somehow of Gerber and Wilkinson. And now let me talk about, yeah, so this is the comment that was made, the curvature bounds is just to control variation of tangent spaces. And since he's just talking about upper bounds, leap sheet suffices. Okay, so now let me talk about the axioms themselves and how to check them quickly. So I yeah, the remaining time, I don't want to waste your time on lots of details, but let me just check a few properties. So the idea is the following, you are going to believe me that I just need to construct my section and understand my maps close to the neck because outside the neck, the genetic is just negative curvature. So everything is fine. I just need to understand how transitions on the neck happen. And to do so, let me just take the example of crossing geodesics. So I'm not going to try to treat the bouncing ones, but the three minutes analogous, let me just look at angles where you're crossing and coming out of on the side. And so my section, I mean, morally, but this is not true, but morally, the section that I'm going to take is just the following. I'm going to take these two curves here alpha and beta, then I can take some disks, SEO disks here on this part. And this will be my map. Okay, so this is basically the cross section. I mean, it's not true. Technically, it's hard to do that, but morally, what I'm what I'm trying to tell you is that to build a map to check channel flexioms, what I'm going to do is I'm going to take to take the vectors based on alpha union beta. And so every time across, I have a map and some disks outside just to complete the picture of the section. So this will be my map. But I'm going to analyze only the map from alpha to beta say, just to give a feeling of what kind of estimate to have to do. Okay, so we are trying to analyze what happens from up to angle. And so I'm going to introduce some notation. So as I told you, I'm going to say from the left to the right, so minus one to one. And then I'm parametrizing this geodesic by some interval of time. So minus CT so that at time zero, I'm really crossing the geodesic in the middle. Okay, so this is the meaning of these conditions. And now I'm going to look at geodesics whose collateral constant belongs to interval, which is pretty close to one. And it's given by a one plus I mean one over n plus one square and the one over n square. Okay, so I'm going to basically divide the set of collateral constants close to one into these intervals. And this is the composition. If you look at this, the way I'm doing this composition is very similar to billiards. And this is not a coincidence. We are really going to mimic what people do for billiards with the notion of what they call homogeneity bands. The idea is that we're trying to get uniform properties like uniform hyperboleity, uniform distortion, uniform growth. And there is no hope to get that globally on the face space. We have to concentrate on bands where guys kind of travel close together. And they feel the same hyperboleity which gets degenerate when they approach collateral constant one. Okay, so that's the idea. I'm not going to compare vectors which are too far. I'm just comparing guys in the same homogeneity band into this thing here. And so the first limit is that guys in the homogeneity band, they take almost the same time about the same time to cross the neck. And the time I mean almost the same time means a time of order n to the power r minus 2 over r where this approximation here is up to two multiplicative constants. But the exponent's correct. There is no loss in the exponents. Okay. And this limit is a calculation with clever row. I mean, you basically use the fact that the clever row is a constant of integration. So this allows you to basically talk about a problem on our first order differential equation. We integrate it to get out the time and you check that to get the correct expression. Okay, so it's a calculus exercise. But of course you see, I mean, this is intuitive. I mean, the closer you get to the syntotic, the more time you travel because it's higher more and more for crossing and then getting back. Now, what is in the bag measure of the set of vectors in a given homogeneity band? So the measure is basically the set of this interval, which is 1 over n to the power 3. Okay. And why am I talking about that? Because this is kind of giving a control of tails of my tower. And so if you use these two facts, you can see that the time, I mean, the vectors is spending a long time in the neck. It's about this series here because you should sum over all homogeneity bands for which this number is bigger than your k to the power something. And this number is k to the power 2. So if you change variables, the time, if you spend time n in the neck, then the measure of those vectors is this 1 over n to the power r plus 2 over r minus 2 plus 1. Okay. And here you see the number that I was mentioning in my conclusion of my theorem. And so for the value, there's a kind of that number in the correlations plus 1, which is normal. For the map, you get always a number plus 1. We saw that in this lecture. For the map, you get a number plus 1. And for the flow, we are going one less. And now we need to understand the map. So the map here, you can explicitly calculate because of the symmetries of the situation. And so basically, we know how much you turn around in the theta variable along the geodesic. And of course, you go from the circle minus 1 to the circle 1. And then the angle that you get in is the same that you get out by symmetry of the situation because you end up, you wrap around, you cross the geodesic and then you can reverse the mechanical system. So you get out to the same angle. So the angle coordinate is not very difficult. It's just the theta coordinate that gets wrapped around. So on each band, you are basically twisting the angle by a quantity, which it depends on the angle of entrance. And the amount of twist is a function that you can control very well. And this is crucial. This is why we stick to rotation surfaces, surface of revolution, because we can really compute this derivative. And there is no error in the error term in the power. It's really this guy up to two multiplicative constants. And for the second derivative, we just need that number bound. But for the derivative, we need a very good bound. And this is why we stick it first to examples. But an important thing is that since you are, so basically, this feature is for you approach the singular set and you twist more and more around this guy. So basically, you are kind of looking at the map of a surface, which is basically made of two components, a hyperbolic map, which is transitions in the negatively covered surface. And then a kind of parabolic map, a then twist, which is strength is bigger and bigger as you approach the singular set. And you are composing these maps, not randomly, but kind of composing them and trying to build a dynamical system, which is iterated function system of these two things, hyperbolic map, infinite then twist. So what is it that is getting bigger and bigger as you approach the singularity? This function zeta, which is a twist that you do in the angular coordinate until you go out out. I mean, the number of turns that you are giving here in this picture. And this this explodes as what happens as you get, oh, as the direction becomes more exactly, it explodes and we can control in a very precise way how it explodes. And this is important for our estimates and to ensure hyperbole. And this is why you are sticking to examples so far. I mean, for the moment. Yeah, but this is the idea. And in particular, we can compute, for instance, how unstable vectors grow. And we get this formula, which is basically, as I told you, coming from here and so forth. So in particular, this gives a very nice uniform hyperbolecy. I mean, you expand at least by some amount. And this amount increases as you approach the singular set, the derivative explodes as you approach, which is good for expansion, I mean, and contraction. So you have this uniform hyperbolecy property. Then you can also prove using similar bounds. Since you have bounds on the derivative and second derivative, you can prove distortion bounds. And then you can prove in particular, yeah, that the log of the derivative, it's varying like a holder function of the points. And this is because we can control the derivative. So the quotient of z to primes by z prime. Okay. And so this one third here comes from the fact that the quotient is really n square, if you divide here. And the interval where this happens is the homogeneity band is n to the power minus three. And so the distortion is n squared by this. I mean, you need to give up this factor here to be able to make a big O there. So if you give up two thirds in the exponent, it remains a one third. And then you can also prove absolute uniform absolute continuity by the same means. Okay. And again, I mean, we are using a holder continuity of the foldation, which is not automatic. I mean, it's automatic in hyperbolic dynamics, but here we are again using Jabra Wilkinson. Okay. They have two theorems, one's giving lip sheets proper along the leaves and holder continuity if you move in any direction. And then as I told you, so these are let's say two axioms or three axioms, I mean, uniform hyperbolicity. Yeah, I have not explained for you the growth lemma, but I said about uniform distortion and uniform absolute continuity. But yeah, you have more axioms to check in particular, have to check that for instance, when you have the singular set corresponding to plural constant equals to one. And then you need to check complexity growth of the singular set. So when you iterate and iterate the set, the weight crosses and the composes the face space is not creating too many pieces and so forth. But yeah, but I think I'm not going to discuss it because this discussion is technical and especially because this discussion depends on a appropriate choice of the section which I did not do today. I mean, I was doing the calculation with this fake section, which is just the next. But actually, in reality, you have to adapt to that. The compose more and yeah, it's more technical. So this is why I'm not going to discuss the other part, because I'm not going to give the correct expression of the map. But you can kind of believe that out of these calculations we can get enough control on hyperbolic to get a map, a surface map with a young tower of exponential towers. And now you need to pass this information to the flow. But to pass the information to the flow is the computation that we already did, because I told you that transition time, so the time to cross is this, I mean, the tail is this, the set of the measure of guys which take time and to cross the neck. It's about one over into this concept plus one. And so if you convert that into the flow, you should take this one out, and then you get the correct, the correct key of correlations. Well, it's not so easy. You have to apply, I mean, if you want to do it properly, you have to apply recent work from 2019 by Ballant Butterlain Melbourne. But let's say morally, that's what it is. And I think I'll stop here. So I'm taking your questions. So thank you. Thank you. Thank you. So I try to say thank you in the language of the organizers. And so let's thank Matheus. Thank you Matheus. Thank you. Do you have questions or comments? We'll have one more question, but I've been asking so many questions. It's really, I mean, it's really just about this last part about going to the flow. I've not, you know, I've worked very little with flows. I did some work with Ian on the law and equations, but I remember that the problem with the decay of correlations is that you need, I mean, I remember that it's, you need actually to show that the flow stretches somehow, right? It's not just, it's not just an automatic issue. Exactly. I mean, usually you need some conditions on, yeah, some sort of stretching because you basically are trying to understand. I mean, when you do the, so let me show some correlation. Yes. So like this one, when you do the Laplace transform, what you see is a kind of exponential i times a temporal function. And then you want this function to oscillate a lot to be able to produce constellations. And in our case, and this is what I mean, you can ensure, for instance, you should ask for the condition called uni, uniform non-integrability of defoliations, meaning if you do a path along the stable of epsilon size, then unstable, then stable. And then what, when, and then unstable, when you get back to the flow direction, you get by distance of epsilon square, say. So the defoliations are not, not only non-integrable, but they are non-integrable in a quantitative way. Yeah. So, and this kind of quantitative non-integrability allows you to run this stretch of things and pass to the flow. But in our case, my claim is that this is automatic because we have a context structure. This argument that goes back to, I mean, this kind of application of context structures to get statistical properties go back at least to a paper of Katalk. And it was used and reused by many persons. I mean, I'm not, I don't know if I'm historically accurate in saying, but I remember clearly a paper by Katalk doing that to get Bernoulli property. Yeah. And so here's the same, I mean, here you have to take advantage that we don't have to discuss much because our flow is already geodesic, so it's context. So it suffices to know the tail. Yeah. But in general, you are absolutely right. We've had, if we had abstract flow, we had also more work to do to check side of conditions, non-integrability conditions. Okay. Thank you. I wanted to make a comment that was not in the slides that the paper of passing from 77, it proves actually the Bernoulli property. Oh, that's right. Yeah. Yeah. I started talking about the DC and yeah, absolutely. Yeah. That's right. Yeah. And it's like when I was telling about the nozzles that I just mentioned, they're geodesic, but yeah, you actually put mixed in more. Yeah. There is another thing that I wanted to point out. This kind of infinite bent twist map that Matteo said is very hard to work with because you have a singular set and if you have, if you get a curve that is transverse to the singular set, the image of this curve of, which has finite length will actually be infinite because its image, it will rotate infinitely often and accumulate in a closed curve. So it makes it very complicated. The analysis, even for this complexity bounds, because you have to have like no triple intersections, but you are dealing with curves of infinite length and you don't know what happens when you perturb the boundary, what happens to this, the new images. So you have the perturbative argument that we make, it's not as standard as we thought and they actually make many mistakes to get to the right way of constructing the section. Yeah. That's right. I mean, so this is why I said that it's what I showed is that kind of fake cross-section because the true cross-section has to choose well the boundaries to ensure these transversality properties that are known as standard and relies on the formula for the map really. So this is why we are sticking to the example, because at least in the example we control everything. So I mean, of course we believe that this could be generalized, but we are not yet there. Do we have any other questions or comments? Okay, so let's thank Matheus again.