 Okay. So thank you, Stefan. So today we will have two hours of lectures, but I will, I mean, I will do one hour and then we take 10 minutes break and then do another hour. So when I was preparing actually that's like for this series of lectures, I was thinking that I will go faster than I'm doing now. So which means that I mean for today's lecture I might not go, I might not go beyond, I might not go to the end of the group of counting these closed geodesics. But I just want to make sure that like, at least what we're doing, like, it is understandable. I don't want to rush too much into getting to the final end of these counting the closed geodesic. Well, as you see last lecture, we were building a specific measure, which is a measure of maximum entropy. Do we have improved it yet? We will just finish the proof today and prove that it is a matter of maximum entropy. And then we will prove that this measure is unique, because you see the vibrational principle it gives you the supremum or you know all event measures so if you find one, the question you might ask whether it is unique. And the proof is, we will see that it's just general proof. I mean, it's an expansion of the general proof by form on using the specificity and shadowing or specification gives uniqueness. So today I will show you how to get specification and expansivity. And then I will, I want to actually discuss the proof of mixing that we did in this paper, which follows by some nice argument, which follows from some nice argument by Babilo for like a negative curvature for instance, when the structure and it uses a very, a very geometric function, which is the closed ratio function. So I want to say for some people who knows a bit about explanation mixing by like the proof of explanation mixing for contact flow by Daniel Leverani, he uses this temporal dispersion function and the fact that it is older, it implies some nice like explanation mixing for the contact and so forth. So today also we see a function that is an analog of that one in this area of geodesic flow, and then the continuity actually of that function will implies mixing. But here we just have continuity for that for that function so yeah, so today we are going to talk about four and five. We will be talking about MMI, like the measure of maximum entropy, and then we will talk about mixing for the MMI, and then I don't know if you will get to the counting, I don't know. Maybe MMI just say few words, or like how to use, because I spent the whole lecture last time to discuss the cross section of this back sense of measures which is key in proving mixing for instance and also in doing this counting. Maybe MMI just, I will just say few words about how the property of the measure is used to prove mixing. Okay, so let me just recall where we were, we have the energy. So, the results that I'm going to prove today like the uniqueness and the mixing, the whole, in our case we were able to prove it for any surface with genius at least two. Yeah, I told you that this genius at least two is just to answer that we have positive topology to entropy because on the tourist, if the, if you have a tourist with that point to get points flat, so do you don't have any entropy there. So, everything that I'm going to talk about today is for surface. Do we have a general results, but that uses some non-trivial assumption, okay. So surface is a point to get points. So if you remember this notation, I just replaced so X is minimal self cover, which is a copy of R2. And remember this notation, this was the idea boundary. It was defined by the equivalence classes of asymptomatic to this grade. Okay. And then, yeah, so what we did is we defined this family. That's, that's essentially one family. And they satisfy some nice properties like this one is going to vary and this family, they are absolutely consistent with respect to each other and the density is given by the good one function. And these measures are full support. And from this page, everybody wants to be fine. D. The new bar on on this space, which is the space of your design by this taking new bar is exponential of H beta P of the data, the new design. The new data. This is exciting. And from this measure, we build a measure now on the internet and longer by using this, the inverse so remember we had this map. So by the way, when I was watching again this lecture I noticed that I was talking about this map, you know this square minus the diagonal. D minus plus. I was talking about the activity and so the activity of this map but what I mean by as x here is this, you know, I portion is the geodesics. Okay, so you use here in relation by saying that B and W are if you want, I'm the same equivalence class, if they belong to the same this was really what I mean, I just take it before with like I'm saying is. Yeah. So, this is what the model is doing. You have to be, you're saying to this point, he might be minus E plus. But for this point also W they declined the same time so I'm pushing the, you know, with a set of geodesics. So, this map was not injective but next time I told you that there is a way to choose a measure of what was for this month. And last time somebody asked a question I promise I was going to think it was about it. So, so this is just from a very abstract result so what you do is you put define an equivalent relation saying that B is equivalent to W. Even only, you know, you have H, B is equivalent to W if and only if. So what I'm going to say is so I should write the picture and then you will see. So, if this happens, if you have two different geodesics so the picture is that we have the whole sphere. They do like this. So, B is equivalent to W if and only if B plus because W plus and B minus because W minus. We just saying that all these geodesics here, all these vectors here, they belong to the same equivalence class. And from there you can define the map Q tilde from Sx to, you know, Sx modulo this equivalence relation, and then you can project this map to the boundary to the to the manifold and then the boundary of the manifold Sx to Sx tilde. And now the, the theorem I did not write, I want to write wrongly the name for instance, like Kura Towsky in. Can you like a little bit bigger Khadim it's a bit. Okay. It's just a very abstract result that uses this. So it tells you that this map will have an impulse as long as the equivalence classes they are compact. And here we have compartments by most of them, like it's bounded they say compact. The equivalence classes are compact. So this theorem. The theorem of this is that there is an impulse, a measurable impulse, impulse for this map and then you can lift it and you will have an impulse of Q tilde. Yeah, so this is just something very abstract. So the way you could define, remember we define this measure me tilde by just taking like a intercept means the inverse of this map, Xi eta, D mu bar of Xi eta, and this is the x squared minus the diagonal. We have some nice properties. Okay, and you project this measure to the, to the money for binding this government and finite and compact sets, so it project down to a finite measure. So, yeah, there were these properties that I told you that this is what gives the fact that this measure has full entropy. You have new P of P R side remember this collection B X row. So raw is big enough. Okay, this is of the order of exponential of minus H distance between X and P. And so this is what we use to prove that to prove that this measure has full entropy. So I will add again that lemma. So new H new it was H top of you want. I mean, it's the measure when you project this. Okay. The measure you get so to prove. So to calculate the entropy of a measure what you do is you take a measurable partition. Let's say we have a, which is anyone. So let's suppose that the diameter of these elements are less than. It is small compared to the injectivity rates of the injectivity rates of this of the manifold. So, and you define this set again. You define this partition by using the dynamics to use your zero to n minus one of feet minus a and an element here alpha is in a and means that alpha is in the intersection of V minus J of the epsilon. So J equals zero to n minus one where V is some, some points in alpha some V. So in elements in this partition is just given by these bone balls like these are a set of vectors. You are in alpha, you stay at a distance epsilon up to time and until time and okay, at least until time and So the entropy here, the entropy of this measure for this system. It is just a limit when m goes to infinity of the entropy of this set, which is the sum of minus mu alpha. Love of mu alpha where alpha is in a and so proving that the entropy is H. So dilemma to prove dilemma it is enough. So let me maybe go to the side. So, to prove that this measure they have full entropy. We have to prove that you prove that new of alpha is less than some constant C exponential of minus H and where H is a topological entropy. Is that stop. Okay, because if you plan that here you see, you get there. Did I do this wrong. So, we just want to prove this plan here. Yes, so we pick an alpha. We want to prove this equality. So let's put everything in the universal car. I'm a big picture because it's going to be so I have the V is a vector says that alpha is in the intersection. So this is V and it has a geodesic. So this is V. And let's take this point. Let's call this point X, which is C V and so remember to know the major of the alpha, you should project it to the you should send these to the boundary and know the major you get here. You know, you should do this. And to do that, let's take another generic vector in alpha so big. W in alpha. That means generic so. W here. So V and W this day at a distance of time. Okay, so that's the definition. Okay, so that's the definition. And so this is Xi, which is that minus. This is that minus. There was a point of that. So let's see that. So yeah, this is that it stays at a distance. I want to draw now this geodesic. So what I want to do is I want to let me say whatever I want to say is if I do the geodesic connecting X, this point X to Xi. This point X to Xi, let's parameterize this geodesic so that this is a point zero. Okay, see Xi X zero is this point. Let's call it Q here. So the first claim is that Q is in a fixed range ball of V. This point is in a fixed range ball. This you can just do it by by by clang inequality. Okay, you can just add W and then, you know, the distance between Q and I of V like the food point of V. Let's say some are not, which is maybe twice the injectivity radius. Okay. So, or also the distance between Q and P is less than zero where P is a point where we did where we did remember when we construct this measure we have this family but we just fix one point and we do the construction so new P this guy. So P is might be different from the V but it's somewhere here. So what I'm saying is that Q is in a fixed range ball that doesn't depend on on end. So the distance between Q and V or key and P doesn't depend on end this by using clang inequality. Okay, so if I project this alpha. So it would be like, I'm projecting here. So, if I take this guy, and I send, if I saw this is what I want to say, let me write. So, this is what I'm saying is the alpha is a subset of the union. So, this is z in P, P. This is our zero z times. So remember this projection of Px. So to know all this all desire. Now it's enough to project X to this to the point in this ball like that set will contains the backward, the backward end of alpha. Okay, because key is in this ball like generally point W this Q that joins X in the geodesic journey exercise isn't isn't this ball. So this guy, they run from this set. And now to know that the other end points is that you just project project is this guy to X and then you have the other end point. So that's what you have here. The fact that this is in a fixed ball here doesn't depend on and is is crucial. So, yeah, once you have this, it's, it's basically, we are basically done. So see that. Now if you use the property, I will have new P, new P of P, B, P, Q, B, Q, B, X, epsilon. This guy is less than by the fact that these measures absolutely continue to inspect each other. This that's between P Q and new P of PR, PX. So he was this point here so if you market you take the other end you have, you know, the point from here to here is just you are changing the base points here. And then, and then this is less than by what we have seen before, this will be less or equals to a constant times exponential of minus H distance between Q and X by this property. And here it's easy to see again that the distance between P and X is like is is is a port of N distance between P and X again by Chang inequality is less than N plus is bigger than N. So if you want the other way around, it's bigger than N minus some, maybe two out of zero. Okay. And you would have that this is a port of N. So this is less than a constant times minus H and this guy is bounded and the measure has a product, product structure. So this is like that me of alpha is less than minus H and a constant. And you have this you, you, you plug in the formula and you get. So, this time. So now, for today, you will move to so this measure has full entropy. We will prove uniqueness here. So actually, uniqueness of enemy. It was used by Bowen that if you have what is called expensive, if the flow is expensive, and you have what is called specification, these two together imply uniqueness. I will say what that is, is imply uniqueness. So, I'm not in Thompson what they did is that they try to relax these properties, like the exclusivity and also the specification, and they will be able to find some nice conditions that you still uniqueness that is this, this argument. So before I let me just define that is expensive. So definition is expensive. Yeah. If this set of point, the set of W. The definition is for general flow, but I'm just giving it here in the, in the, in this case of geodesic flow distance between 50 B, 50 W is less than epsilon 40. So if this set is a subset of, so if this guy is a subset of the geodesic minus s. I'm just telling that if to, I mean, if two vectors, they are the bounded distance absolute. So it's the same. Sorry. Yeah, it's the same journey. Like, you cannot find, if V is expensive at scale, you cannot find another factor that stay in absolute distance over time. That's different from being the same. Expensive set at scale. And the action is set of non expensive vectors at scale. So the second factor that I'm expensive at scale. So for this, I think we could see just some some exercises from last lecture. I mean the second lecture probably. So they were an exercise in the second lecture where we say that if you have negative curvature, this imply that WSSV, we just said WUV is a single point. In particular, I don't think it's as to maybe to check for today would be that if you have this property, then there exists epsilon positive set and E epsilon is anti like the flow is expensive. Yeah, that's easy. And also, remark that the non invertibility of this map that I was talking about, it really is connected to the non-expansivity of the geodesic. Now, E, this map which was so SX under I take explanation between like the geodesics are different classes to minus the diagonal, not invertible. So this implies that you have an excellent zero you have an excellent policies is that this guy is not empty because see this condition. You might have a strict if you have to to two vectors that define the same end points but not belong to the same geodesic. If this map is not in depth in is not invertible. You have, you have that property. Yeah, so this is. Yeah, so what was used by Bowen is that, expansivity is that there exists an excellent zero says that this set is empty. So that's expensive. So, for instance, in negative cover to you have expansivity and not positive curvature. So in particular from this remark is not positive curvature does not imply, you know, expensive, and let alone expensive. Let alone no points. And there's a specification. So specification is just like a shadowing lemon for, you know, for other sort of a mechanism plus so definition. So that was expensive. So given delta positive. You say, so I'll be writing F to be the geodesic law. You say F has specification at scale delta positive. And if every give me any pieces of orbit V1 T1. The end, the end, you give pieces of orbits. So these are in sx minus times. So there exists one orbit that shadows these pieces. There's W in sx and times. I will draw a picture for this. Tn positive says that you have 2j plus one minus 2j minus 2j isn't your delta zero. Oh, give me if there exists. I forgot here. There should be if there exists tau positive. So tau is a transition time. There is tau positive. So we will draw a picture for this is that feet. Tj of W is in is in this set of point that stay upon the distance until be in sx distance between us. This is one single orbit is less than the whole s for us from zero to this to the sum of time. So what does it say this is that you give pieces of orbit. So this is V1 and this is V1. This is V21 V1. I have another piece of orbit here. Another guy here. Another guy here. So this is V2 V2 V2 and so on. So Vn Vn and this is VTn Vn. So this is saying that you can find a single value that stay at the boundary distance here. This distance will be delta. This is W and spend some times outside. So this time is tau. This is tau. The time that is spent outside here is tau is the transition time. And then this is VT T capital T2 of W and again stays at a boundary distance to this geodesic and they state time tau and do this again and then come and share this guy. So this is specification. You have one single orbit that shadow this guy and spending a fixed amount of time outside like this transition time is fixed. This is a feature that you have in hyperbolicity like if it has negative curvature, there is a standard truth for this. Okay. It is using, it's just about shadowing. You have zero orbit you shadow in class transitivity to find these times you use transitivity. So this is something you always have in negative curvature, for instance, or the geodesic flow is transitive and another. But there is a problem like if you want to generalize this phone like expensive video and shadowing and specification implies uniqueness. But if you want to allow non expensive video to prove uniqueness, you should, you know, you should expect the non expensive set to have less entropy. So you should expect the non expensive set to have less entropy, because if the non expensive set has the same entropy as a topological entropy, then you can build a measure of maximum entropy there and a measure of maximum entropy in the expensive set, and then you will have to. So to help to get uniqueness, you need what is called an entropy gap, like there's a difference between the entropy in the non expensive part and the one for the food for the whole for the whole geodesic flow. So that brings this definition entropy of traction. Excellent is the supremum of the ventric entropies of measures, you know that I supported in the non expensive set was zero equals one. So the entropy of measures that are supported in the non expensive in the non expensive set. Now the term by term by the monologue and Thompson. This is 2016 is that you fix scale is absolutely positive. Because I mean they are very precise. So absolutely be very big compared to that. And having an entropy gap, like the non expensive set doesn't have more entropy than the full field has specification at scale. Then the measure of maximum entropy is unique and it is supported in the expensive set. So then there exists a unique. This is a very upset with us so they just do the recording the born coding is this type of specification and because usually the way to use specification is that you want specification at every scale, but if you don't have enough of the city like in our case, we cannot have specification up over scale. There's one scale that we stop and we can have specification there because of lack of capital city again, and. Yeah, so to prove uniqueness, we just need to check those conditions. So, for instance, we will first want to. So first of all, specification in this service is a little different. It's, it's kind of easy just rely on more so them. So specification is given by most of them. So you gave me pieces of geodesics like this. What I do, I found, I find that corresponding. There is a I'm in the service. For instance, it I had mentioned repeat this argument we would need to suppose that there is a background of which there's a background metric for the geodesic flow has specification, for instance, and negative curvature is a good one. So this is we know we have a program. So, this metric, this is the zero, this is the gene. This guy being a result, we can have a specification for this guy. I respect the one and then. So this is a geodesic. Now you go back to most to say that there is a geodesic that say at a boundary distance to this one. So you see the distance might not be small. It just depends on the constant you get from most level. So this is up really big. This delta is up really big. Okay, so this is how we get specification scale. But, so, but to ensure that we have epsilon much bigger than 40. That is something we don't have control on, but this absolutely can make it big how I just by just taking a finite cover. Which is large for the injectivity ranges is big. If you take a finite cover of your money for you can ensure that is big but so. So to ensure. So this one is bigger than 40 delta like to be able to choose such as to know what we do is because it actually is very high and your money quality is small. So, you won't have this like they're not expensive say will be everything so I mean I should start with that so. So, that's one is larger than the diameter of the money for is much bigger than the diameter of M. Then, of SM, then the non expensive set epsilon is is a whole thing. The non expensive is a whole thing is actually is much bigger than because everybody will stay at a distance. Okay, but what we can do is we can increase this game so we can take a finite cover of actually large injectivity ranges. So, so, that's actually something that people call the money for being relatively finite. So, so relatively finite it means that the fundamental group you have a, you know, you can do, you have a sequence of subgroup for the fundamental for which the corresponding injectivity ranges is going to infinity. So, basically we use about that and so, which is true in services actually. And it's true that for services and is finite. I mean, we can take a finite cover and is a finite cover of M of actually large, very large inductivity ranges uniqueness for the genetic flow and SN. You have uniqueness here, you will have uniqueness downstairs because this is just, you know, it's, it's a finite cover. It's a finite cover so you don't have to measure that gets multiplied because the orbits, if you leave them finally because one just to find it. Okay, so you don't go down for even though you don't go there's these measures. So, this is the fact we use to be able to make excellent because we are having this problem that if the manifold is small, which it can be, but this guy. The most delta this shadow this specification that you have from doesn't depend on the cover because this is a universal cover. So this is the universal cover. So, we can. So this allow to take epsilon, take epsilon bigger than 4g delta. Actually, now how to get this the gap. This is also two dimensional two dimensional argument in higher dimension. We don't know how to prove this entropy gap. So, to get this, if so if you have a positive entropy here. So, if, by the way, there is this remark that I forgot to say that. So, an excellent. So, see that sx minus and epsilon and e epsilon. This is roughly the set of expensive. Expensive vectors. This is a set of point for which this guy said of the says that that view. Maybe right. You can correspond to be to desire. P is the food point and I is the plus. That HP. I have to be plus. So the expensive set of these vectors. So, this is what I'm saying is that expensive vectors, the dual spheres intersect only at one point. So this is the. Expensive vectors in this picture that you have to join in this, you'll have the whole street intersect like this, and this is not expensive. This is a non expensive. So, if you're expensive, this is what you get. You can only expensive. This is what you get. So, but if, if this guy is positive. So, actually, we are, we are proving resources is we're proving that this has zero entropy is this is positive. So you have this person manifold, you have this person stable and unstable manifold and be in two dimension. I know that this money for the intersect transversely. So you will be in this picture. You have that transverse intersection between table and like between stable and unstable. These are passive manifolds. Okay, when you have a positive topological entropy, you can have this money for the entire dimension, you cannot have this intersection being transverse because you might have two surfaces that intersect just along the line. You might have two surfaces that intersect along line. So, but in this case, we know these are one dimensional. So the intersection is of this picture, which contradicts that this is supported on the set of vectors that are of this form, because if you remember the definition So this is a simple argument that it for services. Yeah, so what you basically have is that if you if you get at least things we have specification given by most. We have this quantity by going to a finite cover with the we make a month for big we have this. We certainly have this and then we have a unique him and me. So this realize so on the existence of a background metric of negative curvature. Other questions on this. Yeah, so, so I would start the proof of mixing actually saw that the geologic flow with this major is mixing. Mixing doesn't use every doesn't use you just use this every city, which you get from the next. The proof of mixing that I will do here is just using is just using the fact that it is everybody can then and then, and then the, the construction of the major by a sense of a mess. I should maybe call the 10 minutes break and then we stand meeting at the top of the second hour. Yeah, so we have a 10 minutes break now and then. Yeah, yeah, yeah, five minutes past the first the next hour we will start to prove off. Yeah, we started from mixing. Sorry, may I ask a question. Could you repeat why the condition on the expensive entropy implies the possibility of relations. So, so you know you have a major that realize this entropy here. Yes, you take a typical point points on that major so for almost so respect that major V in in any epsilon because this measure is supported in any epsilon. You have WSS, the patient manifold, and yeah, this is actually not quite trivial. This lock V and that us you you love V. And these guys, they intersect transversely. This is a patient. These are two curves. They intersect transversely. So what is left to show is that these basic manifold, they correspond with our definition of these are busy money for you, you just need to check that they correspond with the manifold that you get from the whole sphere like the manifold that they define. Clearly, you don't know where these they correspond to, you know, the set I defined as stable and unstable. But that's not very difficult to prove that this guy, that is law V, the stable set that you get from the whole sphere from who's my lunch. If this is the set that you get from who's my function. Then you know by this intersect transversely, because these are just curves. If they were higher dimensional, you cannot know whether they intersect transversely. These are curved and they are defined as a set of points that, you know, they are converging. Like, if you throw them, it's converting in forward time if you're here. But this is not very difficult to see because this is just I see what I'm saying here. More or less. So, so to see that this guy, this person manifold, so this person manifold law of V is just a set of W such that D 50 V 50 W is less than, okay, some cheetah. And then, yeah, so these are the guys that say for all to positive. It's not quite to go to see that these are the manifold you get from who's my function. It kind of uses the divergence property here again, which was key. So, so see that. So, so, yeah, I want to say that the if you take so V is the point that I'm looking at this is being going in like that. So, if, if I take another back to W here. W, let's say W is in this set. W is in W SS lock V. So I have the geodesic that is, you know, staying at the center by definition. But what happened is that, at this point also, there is another geodesic that is going to end point of V. So this is the plus this point out new. There's another point that's going to do the geodesic to do to V plus. Okay. So, so what I'm saying that these two geodesics are the same by the agents property. There's another geodesic that is, and that is the geodesic that you get from the, like the Boozman function, okay, the whole of spheres like. So, this geodesic that is going to this V plus they are bound to distance to this one. And this is also at a boundary distance to this one because this guy does. So, by divergence property, you have that these two are the same. So this uses that business and let me just. So this is basically telling me that if you have that business property, what are called the Boozman assented are the only assented that's that's how they call it was my assented it mean that the geodesic you get from the Boozman function is just this geodesic. That is asymptotic to this one. So, yes, I understand. Yeah, this is a Boozman. And that is property tells you that Boozman assented have only assented like at this point you're going to have another one that converge to that is going to be plus and, and that's different from this one by that. The Boozman assented are the only assented so that's the only assented this is this is actually something don't you know that uses the divergence. Like any geodesic that stay at a boundary distance to another one is given by the boozman is given by the boozman assented these are called boozman assented so boozman assented. So this is a geodesic that I've given by this back to fit gradient of PV and gradient of PV, like if you follow this vector field, you get a geodesic so these are called boozman assented so this back to this geodesic follows this guy is a boozman assented and diverge property that these are the only The only thing that you need to check is that if you used to dimension you have the transfer of the basic money for now you need just to check that the basic money for are exactly the money for even by boozman. And that uses that and take maybe two minutes and. Thank you very much. So, let's get back. So for mixing. So, I mean, this is a definition of mixing that I'm doing. I mean, it's a user definition. It says that, you know, he is a major and is mixing. So that's the property that. The component is. Is converging to zero for all means. I mean zero functions. Yeah, this is equivalent to the user definition of mixing that says that the enemy that says that the manager of 15 a intersect B is converging to. 50 minutes of a meal. So, um, yeah, so I'm going to do you this this function so I will use this version. I'll use this version of mixing. So as I said, the proof it uses a general I mean argument due to Babylon that was done in negative curvature. And it relies highly on the fact that the measure projected to the boundary is a product measure. The measure projected to the body is a product measure which we have from Paterson's 11 measures. And, yeah, and then you have some happy policy. Yeah. So, yeah, you have some happy policy. The one thing I forgot to say actually, in our proof of uniqueness is that the measure we build the measure that the measure of maximum entropy will be supported on the expensive set on the set of points where the stable and unstable whole sphere intersect at exactly one point. So, and if you have the measure is supported on that expensive set, you have some type of hyperbolicity, even though it's not like uniform or exponential, like you can have this whole sphere contracting exponentially. Like for a time, but you know that the limit is going to zero if you have the, if you have a back to the expensive sets. So this allows us the fact that the measure is supported on that kind of good set where you have a certain hyperbolicity. It allows us to, to kind of do the arguments of Babylon that I'm going to say here. So, there is this abstract lemma. This is very abstract that is used here. So, see that this is just, this is just saying that this function is converging in the weak style to. So, this is abstract lemma is that it says that if you have this function fee. So, that's not converging. So, that's not converging. That's not converging with L2 in the, in the week L2. This is the conveyance week out to then, then there exists a non constant function, constant function. In L2, this is that fee composed with fee Tn is converging to xi in fee composed with fee minus Tn. This is also converging to xi. This is L2 convergence and for some subsequent Tn is a different sequence. So, this is very abstract, this is what we are going to use to prove mixing basically. So, see that if this convergence was point wise, you would expect this function to be constant on stable leaves and so constant on unstable leaves because if this convergence was almost surely. So, you have, if you take two points in the same stable, you would, if you use the fact that stable is contracting. So, that is this part actually. That's that there exists. A bar full measure, full measure, me is a measure we construct. This is that if subsequent W isn't W, SSB, you have the distance between Tn and B as in constant. So, let's just say that if I'm in the in the expensive set, the set where you have, where you have, you know, this model spirit intersect like this. You can choose a sequence of time for you to have this and this is a trigger for if the coverage is negative for instance in the annals okay we have this always and it goes like this. So, I wrote this fact because I was saying something that was going to use this fact that if this convergence was almost surely surely, and this sequence coincide with this one, you would, you would get that he will be constant. And that will be constant in a stable set and also be constant on unstable set. And that's something. So to get that's something useful and to get the almost convergence you use a kind of the chase algorithmic so you, you, if you have this convergence out to you can define this, you know, you can see that this function. You can see that this function converge to that almost surely and plus or minus actually plus or minus almost surely. So, this implies that for instance, if I am in this a, so this is why that is the W belongs to W SS. Then, this is what I said so, and then a similar thing called if you are also in the same unstable so it's done. They won't be used for the one day minus one. Yeah, so, so see that me to do this group we do it by contradiction, we support that. It doesn't converge we have to. So, there's this lemma by a popular that is this function that is not constant, the fact that it's not constant is beautiful. I will say why, you know, it's a non-constant function, which is constant in a full set like if the V, or I forgot to say, if the. For this is true for me almost everything. So, for being a, okay. For all being a something this whole for all being a. You know, there are some technicalities with this subsequence, but here the fact that we can have the subsequent that works for all be that's something good. But you want the subsequent sequence. The subsequent sequence you get from barbillos here, but that's also something you can do because that actually the demo tells you that for any subsequent you have a further subsequent for any sequence actually you have a further subsequent where you do this so there's a way to this, this, this sequence is I'm having but just to, to not be very technical, I'm assuming that I have the same, the same sequence, okay. But there's some small technicalities that that can be fixed that. So, now what we do is we take a lift. So, like, so we suppose that we need to feed. Tell that I write it here again be a lift the universal cover. And I consider this map here. The multi. This is something crucial that gives feet. So I like it as a saying at feet TV. So, okay, I should, I should do something. I should be very explicit here. So, so if you give given to me. So, I'm, I'm now being in this full measure set where I have kind of this type of policy given being a, I can define this map. This is a fact. So now I do this. So this is defined from a to the subgroup. It might be brutal but we'll see what I'm saying. The subgroup that takes you give me V. It gives periods of, of this mark of, of this mark of T is xi equals this TV. Let me spend some, some minutes in this month. So if you take B, you can look at the periods of this month. T gives this map. Okay. You look at the periods of this month. It's a subgroup of half. So you have this map that's defined from a. So think of, okay, this is just a negative curvature. You would take the full unit and a bundle, but I'm just, I just, I just want this type of policy. So I have to reduce to be say A. This is a subgroup of R. V of V is a subgroup of R. The set of periods because it is added and it's a subgroup of R. There is some technicalities that I'm avoiding here that, you know, this map might not be nice in the flow direction. It might not be continuous in the flow direction because this will just help to function. But up to, you know, making it nice in the flow direction by integrating, you know, you could replace, you could replace. You could replace V by this guy, the integral of some epsilon F. And this will be nice in the flow direction just to make this subgroup work nicely. So yeah, so you suppose that this map is nice in the flow direction. So few of these are subgroup of R. Okay. So this map is invariant by the geodesic flow, obviously, so she is easy to see that V of V, it was free of TV, because it belong to the same word, so they have the same previous. V. So he's invariant in the geodesic flow. This is invariant. So by a good city, this is the point where we use a good city of the major meal. So by a good city fees constant for almost every or me almost every week. So me. So basically I did not talk about it, but it follows from uniqueness. Okay. Me being everybody in life. He is constant for me almost every day. So there is one subgroup that works for all fee for almost everything. So in particular, you can write fee at V equals easy. Okay. At the questions about these markets. And maybe skipping from start. Now this is not the time we will introduce the to get the contradiction we need to use this course ratio function that I was talking about in the beginning, which is similar to what was used by Caravaggio, liberally improving exponential mixing to get a contradiction from that. So, I mean, I'm defining this function inside the proof, but so I would say so, so close ratio function. We'll come back to the proof I've made you find this close ratio. So this is a function that is defined in the boundary. So you take four points in the boundary gives a gives a number. So what is it doing, you take this. There are there are the definition of the process of function, but it is equivalent to this one that in the same. So I have your four points I eat up. Eat up. So this guy is joining this to this is doing this to this is here and one here. So what is the cross ratio. So you start with. So there is a, I mean, I will say a problem, but let's suppose that side. These are chosen in the set of expensive factors. Okay, like they are. You don't have two geodesics joining these guys. Like, these are the only two days. It's not necessarily, but just for the future. Let's do that. So if you take this whole sphere, the whole spirit is take start with any point here. You go stable and you end up at this geodesic you take out the whole sphere, going through this point. And you are here, you take now the whole sphere, going through this point. And you're here, you're taking a whole sphere through that point. And this distance you get is the process of function. This is V. If this is V, this is. Fee of the cross ratio. So this distance here. Is. So this is just a lot of if you're familiar with the temporal distortion function, which you go stable unstable stable unstable and you measure the gap you have in the fluid direction. Exactly what this is doing. So I was stable, unstable, stable, unstable, and then I measure the gap. So, so I live this for this part now so I said that these geodesics they belong to the set a, okay, this geodesic they belong to the expensive saves. And then, so the function, if I take V here. But I'm just going to say is that the cross ratio, this guy is a, is a time. Is a period of this map of TV, where V is this guy. Because of this. So, the function five is the same from this vector to this one. So let's say this is V, this is V one. And you go this is V two. And you go here, you have been treated. And you go here you have before, okay. Two. Yeah, V. So, by this thing here, the fact that Xi is constant. So I have side. So why, why is it that is I, V one V, because I, if you want, because V and V one they are in the same. Okay, this is unstable. I say I was saying, but this is unstable. So V and V one, they belong to the same unstable. So you use us to say that is the same. And then it is the same or so in V two, because they belong to the same stable. So this is Xi and V two. And it's the same at the three or so. Which is the same as Xi before, but before is what before is just the procession. I'm writing this guys. You said this guy is a period because it is a period. And, and yeah, so this guy is continuous that this function is continuous. Why is it continuous because I said last time that these forest fields, they depend continuously on Xi. So this guy is continuous. It's continuous. Because I said if I spend this function be, be, be, be, or if you want Xi gives the team that is continuous. You use the continuity of the forest fields, which in a group of two we know the stable for the Asian is holder. So here we just have continuity and that's enough. So how do we use that is that now if I take Xi Xi, this guy is zero. The procession of this guy is zero, but I can change this a little bit, change this guy a little bit this guy a little bit, or just change this to a little bit and I create something that is absolutely a really small. If you take. So this is why that, you know, you can find that this is again using the fact that the major is a product major in the major is not a product major. You cannot, you cannot be changing these to, you know, you can be changing any video. So you find that I'm actually small by, you know, this. So which content is this guy because there's a period. Again, oh, sorry, I forgot to say that a is not zero. Actually, a is not zero. Why, because this was not this map was not constant. Actually, yeah, this was crucial. Why we need this much money is a is not zero. Okay. And it is different from zero because the from the bubble is that my side is not constant. So I create a period which is actually small. And this is the previous artist. So that's a contradiction. So it's, it's just using the continuity of, of this cross relation function and the fact that you have certain have a policy. This guy is certain have a policy and also the fact that me is a product manager to be able to to this guy and if you do. So these are the people but it's all. So these are if you want to remember what is used here is that me is product. This is used to be able to put up this guy individually. So have a policy. I mean, some kind of happy policy is very happy policy on the support of me, which is the set of expensive factors. And then continually, and then continuity of this function, continuity of this function, this question. And every city of course, yeah, because every city was used to say that is exactly one. And the questions about this. So, so I'm going to show you like some steps at least to do certain counting of closed geodesic, some steps you need the steps where you use, where you use this guys like the product measure and mixing. Yeah. So, let's go. So this is about counting. I have an hour which is given very sketch sketchy proof of this. So we are in this energy is a surface. Remember drama was a group of eyes on this. As a matter is, yes, which is it. To do this counting. So, let's say what we want to do is we want to count the number of so problem number to this counting we're going to concentrate the problem, the problem of, you know, we want to identify close geodesics. We use elements of the fundamental group, like elements of it as a metric group. So there is this lemma which is not actually difficult to prove that for every gamma gamma. So this is the geodesic C gamma C equals C, like this is left in vain by the azimuth element by gamma. And in particular, in particular, you have that if you project C C project to a close or to a projects to a close geodesic to a close geodesic. The proof of this, I can just give you the sketch. It's not very complicated. So proof of this is, you know, given gamma, you can define this quantity L equals the infinitum of distance between P and gamma P, where P is in X. It's not easy to prove that this supremum is achieved. So there exists P0 since that L is the distance between P0 and gamma P0. So this you can write as an exercise. So I don't do everything. So as an exercise, this is the fact that this portion is a compact portion. You can choose elements of the fundamental group to bring everything, every point that is going to escape and bring it to a compact set and combine anything on there. So the geodesic joining the geodesic, let's call it C P0 gamma P0, the geodesic joining P0 to gamma P0 is invaded by gamma satisfies gamma C P0 gamma P0 equals C. P0 gamma P0. Why is this fact is just the fact that it just uses the fact that it was an argument. So I have P0 and I have gamma at P0. Okay. So if this geodesic, so this is C, so let's go and see this. So this claim tells you that gamma marks this geodesic in the same geodesic. If that wasn't true, let's say gamma marks this geodesic one as a geodesic. So this is gamma C. If you take a generic point here, T, C, T, and a generic point here, gamma C, T, like the image of that point by gamma. You can see that the distance between C and gamma C T by just using the quality you can see that. And the fact that so this is strictly less than the distance between C T and gamma P0 plus the distance between gamma P0 and gamma C T. Yeah, this is great. And this guy, this one here is L minus, this is L, this is L minus two zero, L minus two. Okay, and this one here, because gamma is an isometric. So this guy is just, it's not the distance between C T and so this is T plus T. And this guy is strictly less than, strictly less than L, which is contrary to the fact that L isn't true. So the picture is that C is marked GZ, is that right. So already given any elements of the isometric group, there is a gamma, there is an X, an X for this element. And so, yes, and then there is this quantity, actually L, the size of gamma is defined as that L is a multiple, you can see that is a multiple of, is a multiple of the period of the length of the exposure, is a multiple, the length. So finding, okay, but this is just general. Now, how we do is now we, we fix a set, but now that we have actually to come, first of all, we define a set, a set, you know, a nice set with some property. I'm sorry, I always forget to look at this camera. There's another camera there, but this is the one I should be looking at. So, we build a set, which now we are going to come the number of closed geodesics in this set, in going through this set and actually after that there is another element to, you know, to come to total geodesics on the, on the, on the whole unit in Bangal. So, there is this set that we define like this. So there is. So, this is Mark, which is called the, the hub mark, which is defined from V from SX, sorry, to the boundary squared minus the diagonal. So, I'm just going to build a nice set. This is using the argument of the Margules argument in counting, which is, he has this detector, this nice set where you have, you know, created by stable and unstable nicely and then you can count in that you can do the counting in that set. But here, we, we, we build that set just using, using this, this map. So, to R, you give me V, it gives V minus V plus and V, V. So, P was fixed, okay. P is fixed, okay. P is fixed, BP. So, this is just, you take a vector. So, this is, you take a vector, you send it to these guys. So, this is V. I have P somewhere here. P is here. At V minus, I know P was going here. So, this is V. This is, this is V minus and this quantity is just this guy here. So, this guy is BP. Now, you can build. So, you have this, you build this guy, V for some theta is, so before that. So, remember, I pick, I pick P. So, I also pick V zero in SX. So, this guy to be an extensive factor. That's a technicality. We will, we will do that and then see that we can define the set, the R, the theta. Oh, sorry, before that. So, you define the set F theta, which is a set of W classes that the angle between V zero and W is less than theta. The integral feature for this set and PT is the set of W minus less than theta. So, this is what, so I have here V zero. I take an angle theta in a project then. So, this set is F theta. So, this is theta. This is F theta and I do the same and this is V theta. Now, the set I want to build, I want to build a set around this vector. So, V theta is just defined to be, I'm just, I'm just trying to think about how much details I should say. Now, and then some epsilon here. This is different. Theta and theta and zero. So, so this is a set that is around here. These are vectors that they send to this set and they project to this set at the boundary and then you take epsilon along the flow direction. Okay, it's just like you have something here, and you do, you do excellent thinking of these guys. This is epsilon. Okay, this is what this is doing. And now what you do is you look at, so being closed. It's, you know, if theta, if C such that C is geodesic, geodesic, such that C at zero is in V theta epsilon. And this is like that there is an element. So it's a close. So there is an element such that is that if you do this, if you do phi minus gamma applied to gamma star of C prime zero. This guy is again so by this is again C prime zero isn't the data epsilon. What I'm trying to say is that if you're capturing close geodesic that I'm going through be, it's like doing taking elements of the isometric for which this set is, is, is kind of non empty. So I'm just trying to motivate the definition I'm going to write now, which is this gamma T. So gamma in gamma says that P minus T intersect P minus T. So be theta excellent, just like P minus T gamma star. Be theta excellent. This guy is not empty. So this is just now one step towards like, you know, can't you close geodesic going to be. So these are the said that if you, if you take the, this is the epsilon. You apply gamma, it gives again is gamma star B. And then you pull it back by the dynamics is going to be something like this. So it's kind of a recurrence like this orbit will be kind of recurrence to this to this set, you will have an orbit that is recurrent to this set. So the one step is so I should really maybe not go very deep into all the technologies here, but one step is, is that this set is in bijection. This is the set of so the T is a one to one correspondent means the set of closed geodesic will be will be be the epsilon of length. I mean, this T plus or minus some maximum. Okay. And what I was just saying is one way like this part is just what I was trying to say here, but the other part is a closing lemma. So to prove this by this injection in this set. This is the index to this set is injection is a closing lemma that is not very difficult to prove that I did not talk about the group. And yeah, and once you have this, this bijection, you, you, if you want to count the close geodesic through this set, you just count this number of guys. And you can see that the major meal of the data, I'm not writing all the details intersect with. So, because this set actually he will write, you can write it as a subset of the union of gamma and gamma of the data intersect. So the measure of this set is like the cardinality time measure of one of these guys. And it's not difficult to say that, like, this guy they have roughly the same measure so the measure of this guy. The part of the measure. Excellent. Excellent. This guy is roughly the cardinality of gamma T times the measure of these guys. And then you, you can use mixing to prove that this guy is converging so this is mixing. This guy is going to measure of the two types, you know, squared. Okay, roughly. And here, that's another thing of the team. Here you use the construction of the man, but that is given by this, but some new measures in particular the product structure to say that this measure is roughly exponential of minus ht times the measure of the two types. So, this part is using products factor like that. Yes. And from here you can, you can see that this cardinality now will go like I'm sorry I'm being very, you know, very precise here but there's no time I just wanted to tell you the two points, the two points mixing. But central one measures they are using these two steps in counting here. And then this is like the ht times the measure of the data. Yeah, and then there is a way to generalize so if I want to summarize everything that I have been saying here we prove we prove. This is two terms in two different papers. So these are joining work is. Clemenaga. You can have people in myself that says that you have energy surface that could get points of Jesus at least two. And the geodesic flow is has a unique MME unique MME meal and that the flow is mixing. Actually, this is better only actually. Yeah. You can prove that it's barely. I mean, the very property from from the construction of from the from the Clemenaga Thompson result, and that is more like we have this and new is fully supported. But you can see it easily by the fact that the person to measure definitely supported on the idea of boundary. That's what we've been like. Well, that's right. And then I saw that me is obtained, can be obtained. And the long clause long clause long clause geodesic. This is not very clear, but I'm just going to write the full term here. And the other result is that in another paper. These are two different. Same. Is that you have in the same setting surfaces like the HD over HD where PT is a set for the classes. And geodesic geodesic because we know it's one geodesic. Thanks. The close to geodesic. At most. So this is known as the, the margulis estimate my duty for typically mixing and also flaws and then there are many results like it's again proved by it's also proved. In the case of one cat zero spaces by. And also there's results for actually flows for instance by. But and, and, and very for actually flow. So there have been many results on this, but yeah. So there are different type of estimate that was studied in the literature, not exactly this one, but this is the sharpest. I think you can get in this counting and yeah I'm sorry this last part was very very sketchy. I'm dressing to read the paper. These are the two things where we need mixing in the passenger when construction. So, I'm going to, I'm going to end here so I'm open for questions. Or yeah, we have to answer questions. So, thank you for. Thank you very much had him. Thank you to everyone who's followed the course.