 I think that's about it, well, that's exactly what I'm talking about. Okay, I understand. Okay, so welcome back everybody. So today is the lecture three, and this is a lecture about Judah's flow, even on manifold without conjugate points, particularly sources. So, lecture three. In this lecture, we mainly focus on measure of maximum entropy via tax and silver measures. I will construct a measure of maximum entropy. Let me just say what this is about. Basically, okay. So, we support that we have mg is our manifold with our conjugate points. There is the usual, I mean, we prove in the first lecture that the universal power X universal power is a copy of R and if any of them is a family. This is a copy of R. And, you know, we want to talk about maximum entropy possible there's a notion of entropy that I mean, I suppose I must people know the definition of the topological entropy for the dynamical system in general, but in this case of geodesic flow. I just want to use something specific like it's not the definition of entropy but it's a formula that is specifically do as it flow. So, I want to start with this. I mean, I mean, you didn't know what it was, but it was already done a negative curvature by money that you have this one that you call the topological entropy extra of the geodesic flow. So, you know, we change the number of the location of the flow is just the limit when are those infinity of one over our love of volume of the P. Okay, so this is the topological entropy for the geodesic flow respect the semantic where for any being in the universal color in it. Okay, this is the definition but it's a characterization and it's going to be a geodesic flow and this is what I want to be using. You know, it will be helpful to think of the entropy as a volume of balls, the growth of balls in the universal color. Yeah, and you're already from this. You can see that in services. Corollary or a negation energy is a surface, the compact surface of genius at least two, at least two, and it's true that the topological entropy is positive. Why, because if I'm on the surface of games at least two, I know there is a magic of negative coverage inside and the magic of negative coverage and we know that the ball they grow exponentially. The things are comparable, like the second one of the problems are comparable. So the volumes of this ball is comparable to the volume of the ball in negative culture. So basically, the pool is that there exists G zero of negative coverage. And you know, by attributes of the matrix, the volume in G of the R is comparable to volume in G zero of B. Okay, and then you see that when you have a background of negative coverage, you always have questions to entropy. And now there is a usual, the very well known rational principle, rational principle. Because the topological entropy is a supremum of all metric entities. So it's top of the team is a supremum of each new P one. The meal is the one in the end. And probability measure. Okay. And this quantity here is a metric entropy. I mean, I'm not going to write the definition, but today we come to the metric entropy for a specific for a specific measure that I will construct and then you will see the definition if you haven't seen it before. So it's a measure. So today's lecture is construct a major meal. That we like to see people. And that man is called me as a man. Okay, so I will not. There's too much time and so construction of the major. So, yeah, so the rest of this talk will be just going to attempt to measure and then calculate the entropy of that measure. Exactly. So some notation. So I like this for this was in the city. I just write it by age. And then I write gamma to be the ultimate to go up. And I mean, it is very standard that I'm not going to prove it here but there is a isomorphism between gamma. And in the first fundamental group of M. Like correspondence like elements of the fundamental group apps on the universal cover as, as a reference and you can write them. As a compact portion by gamma. So come as a group of isometrics and to start. So, see what we want to build is we want to build a person measure that is in pain by the field as. So remember, so I'm just going to tell you the. So, first of all, you want to remember this was last week we define our competition of the universal cover with a boundary. Okay, you take your design grade. Okay, the equivalence classes of your design grade, you want to construct a major that is invariant by the geodesic flow. And if you take a set, you let it by the geodesic flow. This is that they have the same measure. So in particular, this measure should project to the boundary that you could project this measure, you know, you could project this set to the boundary. Basically, what I'm saying is that you have the space of geodesics, which is, you know, one to one correspondence means. Okay. So what I want to say is that if I have an invariant measure here, he should define a measure on the space of geodesics, which is just the product of boundary minus the devil. So, invariant measure defines a measure on the product of boundary on the, you know, measure on x square minus the diagonal, because it just project to the boundary. So what we're going to do is that we will construct a measure on the boundary, and then we will now want to, we want to project it to the unit line and boundary of the universal curve. And how to do that is this part is absolutely one measure so first of all there is this point kind of series point kind of series. And it's defined by this quantity, so you pick it in X and S in R, you define this quantity, P, P, S is just the sum along the isometric group minus S distance between P and gamma. This is a point kind of series. And, you know, it is proved by Kornier that this, if you have a, you know, a Gromov hyperbolic manifold, which is in our case, so Gromov hyperbolic, you can see at this condition. If you take any pair of points, any super points, P1, P2, PR, P3, if you take the geodesic joint in these guys, you can find a fix, you can find the existing R positive, so that if I take a R neighborhood of this geodesic, R neighborhood of this one, I covered the whole, the whole triumphant. And this is true in negative curvature, and it is also true here because of the, of the existence of background magic of negative curvature, because if it is true in negative curvature, you can take any geodesic, you can find the corresponding correspondence between this and the other geodesic in no point get points and you can easily see that this is true. So, Kornier, that Gromov hyperbolic, this is the kind of hyperbolicity that you have in negative curvature, and here you have it in the case of no point get points with a background magic of negative curve. And the Gromov hyperbolic implies that this guy, this point characteristics is finite when S is bigger than H, if S is bigger than the topological entropy is finite, and this guy diverges if S is less or equal to H. So, how do we use this? Will you just point characteristics now to define the family of measures? So, we define the given, this is the usual, you know, when you want to define invariant measures, you define dirac deltas along with orbits, and then the limit measure you get the limit will be an invariant measure. We're doing something similar here along the common orbit, an orbit of an element by the isomac group. So, given P, X, in X, and S in R, let's say S bigger than H, we define new P, X, S to be one over in X, X, S times the sum, gamma in gamma of exponential of minus X between P, gamma, X, and the dirac delta along the lines. So, you're just taking the dirac deltas along, along the orbit of X by gamma, and you normalize by this one. It's not difficult to see that this measure, any limit of this measure when S goes to H will be a measure that is supported in the boundary, because of this way, this way it's going to infinity. So, if we take a compact set on your universal cover, the measure of that set is going to zero as S is converting to H because of this. But, more precisely, we prove this, we prove this proposition, there exists, there exists a sequence S k converging to H, such that for all P in X, new P, X, S k is converging in the weak start to new P, and moreover, we have these three properties. These three properties are very important. I mean, each of them tells you something about the measure we are going to do that in. So, new P, P in X is gamma equilibrium, gamma equilibrium meaning that if I take a border set in the boundary, so as I said, this any limit will be a measure supported in the boundary because if you take a compact set as I said, this way it's going to zero. So, you can think of this measure being a measure defines only in the boundary. So, if you take any border set, you have new gamma P, gamma A equals new P A for all gamma, for all isometric. Okay. So, it's going to be very useful to the measure we construct, for the measure we construct to be to be invariant by the geodesic flow, because know that as a method, they, they commit with the geodesic flow. As a method means that it's a geodesic, so you commit with the geodesic flow. Having this property will help to define the geodesic that is invariant by the geodesic flow. The second property is also very important, which are that these measures I'm not saying a lot. They are absolutely consistent with respect to each other. And the density is given by a Boozman function is exponential of minus H, B P, U Xi. So this is for Xi in boundary events. This is also very useful. This is like, so remember, B P Q Xi, where the Boozman function attached to the geodesic going to be desired. So, so this is die. And this is B, there is this geodesic, it defines a vector here, and then you take the Boozman function, you know, the Boozman function was defined as B P Q Xi, because B P Q, where V is this vector, is the limit when G goes to infinity of distance between C, B, T, Q minus Q. And define this in this lecture. These functions are very useful here as they define the density of these back sensorial measures. And the last property is also very useful, it tells you that the measure that we will build is fully supported. The support of new P is about 100. So this will be used to prove invariance, this will be used to calculate entropy, and this is used to prove that the measure we will construct is fully supported. And this measure will be measured in the open set. Right, so let's prove this two properties here. We need to talk about limit, weak style limit for this measure. We need to see, first of all, this measure should be, you know, the client measure and then like it gives the client measure to the whole compactification of the universal cover. And then you can, if you fix now a compact set, you can find, you can find this subsequent. But now there is another trick, I mean, it's not really, we will do to say that this same subsequent war works for all elements in the universal cover. But let me just say that how, why these families of measures. These guys, new PXSK of X union the boundary is finite. Why, because if you look at the definition of this measure, if you look at the definition actually it is. If you look at the definition, you just use kind of triangle in this one here, exponential of minus S distance between X gamma P gamma X. This guy is that's an exponential of S distance between X times the distance exponential of minus X distance between X gamma X. And here also you have similar bounds to this triangle and equality exponential of minus S distance between PX exponential of minus S distance between X and gamma X. So if you sum over all, and you divide by PXX what you will get here will be one. What you would get here will be one and what you would get here would be, would be, yeah, what you would get here so we'll be one because by definition of the point that I say you divide by this quantity. What I'm basically saying is that new PXS of X union the boundary is less or equal to exponential of S distance between X and exponential of minus X distance between X. Okay. So, so, so this guy being compact, they said being compact so you have at least a subsequent that converge. So if you fix, fix K compact. So there exists, SK, converting H to that new PX, SK is converging in the week start to new P, so they are defined in P as a limit. So for all P in K, all P in K, this is just using the fact that the space of probability measures compact or finite measure in this case we just normalize, but it's a compact, and then you have a you have a compact in the week start topology so you have a you have a limit. If K is compact, you have a limit for HP, for if K is compact, you can have a limit for, for all points in that compact set. But now you want to replace this set K to X. Here you just use the gamma values of these measures. So basically what I'm saying is here by definition here. So by definition, new PXS at A is the same as no gamma PXS gamma A. Just by definition here. So, you see that these measures are gamma equivalent. So, new PXS is gamma equivalent. So it's implied that if a subsequence works for a compact set, it works for gamma of that compact set for gamma is in top. Same here is so same subsequence works for gamma K into K for every gamma. So you have a formula you have and you can write M as the union of K of gamma K gamma and gamma. So you have one subsequent that works for all. And also the limit measure will be gamma equivalent because the sequence would. So A is gamma equivalent because new PXS K is. So it's gamma equivalent. Just by construction. Okay, this is by construction. So this proves that this path. Now to prove this path, this uses something that is not very trivial in not conjugate points. And if you can see it very natural. So, let's write again the definition of the measure. So it would be, you write, you know, PXS is one over PXS. It's now gamma and gamma for exponential of minus S, P, gamma X, and then delta, gamma X. What you do is you add and subscribe to here. Distance between Q and comma X. So you can get a new PXS is one over PXS. And the sum gamma and gamma. So this guy exponential of minus S distance between Q and comma X, and not exponential of minus S. Sorry, I'm forgetting H story. There is an H. Oh, it's not H story. This is S story. So, this is S and S before which hours. So this is now distance between Q, gamma X, minus distance between P, gamma X, and delta, gamma X. So I just add and subscribe this term. What I'm telling that it's kind of easy in negative curvature is the fact that this quantity here is converging to the movement. Let me go for that. So think of this is what you have here. You have P here, and you have Q here. Okay, you have gamma. The orbit. I did not say it, but this guy, the accumulation set the accumulation point of the orbit of any points from the input to cover by his book is, is the whole boundary. This guy is going to be the boundary. So you can think of this gamma X as a point that is converging to something in the boundary. The thing of this is gamma X is a point that's going to the boundary. The thing is, you're measuring. You are taking the difference of distance when this point is going to the boundary. The boundary. Okay. And, and see that if you think of the definition of the, of the postman function, this is exactly going to converge to this guy, which is the movement function. At the geodesic where, so let's say this guy is going to die. So you can see that this distance. So, in the, in the non-quadricate points, it requires a two-perch space, you can just see that the negative curvature of the whole sphere, there are limits of spheres. So you can just think of this point, you take a G circle or sphere around this point, and that guy is converting to the whole sphere when gamma is gone. Gamma X is converting to psi. So, this difference, which is this quantity is redefined to movement function. So the limit when Z tends to die of distance between Q, Z, Z minus distance between P, Z, this is B, P, Q, Z. This is what I'm saying basically. This equality is true. We proved it in, I should know how it's done. So, yeah, I'm going to go to the proof, but this is what we use to see that this is going to be the movement function. The picture is creating a curve so that it's a, it's a, it's a fact. I mean, you can pull it from the back of the sphere. Or another way to see here also is that the movement function depends much more easily on B. If you have this function, it's a movement function, but it depends on B. If you prove that it's continuous, you can also have it. Yeah, so that's just, it's a proposition. So, yeah, so basically that will give you, that will give you part two, because what you have here is a movement function. You have to make this new P, new P, new P, okay. And the last part here, you see, you're supposed by contradiction, suppose that there is a, suppose that the support is not the whole thing. That means that there is new in X, open Z, new P of new zero. So in particular, in particular, new gamma P, gamma U equals zero for all gamma. Okay. Because we have gamma equivalent. Okay, this is going to be an issue this day. Okay. And if you use B now, you see that this implies that new P at gamma, because they are absolutely different. These measures. So in particular, new P. This is for all gamma. So this is something I didn't say, but it's true that this actually is transitive in the boundary, but this is something I, I didn't say it, but okay, I should write here the union, gamma U of gamma and gamma is zero. And this gives a contradiction that these measures are completely respect each other because this would give zero to the full boundary. I did not say it, but the action of gamma in the, in the, in the boundary is transitive. If you take an open Z, you're going to cover all the, you're going to cover all the, you're going to cover the whole boundary. So, this is new P of boundary of X is zero. So you have these three very important properties. Okay, so, yeah, maybe I can erase it now. So, yeah, so what we want to do now we want from this measure, we want to build from these families measures we want to build a measure on the unit in a bundle that is in vain. So one way away to do it is, you know, so now I have this family new people. So they satisfy these three properties. And so now I can define, define, define new bar on the product of the boundary minus a diagonal, which is like the space of your desert by the new bar. So you put the density here. So I will say what is that. Minus. H. I will say what it's going to be. So where the desire is going sometimes that some people call the promo product of the whole spherical distance, which is like this. If I have X union, the boundary of X, I have two points like eta. I have a point somewhere. I look at the whole sphere as a P. And then the whole sphere so that is a psi and eta going to be, and I take a geodesic connecting these two guys. And this distance is beta. So there are some technicalities here in non conjugate points because I might have multiple geodesics joining side to be tough. But you can prove that if you have also another geodesic joining side to be tough. Just do this inside the same. Okay, you can prove that it's not very difficult. You just use the ingredients of this. Yeah, so that's this quantity here. You can also define it by who's my function. This is another definition. You say plus. Yeah, this is the same definition geometrically this is the point. And Yeah, so you have this measure and this is very nice actually because it's a product measure and that's this key in what is going to come next over in the next lecture, but that this measure has a product stretch because it's given by I just an exercise. Take that this guy is going to buy it. Take the bar. Because remember, we want to build a measure that will be very, very good. Yeah, so. And this again another technicality that we are facing in this bar in this case of geodesic. So the problem with that point get points is what I just said here, if you define this map. So he defined from sx. This is the unique and your bundle to the boundary of x square minus the diagonal. So what does this do it just takes the needs be minus the class that's just defined by. So I have the, I just take the two end points. So this is the class and this is the minus. But I'm saying, even in no, no, no positive curvature, this map is not, is not bijected. It's not a bijection. But think of, think of just this simple case. You have this simple surface right here you have negative curvature here you have negative curvature, and you have here the curvature is zero. So all these geodesic here that have closed geodesic. If you look at them, they stay at a boundary distance. Even if you can think about it is there a boundary distance. So they define the same end points. So you might have this picture here. So he is not a bijection. So, and to project this measure to the unit engine bundle, we would like to use this monkey. I mean, if it was a bijection, or we would say your family is this term of expensive. Which means that you don't have this feature. You could just project it using the monkey. Using the inverse of the map. Okay. But here we can do that. But what we could do, which is, you know, very technical, is that this is what we do in our paper. Yeah, in our paper is that there is a measurable choice of an inverse. And that measurable choice that choice is going to buy it. So, there is measurable we needed to make this study like this choice of because one of you want to project it you want to choose the guy. So this choice we have to do it globally in a measurable way. I mean, in a gamma where it was born to live in the measure that we're building. If I'm going to do it. There is a, there is a measurable choice of an inverse. So I mean, so this uses abstract type of unbalanced and stuff, but you can look at the papers. There's a measurable choice of an inverse. So that's just right. And that is gamma. You know, gamma in the sense that, you know, if you do a minus one. Oh, if you do this if you do gamma, gamma xi gamma gamma minus one gamma, but you get is minus one. So, once you have that, you cannot put it is measured to the unit and you might want to do this. We take for a measurable set in the unit and you might want to take me to love. Is, you take the, where minus the diagonal and the lines of a intercept. The length of a intercept means E minus one. So this is this choice. Okay. So this defines a measure on the unit tangent bundle. So maybe that's X. And maybe it's going to end by construction. So maybe by its flow invariant. It's been done by the geodesic flow. And, and moreover, you know, see that this measure is a finite means on compact sets. So if I take here a compact set. So, it's a compact and saying that we told us okay is fine. Why is that because look at the only way that this quantity. Blow up to infinity is that this guy blow up to infinity. The problem of product is going to infinity. And this problem of product is going to infinity only when Xi and data like Xi is approaching it. So if that is approaching data, you see that the problem of product is going to infinity, but that's not going to happen when you fix a compact set. When you fix a compact set these two. So the, the, the projection they are, you know, you have this one is uniform by something depending on the format. So once you have this, and me told up is that my variant imply that it goes through the portion. Me told up. You find, you know, you send to the question defines a major. New one. Do you need to change a battle of your money. Okay, because remember the money for was just combat coaching by gamma. Something that's invaded by the group of eyes. Okay, 15 minutes. All right, so I have to prove now that this measure has. So if you don't know yet what the metric entropy is, don't worry too much, I will, you will see something that, that defines it. So, of me you is H. I'm sorry, I might be very. So there is this. Again, before we do this, there are some preliminaries. So, see that there is. But is that given. Yeah, so see that. So to cut to calculate entropy, the metric entropy, you have to calculate, you know, measure of. And to cut the measure of some of these sets, what we do usually is we project them to the boundary, and then we look at me bar on the boundary. So we're going to define it is this kind of projection. So, even given side, the boundary of x. We define this kind of projection. The boundary in PRP, which is from x minus p to the boundary. So this map is this mean this. This is a publication. This is I hear, and this is, if you take a point. So, you take it on inside x, and you send it to the other side of the boundary, and that gives you this. Similarly, here, this mark. This is a useful tool, you know, we do project the sets to the boundary to be able to calculate the measures. If I take P here, and X here, X, another point. I also draw this here does it. And then this is PR this point here is PRP x. Yeah, that isn't just defined. And this is back. This is why I mean, I don't want to write it. You will see that so that for all x in x bar, so I should have been using using. This guy is a publication. And P in x. There is an answer that this guy contains an open set. PR x of B contains an open set. You project a ball to the boundary. This is an open set. This is almost trivial by again, using most correspondence. So, yeah. This is just saying that remember how open says I define the boundary to find an open set, I fix a vector in an angle, I fix an angle epsilon. I send this, this is the boundary. So this defines an open set. So remember the open set, remember this map that P, which is S, P, X, the boundary. So the topology is given by the, you know, for this map is from a map of this. So this is how open sets are defined. And, you know, you see this part you use this. Because you can do it connected culture. And you do it. And after that, there is this position that measures now is that so tells you that for all row bigger than these are. You just say that if you project these guys because all these sets, they contain an open set, you just proving that this open set is not like getting smaller like there is a, you know, there is a lower bound in the measure of the open set that you get from projecting a board that is big enough. Is that one over me. I should just say this new P of the R side, the X row. This guy is comparable to comparable. I mean there's a constant. You know, from inside is comparable exponential of minus H distance between the X. And you have the same thing. So that is in the boundary. You have also the same thing when you take X under. So I should say, this is true for all. Okay, I should put here X. See, is that new P of the X row is also comparable to exponential of minus H distance between PX. The constant you have might be different from here and there. Maybe I want to do this. I have just done minutes to. So, let me just say in words why, by this is true. For instance, see that B and C, you just get them from, from a. To just a sketch because I'm running out of time and I just want to, I really want to prove that the measure has put entropy bed. So, a is B and C. Why you just use the property that I proved. So you use this property. This guy is exponential. So if you have, if you have. The opposite A will be just given by the interval of exponential of minus H. B, B, U, Z, B, U, Z, A. And you can see that. You can just see that this is the same as, you know, writing, you do this. I said I wasn't going to do the proof but now I'm doing B, B, U, Z. This is of the same order plus distance between. I add and subscribe. Oh, I should. I did something wrong here. So you add and subscribe and see that this is of the order of a constant times the measure you add and subscribe the distance or this is of the order of distance between. So you, what you have is a constant times exponential of minus a distance between. I don't know why I was here where I was using X here. So, I mean, I might have been doing this proper because you add and subscribe this, this quantity this distance and then you see that the other quantity you have will be uniform about it. So a implies B and C, but to see to get a. So I will not do the proof of a, but I really want to have. Or maybe I should just do it and then next lecture I will start by proving that then for being full. Yeah, that's, I think I saw five minutes into two small. Okay, I mean, I would just prove a, yeah. And then in the next lecture. We will just need 10 minutes to prove that. Let's try to prove a is that this says they are not. So first of all, there is this fact that we started with that. So fix the compact set K, compact set X for row bigger than R. R is a fixed radius, and this R will be given by depending on the most correspondence that the distance you have, you know, the area having the most level. There is, there for every accident, there is an excellent way to go. Sorry, there is absolute positive sense that if you pick, if you pick me, I will draw a picture for this. And X in the closure. So you have the C epsilon. I will say what I was going to be the subject. So for being for some being as X, X, and C epsilon is this guy, right? This is being this is being and C epsilon is this open set you get by taking an angle around me. See, absolutely for some being. So the picture for this, but is that. So, this is what I'm saying, you have your partification. And you have a second K. Okay. So this is X bar. And you have X point is in X. You want to find a vector at X. You want to go around that point. So for instance, if I take this year, there are the people that is back to that goal, right in the middle of K. If I project. So, so now, absolutely, absolutely be here. The accident will be here. So, obviously, if I project this set here, it's going to contain CV actually. And this big set here. So this big set here. Will be PX of B. Yeah, just take a back to the book. Yeah, and then, yeah, this is the idea to get this fight. And from this. We know that because the measure is not singular for every accident that I just have that might depend on that. This is how C of C epsilon B is bigger than L. Because these are open sets. This is how open sets are defined. This is how the topology of this guy defined. And this means you'll be fully supported. It tells me that you'll find this, you'll find this. You find this lower bound and the main. The idea is to get to get to this part, what you use is just that this guy contains C epsilon B, and you use the, how to call it, the gamma. This guy, no P, no gamma P, gamma C epsilon B is equals no P of gamma C epsilon B by gamma covariance. So this is gamma covariance. Basically, you have this set here and you have another point. If you project it, that projection will contain one of these, one of these gamma C epsilon. And this guy, they have the same measure that is bounded by below by L. So in particular, you get new P of P R X B P rho is bigger than L. Because it contains, it always contains these sets. So yeah, I'm just out of time. I thought I would. We are just left about, you know, adding these pieces to, to calculate the measure of the entropy of this measure that you constructed. And we will see that it has full entropy like entropy of this measure. So I think I'm going to stop here. So next lecture, what I will do is, I will first use the first 10 minutes to do this estimate. And then calculate the entropy of the measure using what we have, just these three properties are enough. And, and then, yeah, and then I will prove some other properties like mixing and the city of this measure. And then, yeah, let me see what more we can do. So I was up here for today. If there are questions, so I'm going to ask to answer questions. I'm sorry. Can you explain more about the choice, measurable choice of the inverse of E. Yes. So, this is really using the, this is really using the hand banner. So, yeah. So I think. So maybe at the beginning of next lecture, I will start by explaining this, this choice, but it is based on these abstract results of hand banner. And, and I'm sure there's enough time to do it all, but I will, I will, I will talk about it and at the beginning of next lecture. Thank you very much. A question. Okay, thank you. So, see you. So not that I'm sure there is a kind of time for the lecture is at 2pm Italian time, 2pm 04. So the time is due and we will have two hours.