 Thank you. Can you hear me? Thank you, Sebastián, for the introduction and I would like to thank the organizers also for the opportunity to speak in such a nice conference, in such a nice institute and city. Okay? So as Sebastián said, I'm going to talk a little bit about geodesic flows on surfaces of non-positive curvature that are burnally. So let us put the setup of the talk. Well, it's actually here, but let me write it properly. So you have, sorry, M2. It is a sinfinity compact boundary less surface. We assume that M is not the torus. We also assume that the sectional curvature, or here the Gaussian curvature because it's dimension 2 is non-positive. Okay? Well, given this and given this, I have a Riemannian manifold. So I can consider the geodesic flow. It was already actually defined in Keith Bernstock. So let me consider phi. Where does the geodesic flow live? It lives in the tangent bundle. We can restrict it to the unit tangent bundle. So we have a flow on the unit tangent bundle. It flows along the geodesics at unit speed and it sends one vector to the respective vector. Okay? And for the purpose of this talk, I'm also going to consider a measure, which is going to be an invariant measure for the flow. And let us restrict ourselves to two cases of the measure. So either phi is Liouville, or phi is a measure of maximal entropy, just like in the case of Jerron-Busy's talk yesterday. So measure of maximal entropy. Okay? So we have a system and we can ask many questions about the system, whether it is ergodic, whether it has stronger mixing properties. So it was actually mentioned also yesterday, a conjecture that people still don't know whether the measure mu be equal to Liouville, whether this thing is ergodic. So this is an open question, whether mu is ergodic. But it is known that the measure of maximal entropy is unique, has ergodic. This was proved by Knipper. He actually proved in the setting of rank one manifolds, which covers this case. So we know that this guy is ergodic. And the goal of my talk is to go beyond these things. So we know that this guy is ergodic, okay? And provided also that this guy is ergodic, I'm going to convince you, I'll try to convince you, that in both cases you can go beyond ergodicity and prove that actually the measure is Bernoulli. Or the system given by this triple is Bernoulli. So this is the goal of, it's to go from ergodicity to Bernoulli's for these two cases, okay? So I have to tell you first what is a Bernoulli flow. So let us stop for a little bit and remember a few definitions of ergodic theory for flows. And I'm going from the easiest property to the most complicated one, which is Bernoulli. So I think we are all comfortable with the definition of an ergodic system. So actually let me fix here x, the space a sigma algebra, phi t is the flow and mu is an invariant measure for the flow. So I want to remind you what it means to be ergodic, what it means to be weak mixing, what it means to be mixing and going further, what it means to be a k-flow. And finally what it means to be a Bernoulli flow, okay? So ergodicity is the same thing. If you have an invariant measure, then this measure is constant. I'm not going to write the almost everywhere statements, okay? So the idea is that if you have something invariant, this something has to be trivial. It is indecomposable in the measure theoretical sense. Weak mixing is something that is stronger. So I'm going to put this inclusion here. And it says that if you have an eigenfunction, and in this case, because it is a flow, it has to be written, an eigenfunction has to be written like this, okay? The application of the function after the iteration of the flow, it's the function times this thing here, this corrective number here. Lambda is going to be the eigenvalue. So again this, the system is going to be weak mixing if a function that satisfies this is constant. Mixing is the same thing also. So if you get two sets, measurable sets, then the measure of this intersection is going to converge as the time goes to infinity to the product of the measures, all right? So now let us go to the k-flow into the Bernoulli flow. So what is a k-flow? I can give you three definitions of the k-flow which were proved to be equivalent. The first definition is something in terms of mixing. So I won't write properly what it is, but it is a sort of uniform mixing. So because it's uniform, everything that's k is also mixing. And well, we can check also that everything that's mixing is weak mixing, okay? So this is a much stronger condition that we impose on the flow. I can define in these terms or I can define in two other terms. The other one is actually why the number, the letter k comes from, is that the flow satisfies Komogorov's zero one law. And what does this mean? This means that if this letter here that I don't remember the name is a finite partition, then the tail of this partition is trivial, okay? It is to say that tail measurable events are trivial and this is exactly what Komogorov's law, zero one law is about. And finally, and it would be the one that would be more useful for us, it's the definition of a k-flow in terms of completely positive entropy. And what does that mean? Well, every system has a bigger subsystem which has zero entropy and this is called the Pinsker factor or the Pinsker algebra. In the case of a flow, the Pinsker factor of the flow is defined as the Pinsker factor of an automorphism. And the automorphism can be taken to be any time of the flow as long as it's not zero. So you can fix the time one and define the Pinsker factor of the flow as the Pinsker factor of the time one map, okay? And saying that this flow is Komogorov is saying that this thing here is trivial. And if I told you that this is the largest factor of the flow of the automorphism that has zero entropy, well, this thing here is equivalent to saying that every non-trivial factor of my flow has positive entropy. And this is the notion of completely positive entropy. All right, so finally we'll get to the definition of a Bernoulli flow. And I guess that we all know what a Bernoulli automorphism is. A Bernoulli automorphism is just the IID iterations of random variables, right? You have a finite alphabet and you consider the measure on the sequence given by this finite alphabet and this measure, you consider it to be a Bernoulli measure. So it's the product of the same probability vector. And this is what is called a Bernoulli automorphism. And what is a Bernoulli flow? It is a flow such that for every non-trivial time, this automorphism is a Bernoulli automorphism. So that flow is going to be Bernoulli if phi t is a Bernoulli automorphism for every non-trivial t, okay? And it is a theorem that to check this, it is only necessary to check this property for one time. So usually people state this that a flow is a Bernoulli if the time one map of the flow is a Bernoulli automorphism. Okay, so now you know what a Bernoulli automorphism is. But the question is whether they exist or not. So I will give you an example of a Bernoulli automorphism now to tell you that they really exist. So an example, not of a Bernoulli automorphism, an example of a Bernoulli automorphism we all know of a Bernoulli flow. This is called the Totoki flow. And what is the Totoki flow? It is a suspension flow. If you remember Federico's talk last week, yeah, he defined what a suspension flow is. And what is a suspension flow? Well, suspension flow, you need two data to construct a suspension flow. You need an automorphism, so you need a space and a map, and you need a roof function. In this case of the Totoki flow, what is the automorphism? The automorphism is the two-sided shift. And the roof function is a function that is constant on the trivial cylinders here, on the cylinder that has the zero position and zero in the zero position one. So let me write the sigma, although it's totally disconnected, but I'll write it in just like an interval. And I'm going to consider the partition. The left one is the cylinder that on the zero position has zero. And the right one on the zero position has one. So I'm going to tell you what is the roof function now? Well, the roof function is constant here and it's constant here. Here I can make it be equal to one. And here I make it be equal to alpha, where alpha is an irrational number. So constant one on zero, zero, oh, not zero, alpha on one. And the main property is that this alpha is different from zero. Oh, it's not rational. So I have my basis and I have my roof function. And what is the flow that I consider? Well, I just flow at unit speed on this space. When I hit the top, when I hit the graph of the function, I come back to the section according to my map on the basis. So I, for example, if my sequence starts with zero one and I'm going to start here, I'm going to flow time one here until I come here and I keep doing this. Okay? So this is the Totoki flow. Totoki showed that this flow has the K property. So Totoki is a K flow. And Onstein showed that it actually is Bernoulli flow. So yes, they do exist. But the construction is totally probabilistic. So in the basis, you have a Markov chain and you consider the suspension. So now I'm going to try to convince you that actually these Bernoulli flows, they are more, they appear in many other places and some of them were actually not expected during the 70s. So let me mention a little bit of the literature. A few cases in which people proved that the flow is a Bernoulli flow. And what is nice is that in all these cases, the flow is a deterministic one. So they started with a deterministic system and they proved that it is as random as you can expect. Okay? So the first, I don't know if it was the first, but the first I want to mention here was an Osov and Sinai. And they proved that if you have a Riemannia manifold, well, if you are in the case of an Osov flow, so you have negative sectional curvature and you consider the Liouville measure, then the flow is K. All right? I might come back to this. The main tool that they used for proving this was Roth-Linz and Sinai theory of measurable partitions. But, well, that's what I want to say right now. So the second result I want to mention is by Einstein and Weiss in which they develop a condition that is checkable to see if a flow is Bernoulli or not. And they apply this to the situation of geodesic flows on surfaces of constant negative curvature. So phi geodesic flow on surface with curvature constant and equals to minus one and the measure being the Liouville, the Lebesgue measure. So they proved that in this case the flow is a Bernoulli flow. So this is a totally deterministic system, but after a change of coordinates it becomes the most random as you can expect. Okay? All right. So this is a main result because it introduced this notion of very weak Bernoulli partitions and it's a notion that is checkable and if you prove that there is a sequence of very Bernoulli partitions that generate your sigma algebra, then your flow is going to be Bernoulli. So this result was later extended by Rettner exactly by checking the very weak Bernoulli condition and she showed that if you have a nozzle flow and if you have a measure which is good in some sense, in this case it has a Gibbs property, but it has also some other regularity properties that I don't want to mention right now, then what she proved. She proved that if the flow is a K flow, then it actually is Bernoulli. So in this setting there is no space for a flow being K and not being Bernoulli. Okay? This regularity implies that actually has to be Bernoulli. All right. So I wanted to say a few sentences on the approach of Rettner for proving this. So Rettner implemented as far as I understand CNI's program and CNI was interested in analyzing measures, relevant measures for fiscal systems and how did he do that? He realized that if you have a symbolic description of your system, things become easier to do it. So what Rettner did was exactly that. So if you start with a flow or with a system, so Rettner's approach, let me write it here, was you start with your system and you apply a coding procedure to instead of analyzing your system, you analyze a symbolic system. So you have a symbolic model and that will be more precise in a few minutes what it means to have a symbolic model, but things become easier to analyze in the symbolic model. So at this level here she implemented these properties plus Onstein-Weiss idea. So Mil being a good measure and she used Onstein and Weiss. Actually I'm going to write here Onstein's theory because that's how it's known nowadays. And she was able to get the Bernoulli property, to analyze the Bernoulli property. So why am I mentioning this? Because I'm going to write a result right now or in a few minutes. And the approach for proving this result is exactly as Rettner's approach. So I'm going to give a setup on which we are able to say many things about the system. It's going to be Bernoulli or basically a Bernoulli times a rotation. And the way we are going to prove this is exactly as this. You start with your system and you get a symbolic model for the system and after that you use in our case instead of having a measure like that you're going to have an equilibrium measure. So we are going to use thermodynamic formalism plus this very weak Bernoulli condition to get the Bernoulli property. So what is the result? We prove a theorem following the same approach of Rettner. And what is our setup? So our setup unfortunately at the moment we are not able to deal with all the dimensions. So you have a three-dimensional sinfinity closed compact without boundary manifold. You have a flow and I have two assumptions on this flow. The first assumption is that it doesn't have fixed points. So without fixed points. And the first assumption is a regularity one. It is the assumption that the vector field that generates this flow is at least of the order c1 plus beta. If x is the vector field of phi. And I assume that this is of class c1 plus beta. In terms of the vector field saying that the phi doesn't have fixed points the same as saying that the vector field does not vanish anywhere. So you are in the situation. Well, we have a dynamics. We need a measure. So we're going to have mil. It's going to be an equilibrium measure in the sense of the Hombuses talk yesterday. So it's an equilibrium measure of a holder potential. I'm going to assume it as ergodic. And I'm going to assume it has positive metric entropy. Okay. So this doesn't seem similar to what has been done before about another flows. But note, as long as I am assuming that the positive metric entropy is positive, well, by who else in the quality, since we are in low dimension, this means that I have, well, I have a flow. So I always have a Lyapunov exponent that's equal to zero is the flow direction. Having positive entropy tells me that I have a positive Lyapunov exponent as well. If I do the same thing for the inverse flow, well, the inverse flow also has to have a positive Lyapunov exponent. This means that the flow actually has one Lyapunov exponent, which is zero, another one, which is positive, and another one, which is negative. So I am exactly in the situation of non-uniformly, non-uniform hyperbolicity. And I'm happy because if I have non-uniform hyperbolicity, I can use passing theory. Okay? So we are going from something that's uniform hyperbolic to something that's non-uniform hyperbolic. All right, so this is the setup of the theorem. And what is the theorem? The theorem is a joint work with Le Drapier, François Le Drapier and Omri Sariq, is that in this situation, my triple here, m phi mu is either a Bernoulli flow or it is the product of a Bernoulli flow with a rotational flow. Okay? That is a second part of the theorem that if in addition we impose an extra condition on our flow, and the condition is that the flow is a red flow, if in addition phi is a red flow, I don't want to define what the red flow is right now. If I have time towards the end of the talk, I will. So in the case of the red flow, you should think of the red flow is the most non-integral situation. So a contact structure, giving a contact structure is just like giving planes, in this case, you are two dimensions, so you give planes, tangent to the space that are non-integral, so you don't have an integral foliation of these planes. And a red flow is something that is associated to this contact structure. So in the case that you have the red flow, you cannot have the situation of Bernoulli times rotation, you have to be Bernoulli. So if in addition phi is a red flow, then only the first case happens is Bernoulli. Okay? In the geodesic flow is an example of a red flow. So in our case, we don't have Bernoulli times rotation, we have to have Bernoulli. All right. So now let me mention to you this thing that I just erased. How do you use this same sketch, the same scheme of retina to prove this theorem? So the first part as I wrote here is that you have to have a symbolic coding of your system. So remember that the system that I'm dealing with is this triple here. I have this three-dimensional manifold, I have a flow without fixed points, and I have a measure with positive entropy. So this first part here was made in a joint work with Omri Sariq is that if M5u is as above, then there is a symbolic model. And in this case, then there is what we call topological Markov flow. And I'll define what it is and a map which is actually holder. Well, I need some notation here. Topological Markov flow is going to be sigma r, sigma r. I'll define this right after I write the theorem. There is a map pi r from sigma r to the manifold such that it commutes the diagram. So pi r composed with sigma r is equal to pi composed with pi r. And you should think of the diagram here exactly as Pablo Carrasco said last week. So what is the goal of symbolic dynamics is to obtain some extension that intertwines the dynamics, but the extension itself is not much more complex than this thing here. So in the sense of not being much more complex, I mean that it is basically finite 2 1. So pi r is finite 2 1 in a set of full mu measure. So why is it good to have an extension which is finite 2 1? Well, we all know that whenever we have a measure here, we can project the measure here. But when we have a measure here, it is more complicated to lift the measure. Well, in some situations, you can lift the measure, but the measure that you lift is going to be much more complicated. If you have a finite 2 1 extension, you are always able to lift your measure in a way that the metric entropy of the lifted measure is the same as the metric entropy below. Why is that? You have this Abramov-Rochlin formula which tells you that the entropy on the top is the entropy below plus the average of the entropy on the fibers. If the fibers are finite or countable, they carry no entropy, so the entropies are the same. All right? So this is one of the main goals of constructing these finite 2 1 extensions of your dynamics, because you can analyze ergodic theoretical properties, not here, but here. You can lift measures. I just want to mention two corollaries of this theorem. One of them is again in the setup of Jehovah's Witness talk yesterday. So one of the corollaries that we get, we don't get finiteness still, but we get countability of the measures of maximum entropy. So there are, at most, countably many measures of maximum entropy. And the second corollary is in the terms of counting the number of closed orbits. So the symbolic model is much easier to analyze closed orbits. And by doing that, we are able to prove that the number of closed orbits of length of period at most t grows at least as a constant times e to the t times the topological entropy divided by t. Okay? I have a measure that's positive. So the topological entropy is also by the variational principle positive. So this is a positive number. So it grows roughly as an exponential after you divide by this t here. All right? And we cannot expect much more in this setup, because our conditions are very weak. I mean, you could have countably many closed orbits, and this would, well, this tells you that you cannot expect to have an upper bound for this thing here. All right. So this is the symbolic part. And what does it allow us to do? It allows us to consider this measure, mu, here, on M, and lift it in a way that the entropy is preserved to a measure nu on sigma r. But what is sigma r? I still didn't define what sigma r is to you. So before writing this, let me tell you what is sigma r and sigma r. Okay. So let us say that you have a graph oriented graph with countably many states, countably many vertices. You can consider the paths on this graph, and this gives you a symbolic space, which I'm going to call sigma. So paths, two sided paths on G. So what is the graph that defines the two shift? Well, you have zero, one, and you have all possible transitions allowed. You can consider a more complicated graph with countably many vertices and do the same thing here, paths on G. And there is a dynamic associated to this, which is the left shift. And this is a topological mark of shift. What is a topological mark of flow? Well, it's just like we defined on Totoki's flow. You need a roof function, so you need a function from the basis to the positive reels. So you can consider the suspension space and the suspension flow. Sigma r, the suspension space. What is that? All the points below the graph of r with the identification that x r of x is the same of sigma of x zero. And sigma r here is the suspension flow. You flow at unit speed vertically upwards in this space here. When you hit the top, you consider the identification and you come down. So this is our model. It's a suspension, but the condition that we have to put is that instead of having finitely many states on our symbolic dynamics, we have countably many. Okay? But this is not bad. This is actually good. So what I want to do right now, well, I want to tell you more or less the scheme of the proof of the theorem. So we start with a measure on n. And because we have a finite one extension, we can lift this measure to the space sigma r. And it is easy to check actually, because the entropies are preserved that if this guy is an equilibrium measure, this guy is also going to be an equilibrium measure. Okay? So we are with an equilibrium measure with respect to the flow, but we can consider the flux measure. So every measure here is associated to a measure on the... Yes. I'm proving the theorem that I just erased here, that I want to get Bernoulli-City or Bernoulli-City times rotation. So this is the first part of the proof. You get the symbolic coding. Now you need to do the Einstein's theory and thermodynamic formalism analysis to be able to prove Bernoulli-City or Bernoulli-City times rotation. Okay? In the end, we are going to prove that this thing is Bernoulli. And two Bernoulli flows, they are the same. So all flow is going to be the same as Totoque after a change of phase. Okay? Okay. My time is almost done, right? I have five minutes? Three? Four minutes. Four minutes, okay. So you get this measure which is leaving the suspension space and you can project it down to the flux measure here on the basis. So you are with a measure on your topological Markov shift. This guy is an equilibrium measure. I just told you that this guy is an equilibrium measure as well. There is a lame of Perian polycott that shows that if you have an equilibrium measure in the suspension space and you project, what you project is still going to be an equilibrium measure for the basis with respect to another potential function. But the conclusion is that we have a topological Markov shift and we have an equilibrium measure on the basis. So Perian polycott tell us that this guy here is equilibrium measure. And what is good is that Boozian-Sarig analyzed equilibrium measures for topological Markov shifts to accountably many states. So we have two properties of these measures that it has local product structure and the Gibbs property. Some Gibbs property. So this is the thermodynamic dynamical formalism part that I need. And this allows me to, to the following sketch of proof. We can see our suspension flow. I'm going to draw it like this now as a partially hyperbolic system. And the tool that we are going to use to start to analyze the ergodic properties of this flow is something called the olonomy group. It has been used by Brin, by Pessin and by many other authors in the literature. What does it do? Let us fix a point here and let us consider all SU paths closed centered at this point with respect to the basis. You can lift this SU path with respect to the basis to SU path with respect to the flow with what happens when you lift. Well, you do this, you do this, you do this and you do this and you come back to something here. Not necessarily X again, but some, but X plus some displacement vertically. Let me call this vertical displacement to be the P of my curve gamma that I start with. And I can look at all of these P of gammas here. So the olonomy group is the closure of all of these P of gammas where gamma is a closed SU path at X. Why do I do this? Because I have to understand the ergodic properties in the flow direction. And how do I do? I do by means of the hyperbolicity in the basis. So for example, if I have something that is a constant suspension, these guys here, they are always going to be multiples of the same number. And this is what we analyze. Analyzing this G here, how do we analyze this G? It is easy to see that G is an additive subgroup of R. And it is closed. What are the closed additive subgroups of R? They are either zero or discrete, non-trivial, or they are the whole of R. If it's zero, we can apply Katak Kononenko's construction for Liftschitz theory, Kononenko, to conclude that actually the roof function has to be a co-boundary. But roof functions cannot be co-boundaries, right? They have to be positive everywhere. So how can it be a co-boundary? So this does not happen. If G is discrete, non-trivial, again applying Katak Kononenko's construction, we conclude that R is columnologous to something taking values on this closed subgroup. So in this case, our flow is almost the constant suspension. So let me conclude here. So our flow is something that the roof function is a constant multiple, is a multiple of some constant number. And that is a trick of bowing that allows us to conclude that actually suspensions like this are constant suspensions. So what do you do? You just change your section by completing this thing here. And if you see your flow with respect to this section, well, how long does it take for it starting in the section and go back to the section? It takes a constant time. So this thing here implies that by the streak of bowing that this thing is a constant suspension, which is almost Bernoulli times rotation. So if you apply, again, non-steam theory, we obtain that in this case what we get is Bernoulli times rotation. So we are already in one of the cases of our theorem. We just have to analyze what happens when g equals r. When g equals r mean what that means is that you can go in the full direction as much as you want by doing sq paths. There is the Rochlin-Sinai theory, and that's when I say it again, of measurable partitions that says that in this case you can actually prove the k property. So having this saturation in the full direction tells you that the flow is a k flow. So we only need to analyze what happens when it's k. Remember Ratna's theorem that she allowed us to go from k to Bernoulli. So we apply her ideas in this case are these properties of Boozian-Sarig, so thermodynamic formalism, plus the notion of very weak Bernoulli partitions. So I'm going to put on this thing here to conclude that if the flow is k, then the flow is Bernoulli. So this is a rough scheme of the proof that goes with the with the head of this Holognomy group. This Holognomy group allows us to get three very rigid situations in which starting from them we can conclude that this case does not happen. In the second case what we get is Bernoulli times rotation. In the third case what we get is a Bernoulli. And I think I'm going to stop now. Thank you.