 Let me start by thanking the organizers for receiving me back again in Trieste. It's a pleasure to be here. Everything that I'm going to talk about is a joint work with Federico Rodríguez Hertz. So I'll be working in the following setting. I'm going to consider a compact metric space and F on theomorphism of it. And I will be interested in the thermodynamic formalities that you can associate to these systems. And in particular, I'll be paying attention to the concept of topological pressure. Instead of giving you the definition by n epsilon balls and using exponential sums, I'm going to save some time and I'm going to use directly the variational principle. So let me denote by Pfm, the set of all F invariant Borel probability measures of m. For a given continuous map, just a real value map which in this part of the theory are usually called potentials, you define the topological pressure as a following quantity. You take an invariant measure, you look at the entropy of the map F and you add the integral of phi respect to nu and then you take the supremum among all possible invariant measures. That's the topological pressure. Then let me define what is an equilibrium state. An equilibrium state for the potential phi is simply a measure such that the previous supremum is obtained, namely for the topological pressure of F is equal to the entropy of F plus with respect to nu plus the integral of phi with respect to nu. You can think of that you can define in some sense that thermodynamic formalities, at least in equilibrium is precisely the study of equilibrium states. And we will be interested in questions as the following, do they exist? I give you F and I give you phi. Can you find an equilibrium state? Well, it happens that this part of the theory is in some sense the easiest to deal with due to very general methods. I'm going to put here the name of Bowen and it's meant to be in an illustrative tone. There are many people who have to work in this, I cannot give you a complete bibliography, but it's just a sample, okay? General methods, what I mean by general methods is give you an example if you were yesterday in the talk of Radu, he looked at this map. You fix F and you look at the map that for an invariant measure, it gives you the entropy of F. And he was talking about what happens with this guy's upper semi-continuous. Okay? So if this map is upper semi-continuous, adding an integral, fix integral, or an integral of a fixed function is also upper semi-continuous because this function is continuous, whatever the measure. Now, the other probability measures comprise a compact metric space and you have an upper semi-continuous function of it so it has a maximum. So in this way you prove very simply that equilibrium state exists, but it doesn't tell you anything about the equilibrium state. Now, if you have existence, you may wonder about uniqueness. Here, you also have general, but general methods, but they're a little bit more restrictive. You have to put some conditions, otherwise you won't hold. And here, again, you have the very classical work you have the very classical work of Bowen or more recently, Clemenada and Thompson have been working in this part of the theory. And then, once you have existence, uniqueness, you discuss uniqueness, you are interested in properties or description of equilibrium states. And for this, I mean the following, say, if your equilibrium state mixing, is it Bernoulli? That is, is it your system homeomorphic to flip in a coin? Maybe your coin has many phases, maybe your phases are not equi-probable, but still this is essentially most random system that you can think of. And also the chaos correlations that was discussed previously in the lecture of Federico. I'm writing properties of description because to have these properties, you need to know something about the equilibrium state. The general methods usually don't tell you anything. Yeah. Now, I'm going to leave the general theory and I'm going to concentrate in the part of the theory that I'm interested in, which is a smooth algorithm theory. Smooth algorithm theory, I'm going to assume that M is a manifold and F is a morphic. And I'm going to tell you about one of the most important theorems in a smooth dynamical systems. It's a following. If you have an hyperbolic attractor of F and I'm assuming transitivity, then for every held potential, there exists a unique equilibrium state for the restriction of F2 to lambda. Now, I'm asking a little bit more than continuity, but that's not too much, right? It's just healthy. And if you assume a formal property that is that F restricted to lambda is topological mixing, then your system is metrical isomorphic to our relationship. Names that are related. This first part is essentially due to senior, real and Bowen. And for the last part, you need the arguments of the Orson theory that part of the essential is Orson and Ways and Bowen also. Okay, this is a very important theory. Give me a couple of examples. If Phi is identical equal to zero, then use the entropy maximization. And if Phi is minus the logarithm of the determinant of F restricted to the unstable bundle, then use it as SRB measure. Here, you need to assume a little bit of more differentiability because you want that potential to be healthy, otherwise it's not. But as you can see, we already have two cases, very important cases, very popular cases, a lot of research going on in this part of the theory. What is the main idea of the proof? The main idea of the proof is following. Use Markov partitions and you really use your system to a subchief of finite type. You study a subchief of finite type and then you go back to the manifold using the semi-conjugacy. That's nice and everything, but you need the Markov partition. In the full generality that I wrote, you can do it for some cases without passing through the Markov partition, but not for all of them, as one can expect. Since this is a foundational theorem, there are a lot of generalizations and people working on that. And I'm going to just highlight the case of, and also flows, and I'm not going to go further. Okay? This is in a Royal Bowen, essentially. Maybe Royal Bowen. Let me remind you of the following. Some hyperbolicity is usually required for the general consensus. You either assume something about your map, or some part of the map, or you assume something for the potential. And in the next week, you are going to have a course by Yuri, which he's going to assume a very mild condition of hyperbolicity. And he's going to work with that to obtain Markov partitions, in some sense, and work with, get the equilibrium states. And now, let me discuss an important example where the available methods fail. Those are diagonal actions on locally homogeneous spaces. I'm going to rely here on the course of Professor Muhammadi, which I have seen that you have been discussing diagonal actions, at least on quotients of SL and R. I'm not going to work in the most general setting. I'm just going to contain myself with giving you one example. Take the three SL, three R. And take the diagonal group, the matrices with positive entries, the two group of G. And consider a co-compact torsion-free lattice there, in G. That means you already know what co-compact lattice means, torsion-free means you can think it in the right way. For me, what is important is that G mod gamma is a manifold. You may not even need that, but let's assume that. Then you have a natural action of the diagonal group on the quotient, which is a manifold. And this is an analysis of action. And what is an analysis of action? There is an element, at least one element, not all of them, but at least one element that leaves a splitting of that form, ES plus EC plus EU invariant. For some metric, DF and EU is a expansion. EU is an unstable bundle. Of course, DF, restricted to ES, is a contraction. And EC is tensioned to the orbit of the action. Now, in the orbit of the action, you are essentially a translation. You are a multiplication by a group element. So in there, you don't have any publicity, okay? Essentially, what they say is it's a little bit of an exercise, but not too hard, that you can choose the metric in M, such that F as an isometry on the orbit for the action. And then you say that F is a center isometry, and the definition should be evident. So you have three bundles, ES, EC plus EU. In ES, you contract, EU, you expand, and on EC, you are an isometry. The differential is an isometry, okay? Now, what is the difference that you may interchange leaves? In the orbit foliation of a new group action, the leaves are fixed. Here, you may change them, okay? So this is an example, and this is an example of partial hyperbolic dynamical systems. And again, the remark that I was trying to make is, you don't have any publicity on the center. You are an isometry. Anybody on measure, you look at the Lyapunov exponents, restricted to the center, they are zero. You are not going to get any hyperbolicity. This is just congenital facts. Believe me that this is true. All those bundles are integrable. ES, EU, EC, EC plus ES, EC plus EU, which are center-stable and center-unstable. And you can integrate it by its center, F, invariant, foliation, which I'm going to denote W star, where star is SU, CC, CS, or CU. And now I'm going to state our main theorems. Suppose that you have a center isometry of class C2, and assume that W, the stable and the unstable are minimal, that these all leaves are dense. This is the topological mixing condition that I was talking in the previous theorem, okay? And suppose that you're given a holder potential, then the following happens. You can find an invariant measure and five minutes of measures along the unstable, the stable, the center-stable, and the center-unstable, which satisfies the following. Well, the probability new phi is an equilibrium state. And for every x, the measure, say, look at the measure in each unstable leaf, you put a random measure. And you say do the same in every stable leaf, and you do the same in every center-stable leaf, and every center-unstable leaf, those are given. That's a part of the theorem. And it has some properties. So if you have a measurable partition that refines, that means that the atoms are conditioned to the unstable leaves, say, the conditionals of your equilibrium state are equivalent to new ux. Here I mean the following, say, suppose that you have eufoliation, you take a box, and you apply fubini, and you get measures here. Those measures are going to be, after normalization, are going to be proportional for some positive function to the new ux and new ux. Here is this partition. I look at the conditional measure here. This is equivalent to new. Then some other properties for every epsilon sufficient and small. If you look at the ball size, the center-infection size epsilon, you have product structure with respect to the unstable and the center-stable, that is that your measure there is equivalent to a product. And then you have what is called the Gibbs property. But let me be precise here. Consider epsilon, say, for x epsilon n, look at the following. You look at the points in the locally stable manifold of x, such that the distance from x to y are less than epsilon for the n-first iterates. This is just the intersection of the n-epsilon, ball and ball of x with respect to the local unstable of x-epsilon. And then the measure of that set is comparable to the virgo sum of phi of x minus n topological entropy of phi. Okay, so you come for every n, that's for every n. The bounds depend on epsilon, but they don't depend on n, neither do they depend on x. This is called the Gibbs property. Is this in the unstable manifold? What, did I write wrong? Ah, yes, yes, yes, sorry. Yes, thank you, yes, yes, absolutely right. It's the new UX, sure. Now, what happens? Why I'm writing that? I'm writing that because of the following. Along the center, you are going to have diversions because you are an isometry. Each time that you add a virgo sum, two points, the difference between two points, this difference is going to maintain forever. So you're not going to be able to get something like this. You get something which is exponential with n, but that's the term coming from the center, which we already know that you have. So you have that. Now let me give you, discuss a couple of more properties. F is a center isometry. C2 and unstable and stable are minimum. This is a center isometry. So you act, it's a partially hyperbolic. In the center, you act as an isometry. And then the center isometry. So you can think of the regular element of an enosofaction. That's my main interest to find. So let me now discuss Bernoulli, which should be Bernoulli property and uniqueness. Time t of an enosoflows, yes, precise. So along this, you have unstable, stable, and along the flow direction, you add an isometry. For an enosoflow, it was already done, but you need to pass through the Markov partition. That's essentially Bowen. But he gets Markov partition for the flow, not for the time t. If Markov partition is for the flow, no, it's the flow. It's for the autonomy of the flow, in fact. If you want, I'll discuss you in the beginning, in the end. But if you wanted, for the case of flows, it's no. The problem is higher rank actions. Now, let me, there are a couple of conditions that it will tell me, it will leave me straight. So I'm not one to mention what are the conditions that I need, but if you put a couple of conditions, simple conditions, you get that your system is metrically isomorphic to a normally shift, okay? And what about uniqueness? Well, assume either that the dimension of the stable and the unstable is one, or that you have an ergodic automorphism of the torus. That's another example. Ergodic automorphism of the torus are partially hyperbolic, and the center direction you act as an essentially isometric. Well, should be an isometric, I guess. I should put an ergodic automorphism of the torus, such that irreducible, maybe. Then, the equilibrium state is unique. And what is working process, what we are doing, we are almost complete, maybe, is that uniqueness also holds in the homogeneous cases. For example, the weight chamber flow, if you know what it means. Okay, but I'm not going to talk about this, I'm going to, how much I'll do I have? Five minutes. All right. Okay. Okay, that's fine. Let me then speed up a little bit. Let me remind you a couple things of SRB measures. In the general settings, an SRB measure is a measure which is absolutely has unstable conditionals, absolutely continues with respect to the back. This can be made precise in very general framework. I'm not going to do it, I'm going to work, keep working with center isometers. But here you have two important works which tell you what is the behavior of these type of measures. And the first is SRB measure exists. And this is due to the foundation of work of Sina and Pessim. In principle, that's not obvious at all. For the hyperbolic, you have a mark of partition. For other systems, you don't, they exist. And the other thing is that are the sophisticated argument of Le Drapin and Young, that if your measure is an SRB measure, if and only the entropy of F is the logarithm of the unstable, is the logarithm of the unstable Jacobian. Now, take everything to a side. That is that if the entropy, that's Pessim formula. So you have equality in Pessim formula. If you have, you put everything to the left. So you have that NS new has absolutely continuous conditional measures along unstable. If and only if the entropy minus the logarithm, the integral of minus the logarithm is zero. So it's a equilibrium state for that potential. That's what I wrote in the beginning. And here I'm using implicitly that unstable manifolds coincide with Pessim's unstable manifold. There are no expansion whatsoever along the center. All invariant measure has zero the opponent's phone is along the center. So let me try at least to tell you what is the theorem that we are proving, or we have proof, which is you have a similar characterization of what are equilibrium states in terms of the measures. So you have the following. New is an equilibrium state for your potential and phi if and only if your conditionals along unstable are equivalent to new you. Those are the families that we constructed along the unstable. And now a measure, an invariant measure is an equilibrium state if and only if, if you disintegrate, you look at the conditionals, those are equivalent to new. This is, well, let me go to the, this complete the theorem and make some remarks. This, you have this in for the stable. So the conditionals along the stable should be new as. And conversely, if you have an invariant measure which has conditionals absolutely to continue with respect to new, new you, then you use an equilibrium state. And in particular, they are equivalent to new, the conditionals are equivalent and along the stable you also know them. So if you think about this, this is a complete analog of the case SRB. Your equilibrium, you have, you know the disintegration along the stables and you characterize the equilibrium state. You can hear in this case, what is the point is that these families, they provide you the reference measures to which you can compare. For the SRB, you already have Lebesgue. Lebesgue is always there. But for this case, in principle, that was unknown, right? What do you compare? Well, you compare it with new you and new us. That's what you compare. And this, I think it is even new for the case of, I lost my morphism. That's why the decision is I haven't seen it anywhere else. So I don't prove, and probably I don't have too much time, I guess, all right? Couple of minutes to go into the proof. No, maybe not, maybe not. Let me see if I have. I don't think that I'm going to, I'll leave it here, I'll leave it here then.