 As the last speaker of this conference, I have a pleasant duty, but it's really a great pleasure to thank all the organizers of the conference, Jean-Christophe Yokoz, Marcel Aviana, and of course, our lovely Stefan for organizing this great event, very important, very exhausting, and we just want to thank you very much for doing this great job. I mean, having three weeks of school with so many participants, so many lectures and talks, essentially without any flow, working absolutely smoothly is incredible job, and I would like to thank you all for that, and Stefan. So my talk is kind of a survey-like talk, where I would like to express certain visions rather than present and discuss any particular results, but of course I will sum, but I'd like to kind of paint a bigger picture rather than concentrate on proofs of some particular statements. So what I want to talk is building thermodynamics beyond uniform hyperboleicity, and I would say maybe 10 years ago, maybe 15 years ago, I would probably not dare to give such a talk, because not much I would have been able to say. And within these 15 years or so, there were so many interesting results that actually made the whole topic attract so many young and good mathematicians, and we've heard quite a bit of talk. I mean, this conference is an example of that, where in many talks, we are devoted to thermodynamical formalities for system which are not uniformly hyperbole, starting in a sense with the first talk by Keith Burns, talking about a wonderful example of this, and I will mention its geodesic flows on services of non-positive curvature. Now, I'd like to apologize that I will have to start with something very, very basic that you've seen already several times in other talks, but I need to do that in order to set up some terminology, and most important to stress certain points which are very important when we go from uniform hyperbole to non-uniform. So, first of all, what is thermodynamical formalism? You start with a compact metric space of some metric D, and you consider a continuous map of finite topological entropy, and a function that we call potential function, which I only assume to be continuous. Then I choose it, consider the space of all F invariant barrel probability measures, and I call a measure equilibrium is if that expression which is defined in this way is obtained on a certain measure. Now, what is E mu phi? It's free energy of the system. So once you have a measure, then the expression entropy plus integral is minus sign is from the point of view of statistical physics represents a free energy of the system with respect to a given probability distribution or state of probability state, which is mu. The minus sign means that what you want to find, you want to find a measure for which it minimizes the free energy. Then you turn things around and you call this expression topological pressure, and the reason it's called topological pressure can easily be seen from the talk by Felix Przetticki not too long ago where he defined topological pressure using avoid in any measures. It's a pure topological way to define the measure. And once you do that, then this not become not the definition of the topological pressure, but a statement which is known in statistical physics as a variational principle. And then one has to prove it. And so this equality is actually consist of two, proving variational principle and then establishing existence of equilibrium measures. Now, first of all, it's easy to see that one can use not instead of the space of all invariant measures, only the space of ergodic invariant measures. You get the same in film or supremum if you want. So it's the same thing. And I would like to stress that when we talk about equilibrium measures, there are two problems. One is whether such a measure exists. And this is rather simple, simple problem because all it requires if you look at this expression, this is a continuous, the integral is a continuous function in the space of invariant measures, which is compact. So if you have semi-continuity of the entropy, then you guarantee existence of an equilibrium measure. And there are plenty of war and some reasonably mild conditions on the F which guarantee semi-continuity of the pressure. Outside of uniform hyperbolicity, this is a problem. And it's very well-known problem. There are plenty of results on semi-continuity of the entropy for various classes of system which are non-uniform hyperbolic. And this all together will guarantee existence of an equilibrium measure. Now, uniqueness of an equilibrium measure is where everyone got stuck. Uniqueness from the statistical physics point of view corresponds to phase transition. In the system, I'm not in a position to explain why it is so and what it means. You have taken for granted. But once we accept this, you understand that this is one of the major problems in statistical physics and hence in the semi-dynamic formula is to find out, to find conditions that would guarantee uniqueness of the equilibrium measure. So one case which is very well-known is a case when you have a smooth compact remanium, the excess smooth compact remanium manifold. F is a diffeomorphism. Lambda is a locally maximum hyperbolic set. And we assume that F on lambda is topologically transitive. Phi is a colder continuous function. Then it's easy to see that it's well-known that the entropy is semi-continuous. So it guarantees existence of equilibrium measures. Then one can, in fact, prove uniqueness of equilibrium measure. Studying ergodic property and prove that the measure is Bernoulli up to discrete spectrum, meaning that if one assumes topological mix and then it's just Bernoulli, no discrete spectrum. And moreover, it has exponential decay of correlation and satisfies a central limit theorem, meaning that it has the best possible ergodic properties one can imagine in a sense. So this is kind of an ultimate goal. And the reason I mentioned this is that I would like to prove all of those properties in the non-uniformly hyperbolic settings, which I don't know how to do. But there are plenty of good ways of thinking about that and obtaining certain results in this direction. So let's try to see what one can do. Well, in the uniform hyperbolic setting to prove all of these results, what one do is a well-known fact. You build Markov partitions, then you use symbolic representation of f on lambda as a sub-shift of finite time. Then you say, well, I can easily prove all of these results for sub-shifts of finite time. And therefore, I will then go back to my system and that's how I obtain all the results for my system. So that's a way. You build a symbolic representation and then study symbolic model and then go back to your smooth system. Okay, let's see where one can do anything about this in the non-uniform hyperbolic setting. Before, I would like to stress one more thing. Well, what the previous slide says is, there is a, if you choose a herder continuous potential, you are fine. Now, do we have any good examples of herder continuous potential? Of course, you can consider herder continuous function, fine, but you can choose one. The question is, can you choose the one that will bring you to a good, very good, well-known measures? Well, that is achieved by considering the geometric, I call it geometric T potential because in many literature, it's just called geometric potential. It's a little bit misleading because it's a family of potentials. It's not just one. So that's why I propose to call it geometric T potential for each T and it's given by the well-known formula where the EU is just a stable direction. And because, and this rests on the fact that unstable subspace depends herder continuously in uniformly hyperbolic setting on X, you obtain that this potential is herder continuous, then you multiply by T is still herder continuous or for every T, it's herder continuous and falls into the previous theorem. For every T, you have an equilibrium measure and it has all the nice properties. Very good. Then, what is that? I don't know how to, yeah. So what you do, you consider the pressure function again something that was discussed here many, many times and it has a graph which is well-known and all I want to stress it's real analytic function in the uniformly hyperbolic setting. At zero, it crosses at the topological entropy and this point is very interesting one, I call it T naught, this is one and sometimes T naught is one, sometimes not. It's less or equal than one all the time. For a hyperbolic set, it's some number. If it's in the two dimensional case, it's number is known to be a householder dimension of the intersection of lambda with an unstable leaf and the corresponding measure, equilibrium measure at T naught is a measure of maximal dimension and if the hyperbolic set is in fact an attractor and there is a definition of it on the slide, then T naught is actually equal to one and the corresponding measure is a well-known SRB measure for the system. And then, so this graph includes the two most important measures which is an SRB measure as well as measure of maximal entropy as being an equilibrium measure for the certain potential and therefore, it's not just that you want to build some, maybe even nice class of potential functions but you want to build the one which contains the geometric potential and that's what I would like to stress very much because this will be kind of a guiding principle for us when we move to non-uniform hyperbolicity. Okay, let's now try to go away from uniform hyperbolicity. Don't look at this slide for a moment. Look at me, I will tell you something interesting. The point is that when you want to go from uniform hyperbolicity, there is two ways to go. One is you keep the map uniform hyperbolic and that's not on the slide but you change the function, potential function from herder continuous to something else. For example, just a continuous function and you wonder, is that true or not? The answer is due to some recent results by Omri Sariq you can build actually a huge class of potential functions which are not herder continuous however, for which you can claim uniqueness of equilibrium measures. They are much more sophisticated to explain and I'm not going to do that because that's not my main goal. However, you can do that. On the other hand, there is a rather simple example which was in my paper with one of my students, K Zheng where we just consider the best map in a sense to X mode one and you consider a potential which is smooth everywhere except one point where it loses herder continuous. It's not even, it's continuous of course everywhere. It's smooth everywhere except one point where it's not herder continuous and then you have phase transitions. So, as soon as you move from the class of herder continuous you can phase transition. On the other hand, there is a big class where you have nice unique equilibrium measures. Now, I'm not going into that direction. My direction is to try to keep the class of potentials reasonably good, however, and describable in a sense but go away from uniformly hyperbolic systems. Now, when you do this, there are several possibilities and I describe them in this slide. I'd like to tell you that I'm presenting it just for the sake of mentioning certain very nice results. The list is very far from being complete. The list of people I mentioned is very far from being full. The list, so please keep this in mind when you read this. So, the examples where one can move away from uniform hyperbolicity is one dimensional case and there's unimodal, multimodal, mass we've already studied by this group of people there are more and many than that, that's one. And they proved existence and uniqueness of equilibrium measures. Most important, it includes the geometric T, potential for some intervals in T. Then you have maps within different fixed point. Again, the list of people not complete who studied this, what they proved is that you have the pressure function to be like that. This is your pressure function. And this point is where one has a phase transition. You have two measures, one is an absolutely continuous invariant measure and another measure is a direct measure that sits at the neutral fixed point. Then there is nice results about polynomial and rational maps where the graph is again something like that, but you stop somewhere here, you go up to somewhere here, T naught, T one, and here the pressure function is analytic, but at these two points, sometimes they are infinite, sometimes they are finite, you can have infinite, you don't have phase transitions when they are finite, you do. And this is due to Prititzky-Lidlier and the fact that you have some finite T naught and T one is due to Makarov and Smirnov, and again, the list is not complete, of course. And finally, I want to mention results by Buzisarek, Elvis, Luzartepin, Herra, Levera, Varanda, Sviyana, and others who studied non-uniformly expanding maps, some with discontinuities, some are smooth maps. This is a very interesting topic, it's being developed, it's far from being complete. You've heard some talks on this topic from Olivero, from Buzi, et cetera, but again, it's proving something along this line. It's not complete. The best known results are in one-dimensional dynamics and rational maps, the last one is under, I would say investigation, but pretty much is known, in fact, about all these cases. Now, what I'm going to do is to go to another situation where X is, and that's what I'm actually the whole talk is about, when X is a smooth manifold, dimension is bigger or equal than two, which exclude one-dimensional case. F is a C1 plus L, and I will actually assume it's C2 defilmorphism, which exclude any discontinuities possible. And then I consider the set of points which are Lapunov-Peron regular, usually they're called Lapunov regular, I proposed to call them Lapunov-Peron because contribution of Peron is at least as important as of Lapunov, but somehow people used to forget that. Points for which famous oscillators, multiplicative ergodic theorem holds. So you have all the nice properties along the trajectories of this point. Then you consider the set of ergodic measures which gives positive weight to this set, meaning that every such measure is a hyperbolic measure. And then you want to, you have a potential, and then you want to find equilibrium measures which sits on this set gamma. And here we, again, answer the two questions whether such measures exist and whether they are unique. And already existence is a big problem because this set of measures is not compact. Because we see it on an all compact set gamma. And the question is where, why the entropy function is semi-continuous? Well, there are some results in this direction for some particular map, but there are no general or as general results as we saw in the uniform hyperbolic case. However, that is a very interesting question to discuss. And then, of course, the question about uniqueness. Now, say it again, please. As the points for which you have oscillated multiplicative ergodic theorem, the trajectories for which this theorem holds. And for every measure, this is a set of full measure. Okay, so I have troubles clicking on this. Maybe I'm a little too away from here. Yeah, so what I would like to say is that once we move to non-uniform hyperbolicity, you may want to change the whole setup of semi-dynamic formalism. And what it means is the following. That you want to consider not the class of all the invariant ergodic measures sitting on this set, but the ones with some big entropy. And of course, there is a number H which is threshold. And you have to define this H depending on the system. Well, how you wanna do that? Normally what happened is that you have a set S of points which are called bad points. Now what it means, I will comment a little bit later, but you just say it's a bad set. You don't like this set, so you call it bad. And therefore, you want to, and you study the topological entropy on that set. This set may or may not be compact. It's often compact, but it doesn't have to be. And, but it's usually not invariant. And then you consider the topological entropy, you measure the topological entropy, and that gives you a threshold. That's what you call a threshold. Now, that concept was, I guess, I mean, Jerome is here, he will either conform or deny it, was first brought up in one of your works. That's what I said here, I hope that I'm correct. But anyway, I mean, I learned it from your paper. So, what happened here is that now you want to, you can even move it a little bit further, and you say, I fix some number P, and I study all measures for which the free entropy is bigger than this number P. And again, P is a threshold, and how you define it, you find a set of bad points for which you say, well, I don't like those points. I look at the topological pressure of the function P on that set of bad points. That's my threshold, and I consider only those measures whose entropy is bigger than P. Well, what is a bad set? How do I define a bad set of points? There are several situations which I can use just to illustrate the idea. One is when map has some singularities. And those singularities, of course, constitute a bad part of the system. For example, if you have a Lorentz attractor, it's a uniformly hyperbolic system everywhere except for discontinuities, whereas the map, the derivative are not even continuous. Which means that effectively, the system is non-uniformly hyperbolic, because every trajectory that comes too quickly to the discontinuity set is not uniform, has zero, may acquire zero leponoff exponent. So you say, this is a bad set, responsible for producing zero exponents. Therefore, if I look at the topological entropy of this set, and I say, well, I only consider measures that do not see that whose an entropy is bigger. What does it mean to me? It means that you start from some measure sitting away from this set of discontinuity, move this measure by the standard Bogolubov-Krylov procedure to obtain an invariant measure. If the entropy of the final measure is bigger than the topological entropy of that bad set, it cannot sit on that set. So that would guarantee that the good measures that are interested in will avoid the bad set and therefore will sit on a set of positive, of a set where leponoff exponents are non-zero. That's how you want to treat the bad sets, and introduce bad sets and treat them. And this can only happen if the topological entropy of the measure you are dealing with is high enough. Now, when the system is smooth, what are the bad points in the bad set? Well, there are points which in a sense are responsible for obtaining zero exponents. So there are points where not only leponoff exponents are zero, but the trajectory can spend a lot of time nearby those points so that they can spoil leponoff exponents for other trajectors. And then you use more or less the same type of arguments. You want to deal with measures whose entropy is high enough so that it cannot sit on a bad set after all. Now, the problem with all that is that, maybe I should stay from that side. Yes, that the set S of bad points, even if it's compact, it's almost never invariant, or may not be invariant, or may not be compact. So the question is how do we define entropy, pressure, et cetera? Well, for this, you should use an approach that I brought up in some of my papers, and it's also in one of my books, that uses a definition of topological entropy and pressure as dimension-like characteristic of the system, this is a so-called carousel-dory approach. And this works for, you know, when you describe them as a dimension characteristic, it doesn't have, the sets on which you define it does not have to be invariant or compact. So we can use this definition of entropy and pressure for sets which are not necessarily invariant or compact, which are the bad sets that will give you threshold in one way or another, and then you set up the whole thermodynamic formalism by looking at those measures whose entropy is bigger than the given threshold. So that's some kind of an approach which I have seen in certain results in certain papers, but this is very promising one which I kind of advertise and I hope it actually will work in those settings. Okay, so let's move on. Now, how do I wanna build thermodynamic formalism? Well, again, follow the uniform hyperbolic case. Start with the symbolic models. What are symbolic models for non-uniform hyperbolic system? Well, one of them, of course, is a countable subshift system with countable alphabet. So it's a countable set of states or subshift of countable type, whatever you call it. And here there is a list of people, again, not completely list, but I kind of named the most important papers who worked on that and created certain conditions on the matrix, the transition matrix A, on the potential, et cetera, that would guarantee existence and uniqueness of equilibrium measures which are Gibbs measures in this case. And the most advanced one, in a sense, at least for my purpose, is a work by Omri Sarek which not only give you sufficient conditions, but the conditions which are essentially necessary. So in a sense, you can do better than his conditions. And what it means to our purpose is that if you have a potential function on this sub-shift of countable type, and this potential is bounded from above, I'm gonna understand that. It has summable variation. It means that you can sum up all the nth variation of the system of the potential. It has finite Gurevich-Sarek pressure. Again, the term that I introduced, Sarek used the term Gurevich pressure. Of course, it's obvious. So I propose to call it Gurevich-Sarek pressure because Gurevich indeed introduced this notion, but he only considered potentials which depends on finitely many coordinates. And Sarek extended to infinitely, to any potential. And that's not a trivial extension. So I think it would be correct to call it Gurevich-Sarek pressure. No, each of them agreed, after all, to accept this term. And it's just a way to define the topological pressure in a pure topological term similar to the one that was described by Felix in his talk. Just do it for countable shift, Markov shift. And this is the most important requirement, the so-called positive recurrent. And I'm not in position to describe what it means, but what Omri's work produces is that a potential can be either recurrent or positive recurrent or neutral. So that's complete classification. And in each case, you can say something about uniqueness and existence and uniqueness of equilibrium measurements. And that's why I'm saying that his results give not only sufficient but essentially necessary conditions. So positive recurrence is what needed. Now, this is a good symbolic model. And the question is, can it be used to study some smooth examples? And that's what I'm interested in. And a smooth example is, of course, you just use again another result by Omri that if you have a defiomorphism, C1 plus half a defiomorphism surface defiomorphism, then it allows countable Markov partition. Once you have countable Markov partition, you have a symbolic model that I just described. You have a class of potential function for which there exists unique equilibrium measure. So far so good except that we don't know much about this class of potential functions. For example, does it contain the geometric T potential? And that's an interesting question. Now, if you look at the geometric T potential for non-uniformly hyperbolic systems, then remember it's minus T log differential restricted to the unstable subspace. Once stable subspace is not a continuous function. So the geometric one potential is not a continuous function, less herder continuous. So it does not immediately at least fall in the category that is well suitable for symbolic models. Now, under certain conditions, one can still do something using this symbolic model and study this potential, but not much is known about that. So I'm just kind of skipping it. I haven't seen any formal preprints or anything. I just heard some discussions which I believe are correct. However, we may have to wait a little bit before I can fill in this piece. So we go next. And another scheme is a so-called well-known towers, symbolic representations by towers. I call it inducing schemes. It's a very general symbolic description which starts with an inducing domain which is the same as a base of the tower. Then a collection of disjoint barrel sets which I call basic elements whose union is this inducing domain, this base of the tower. Then there is an inducing time to each basic element you associate at time t which is the height of the tower. So the tower itself, you think about the tower as this is a base, this is a countable partition by basic elements and for each element you have a time, the height of this tower and it goes like that. And the time is, now it may but doesn't have to be the first return time to the base. So it's some induced map. And then on the whole base you have the induced map which is just the return map under time t to the base. Now, what happened, what often happened is, I mean if I do this way that what I require that this map on the base is essentially a full Bernoulli shift on countable set of states. It almost never happened exactly. It is true except that it's true for some invariant subset within this symbolic space. But this invariant subset is big. In what sense it's big? It's compliment, I mean if I wanna say it's big, I wanna say it's complimented small, right? You would accept this as a definition of being big. So to say that complimented small, I'm saying that it does not support any measure that gives positive weight to an open set. Now, why I'm choosing this strange way to define a small set? Because every Gibbs measure on symbolic space always give positive weight to any open set. So in other words, the compliment does not, cannot support any Gibbs measure on symbolic space which are candidates for my future equilibrium measures. So if I have an equilibrium Gibbs measure on symbolic space, I project it down, I will never lose it because it will never go to some bad part of my base. And the next property which is one of the most disturbing is that there is a number H such that the number of those basic elements with the same height, with the same inducing time is bounded by this H, exponentially with the exponent H. In other words, I just look at the collection of basic elements with a given inducing time. And this will be this one, maybe this one, maybe that one, et cetera, and I count all of them and the number grows as less than some exponential function with some H, where H will be my cut-off value. Now, what is shown for this symbolic model is the following, that the class of potential is the very same class that was described by Omri Sarek. It was described for a sub-shift of finite type. Here it's the same one works for symbolic tower mode. Now, unfortunately, the problem with tower is that not every measure can be lifted to the tower. The measure that gives some weight to the base does not necessarily lift to a finite, every measure can be lifted, but the measure you get may or may not be finite. So to have a measure finite means to be able to lift it because I don't wanna consider infinite measures. And it turns out that the condition we just discussed, which I didn't like very much, which is this one, guarantees that every measure can be lifted assuming that it's entropy bigger than H. So my H is a threshold. If I consider measures with high entropy that are lifted to the tower and among them, so I'm considering my set of measures is this one. I consider only measures with high entropy. Within these measures, I have unique equilibrium measure for the tower mode, for tower symbolic mode. Smooth applications of that are young defumorphism. And everybody, in a sense, I guess everybody here have heard about that, at least the name. What it means that you have a set which is the base of the tower, the base of the tower is a direct product of two counter sets. So it looks like the following. You just have something like that. These are local stable manifolds through some points and these are local unstable manifolds through the same points. The set and the intersection has direct product structure. That's the base of my tower. What I require is that if I consider intersection, this is unstable guys, these are stable guys, I'm considering intersection with unstable leaf, it should have positive measure. Leaf volume should be positive. Well, then there is a whole construction due to Leysank, which was described in her famous work, which says that if we have all of those, if we have this defumorphism, which has this set with a direct product structure, and if you choose the induced time, which does not have to be the first return time appropriately and it satisfies certain properties, then what we proved is that if you choose geometric T potential with T naught to some negative number, so we are only talking about this interval. So that is the interval I'm talking about, which is some T naught and this one. It's T naught is maybe not too far from zero, but maybe far away from zero. This is number T naught, this is one. For all T in this interval, there is unique equilibrium measure, and this measure has fantastic properties which are exponential decay of correlation at central limits zero. So that's an application of that symbolic model. Now, the particular examples, which I want to describe is a hand-on map at the first bifurcation. It's a work of Saint and Takahashi, and Takahashi talked about this in much more detail that I can present now. And another example is the katok map. The katok map is a map which is, I mean, I believe people here more or less know what I'm talking about, but if not, you start with a linear and also automorphism of the two-dimensional tors, two matrix, two one, one, one. It has one fixed point which is orange, and around this point, what you do, you slow down trajectory. So the trajectory in the standard way goes along the hyperboles, and this is a standard trajectory. What you do, you change the time, so that the trajectory starts moving slower, and you have many more points here. That makes the point zero to be a neutral fixed point. The Laplanoff exponent at that very point are zero. It's very similar to one-dimensional case in that sense, and what you get, you get Laplanoff exponent on some trajectory is zero, but almost every trajectory with respect to area which is preserved in this case are non-zero. So it's a good example of a non-uniformly hyperbolic map, two-dimensional, where it lies on the boundary of a nozzle. So if you look at the set of all a nozzle, defiomorphism, it's an open set, that the one I'm talking about lies on that boundary. And that's the map. The question is, what is the thermodynamics of this map? Well, you build symbolic representation for the Katok map using tower construction that I just described. And by doing so, you obtained that this map, if you look at the geometric potential, if you look at the geometric potential, that there is this number T naught such that for this interval, you have unique equilibrium measures. You can add a little bit to that by saying that the smaller perturbation, I mean, if I start doing this guy in a very small neighborhood of this, not far away, but in a very small neighborhood, I can push number T naught as far as to minus infinity as I want. Whether I can push it to minus infinity for some fixed R naught is a problem, I don't know. But that's what this model does. Okay, now, there is a third model. So unlike the uniform hyperbolic case where we have sub-shift of finite time, in non-uniform hyperbolic case, we saw two, this is a third one. And the third one is basically due to Clemenhaga and Thompson. And it's a symbolic model called non-uniform specification. So what it means is that you deal with a finite alphabet. You call a word and you choose a sub-invariant subset and it can be any. And you say the word is admissible if it realized by some point in, you can find this word for some point in this set A. Then you say that if I take any two points in this set and I can connect them through admissible word so that the lengths of those admissible words are uniformly bounded and I say I have uniform specification. And non-uniform specification is a much more complicated property. But roughly speaking, it means that almost any two points should be connected. So when I say almost any meaning that there is some bad set of bad points. This is a bad set that I was talking about from the very beginning, which you cannot connect. So you want to kind of avoid this set. So that's a symbolic model. And what they prove, they prove existence and uniqueness of equilibrium measures in the case of uniform specification for herder continuous potential. And then what Clemenhag and Thomson did, they pushed this to non-uniform specification. And I'm not in a position to discuss this in details. I mean, it's a story that deserves at least the same length stocks as I have. So it's a very interesting results. It's kind of a new symbolic models for, I mean, in the uniform hyperbolic case, specification is a more general property than a sub-shift of finite type. But it's not that much more general. In the non-uniform hyperbolic, it's a huge difference. Now, what are smooth examples? Well, there are basically three examples. One is a Manier example on three torus and Bonatti-Vian examples on four torus. And again, I'm sorry, but I will not tell you what they are. But it's easy to find what they are. And it's a very famous example. And what they proved is that all are derived from unorth of ease at starting point and then you do something, some surgeries, et cetera, to obtain both of these examples. And for herder continuous potentials, there is a one open sets of such maps for which you have unique equilibrium measure. But that result does not necessarily include geometric potential. So the second one does. So what they say is that if you look at C2Maple, many examples and Bonatti-Vian examples, then if you look at the geometric T potential, but be careful, it is defined not on a unstable but central unstable direction. Then you have some interval on which you have unique equilibrium measures for T in this interval. And by the way, I want to mention you that when I say that you have an interval on which you have unique equilibrium measure, it always accompanies. And all this result that I mentioned is accompanied by the fact that the topological pressure is real analytic on those intervals. So as long as you have uniqueness of equilibrium for a T in certain interval, it comes, not automatically, but it comes with analyticity of the pressure factor. And the final result I want to mention, which was described by Keith Borce in his talk. And I can only just say a few words on that, that you have this for the geodesic flows on rank one surfaces of non-positive curvature and alone. And I have to tell you that, and that's my personal view, that from, I mean, among the examples that I described, I consider the Catoch map, the non-map at the first best vocation, the Catoch map and the geodesic flows as, in a sense, real example of smooth systems for which we have results on existence and uniqueness of the equilibrium measures for the geometric T potentials. Other examples are, in a sense, artificial built to produce the results of this type. The other ones were not produced for that reason. So that's, and again, it's a personal view. You don't have to share anything like that with me. Anyway, so we move forward. And what I would like to advertise today, and in a sense, it's one of the major things I want to mention is that the uniform hyperbolicity teaches us that you have to do symbolic dynamics first, do thermodynamic formalities on symbolic map system and move them back to whatever smooth system you have. Now, you don't have to do that at all. In fact, I strongly believe that every single result obtained in this way can be obtained without building any symbolic models. And that's what I'm trying to tell you here. Now, so it's a non-symbolic approach. So what I would like to, I start this approach by saying that the same geometric T potentials, the things that I already told you that the function phi T is measurable forever. For a given T, it's a in general measurable function. However, you can still build the pressure function on that set. And it will be a very nice defined function, except that you don't know whether for any T you have equilibrium measures. Well, what you do then is, so what I would like to do, I would like to build some interval on which my function has unique, for every T, my function has unique equilibrium measure. I would like to have T equal to 1 to be included into this. But I told you that T equal to 1 is an SRB measure. If I can do it, I should be able to build an SRB measure this way. Well, which means that the set lambda, which is the port of this measure, must be a topological attractor. It could be the whole manifold, of course. It's not an excluded. Well, then by Le Drapier and Strelzin old result, we know that if I have an SRB measure, it must be unique due to the entropy formula. Every measure with known, remember, I only talk about hyperbolic measures. I don't talk about any other measures. So in the class of hyperbolic measures, if the measure satisfies the entropy formula, it must be an SRB measure. So if we have it, it's unique. It doesn't say anything about other T's, but at least for this one. Well, so what I'm going to do, the way I want to proceed is to first build an SRB measure. Because this is, I would call it a reference measure. This is a measure I can start with. Then I want to build, extend it to some interval and prove the uniqueness and existence and uniqueness of equilibrium measures. And then I want to study differentiability, which is the same as analyticity of the pressure function. That's the three problems to deal with if you want to avoid any symbolic approach. So the question is how to study the SRB measures. Before doing this, I want to tell you that there is an old but doesn't diminish the importance of the results. We describe the properties of SRB, ergodic properties of SRB measures, which is that every such measure has at most countably many ergodic components on which of them the measure is essentially Bernoulli up to a discrete spectrum. And then the question is how do you prove ergodicity? Well, this is an approach which works sometimes very rarely, but there are examples where it works. You say, well, what if I can prove that every ergodic component is open? I don't know why, but if there is a way to do that, then I can say if my system is topologically transitive, I get unique equilibrium measure. And there are examples of this where this, but where this approach can work and this way you get ergodic unique SRB measure. Well, what about ergodic properties of the equilibrium measures? That's a unknown field. We don't know. I mean, if it comes from symbolic dynamics, then you say, well, then I can get some properties, maybe a decay of correlation. That's where symbolic models were very much useful. However, what if you don't have them, how you wanna proceed? And I wanna tell you that there is a way to think about that, which actually comes again from uniform hyperballicity. Every equilibrium measure has what one can call direct product structure, which means that you can read it here, but you can look at the picture. You have a point, you have unstable leafs through this point, you draw all stable leaves, you draw another unstable leaf through this point, and you look at the halonymy map by stable leafs from here to here. If this halonymy map is absolutely continuous with respect to conditional measures, which do not have to be leaf volume, you don't know what, I mean, you have a measure, the measure gives conditional measures. They can be ugly, we don't know. However, if you can prove that the halonymy map from one to another is absolutely continuous with respect to these measures, you say my measure has direct product structure. Now, what I do this, I'm sorry, yes. So I have this halonymy map, I define this guy, and then I say that I define the direct product structure, and here is my conjecture. If you have a hyperbolic ergodic equilibrium measure for the geometric t-potential, not for some funny potential, but for geometric t-potential, then it has direct product structure. I don't know if it's true, I hope it's true, there are some indications that it's true, otherwise I wouldn't state this as a conjecture. There is actually a rule which I learned from a talk by Schwartz. He said, you can call a statement a conjecture if you know how to prove it, but you are lazy to do that. If you don't know how to prove it, you can call it a problem. So I should have called it a problem because I really don't know anything about that, just some kind of crazy ideas here and there, so you can say this is a problem, but I actually do believe it's true, so that's why I decided to call it a conjecture. Now, if this is actually the case, and I believe right from that property, and that's why I want to fight for that property, you can extract a lot of ergodic information about the measure. Not necessarily ergodistic, but you can get countably many ergodic components. You probably can go further than that and extract Bernoulli or something like that. Maybe you have to add a little bit, at least for the geometric tip potential, I believe that once you establish that, that's the path to study ergodic properties of equilibrium measures, but that so far is at all unknown. Now, so how to study SRB measures? How to build them? Because this is a roadmap to study equilibrium measures. Well, what you do? You want to start with some measures that you call a natural measure. Then you push it forward, you apply the Bagalubov-Karolov procedure, you take a limit, you say this is, it sits on your attract, I mean remember we are talking about an attractor, it has to sit on an attractor, you say this is a natural measure on attractor, and if I can choose this measure appropriately, maybe I can build a nice, maybe it will be a nice SRB measure. Well, there are some pros and cons in doing that. If you do that, that you, I mean following this, every procedure that we know so far of building SRB measures can be obtained by that Bagalubov-Karolov push forward. Even if it's obtained symbolically later on, you can actually prove that you can do a push forward procedure to get the same measure. However, if you don't have symbolic dynamics, so far we don't know how to establish higher properties like exponential or whatever, decay of correlation, center limit, center limit, et cetera. It's in the work, I know some people working on that, but there is nothing I can report on that at the moment, and it's a great problem in my view. Well, you can do it differently, you can do symbolic representation, and that's how in the uniformly hyperbolic case it was built, and then you can use in the, for non-uniform hyperbolic system, you can use either SARIC and build countable Markov partition or Yank tower and build the tower approach, or maybe non-uniform specification, whatever you can do, and the pros is that if you can finish this project, then you end up with decay of correlation and center limit theory. The cons are that there are examples, especially in a partially hyperbolic business where there are no Markov partition, no tower with nice properties, et cetera, et cetera, at least as we know of, and then you got stuck and then you may be back to this non-symbolic approach. Okay, so here's how to build SRB measures without any symbolic approach, and what I'm saying here is that you have a hyperbolic attractor, I'm sorry, you have a typological attractor, and you took a neighborhood of this attractor, and I assume that I have a set D. Well, what kind of set D I have? So now it's a collection of assumptions. One is that it's forward invariant. Second, that I have two cone families on that at every point on this set D. I didn't tell you the set D is open, closed, or anything which it may not be. You can think of this set as a counter set of positive volume, but it's really a counter set, and I did not tell you that the cones depend continuously. They may not. It's a measurable family of cones in general. Then what you do next is, here is the most difficult part of the whole business. Namely, you define two numbers. You have a cone, you can consider the smallest expansion. Well, it's an unstable cone, you expect, you call it unstable because you expect it to expand. So you consider the smallest expansion, and then you take the biggest contraction in the stable cone. They call it effective expansion and contraction coefficients. However, you choose a cone which you call expanding, you don't know if expand, it's actually can contract, at least along the certain period of time. Then it may start expanding and eventually get, say, positive Lapland of exponent. But at every given moment, you don't know if it actually expands. So you introduce a number which is a difference between lambda s and lambda. You call it defect of hyperbolicity. And then you introduce a number lambda x, which is what we call effective expansion rate, or effective hyperbolicity. Because this is, I mean, if you compare these definitions, you realize that if this number is positive, you actually have expansion. It's kind of a local one-time domination, if you want. And the defect measures how much a far from domination at a given point the process is. And then, of course, there is an angle between them. And you want to know for a given threshold that the angle, in average, is positive. So the angle does not just simply go to zero with an exponential rate. You want to avoid that. So that's what you require for, that's what you have on the set D. And then the major requirement is that if you look at, if you take the difference between this effective expansion and contraction rate, take the average asymptotic of this number, you require that on the set D it is strictly positive. It's a serious requirement. And then you require that regardless of when alpha, the threshold for angles go to zero, that goes to zero. That's all, if I go back for two slides and I join all this slide, what I wrote on these slides, that define the set D. If this set D, if I have a set D of positive measure, then I have an SRB measure on this attractor. We call this attractor chaotic. So that's a way to build SRB measures. And what I claim is that every single known approach to build SRB measures can be obtained from this result. And there are situations where you can build SRB measures using this approach, which cannot be covered by other existing methods. Well, how to prove it? How much time I have? Almost none? Two minutes? Okay, thank you. So then I don't tell you how to prove it. You will read the paper. So I have to go through this. I just want to say one, maybe I say one word. Maybe I say one word. The way to prove it, maybe instead of all the slides, I just show you a picture. The way you wanna prove it is that you have an attractor and you have a set D, which is set of positive measure. Who knows what it is, but it's invariant forward invariance. You can approach attractor. You take a point, you draw a nice admissible manifold through this point. You have cones, you have admissible manifolds. You consider any function which is positive on that admissible manifold. A pair, admissible manifold and a function is called a standard pair. Because you have a set of positive measure, you can build a positive measure of those standard pairs. You just fill it in. Then you take all of those density functions and move them forward by the standard Bogolubov-Korlov procedure. You prove that it converges to an SRB. A subsequence converges to an SRB measure. So that's the way to build it. And if I can have one more minute and I scroll the slides, I don't wanna talk about anything else here, but one particular bit. I'm saying that you can change this procedure a little bit. Namely, you move functions using not the standard push forward, but essentially a real operator with a given potential. So an SRB doesn't have any potential. If you have a potential function, you build it in the procedure of pushing forward. And that's the way to do it. And then you take, I mean, this is a normalizing factor. So the standard, it's an adaptation of the real operator to the situation. And that's how you wanna push those density from standard pairs using the potential. And if you do that, then it's a conjecture. I don't know how to prove it. I believe it's completely true. Then what I'm saying, oh, I'm sorry. That's what I'm saying, a problem. Now I call it a problem because I don't really have any idea how to do that. That assuming that if you do this, okay, forget about that. I just tell you on the picture. If you do this push forward procedure adapted to the real operator, then the limit is a subsequent converges to give you an equilibrium measure. And this approach has no symbolic dynamics built in. It's absolutely avoiding any symbolic dynamics. Thank you very much.