 Well, it's nice to be here. I just want to say that after I moved from Russia to the United States, my first conference I attended was conference in Trieste. And it was 1991, organized by Jacob Pallas and Yakov Sinai. And it became kind of a tradition for several years. So I'm actually very happy to see that this tradition has developed into substantially more interesting and broad, like those schools and conference and thanks to the organizers again, Stephanie, first of all. So it's nice to see that it's not only life, but it's also doing pretty well. So the topic of my talk is thermodynamics of the Katoch map. And I would like to say that before I proceed that, last year I gave a talk here at the same room about building thermodynamics for non-uniformly hyperbolic systems. And the reason I kind of advocate in thermodynamics is that from my point of view, there are a number of very interesting, still very open problems in non-un hyperbolic dynamics. Just to mention one of them is the generosity of systems with non-zero Lyapunov exponent in class of smoothness C2. Not C1 for C1. It's pretty much known, but it's completely different world. There are other very interesting problems, still very much open in non-uniformly hyperbolic dynamics. Another one is the coexistence of dynamical system with non-zero exponents and essential coexistence and systems with zero entropy. So the space is split into two. And I will mention this during my talk a little bit. So the space is split into two. On one part, the system is non-uniformly hyperbolic. And this part is open and dense. The other part has positive measure, but has zero entropy. So that's another great problem. And the third problem, I mean the list of course goes on, but the third one is thermodynamics for non-uniformly hyperbolic maps. And fortunately, during the last few years, there has been substantial progress in understanding how to build thermodynamics for non-uniformly hyperbolic maps. And today I will only explain one example where it was done. But there is a theory behind this example, which I briefly mentioned. And there are other methods now available that, for example, von Kleimenhager talked during his talk on non-uniform specifications and things like that. So at the moment, there is a collection of techniques which was developed that one can use to affect thermodynamic formalism. And today we'll talk about one example of this type. So the map that I'm gonna talk about is the Katok map. It was introduced by Katok in his very famous paper in Annals in 1981. I believe that most of you might have heard about this map, but I will briefly go over it just to explain what this map is about. And this was the first example of a diffeomorphism with non-zero Lampunov exponent. So it starts with a two-dimensional, map on the two-dimensional torus, which is just a hyperbolic automorphism. The Kat map, as you recall, we called it given by this matrix. And the idea is that this linear hyperbolic automorph has a point zero as a fixed point, of course. So what I would like to do, I would like to perturb this map in a small neighborhood of zero, leaving the rest unchanged. The perturbation that I'm gonna make is localized and the neighborhood will be very small, but the perturbation itself will not be small because if it's small, then you end up with an loss of diffeomorphism. And I want to construct a diffeomorphism on the torus, which is not uniformly hyperbolic. So to do that, I choose a function psi, which is a C infinity except at the origin. So you should think about the graph of this function as being the following. This is the graph of the function psi. So it's one when you exceed some number R naught and then it's C infinity except for this point. And the reason I want it to be like that is that the last requirement that the inverse of the function is integral. So I use this function to slow down trajectories of the original map in a small neighborhood of size R naught around zero. So my next step is to consider the disc, choose R naught such that we have this property and T is a linear map which is given by the matrix A. So I distinguish between the matrix and the map. And then the main trick is that in the small neighborhood of the origin, the trajectory moves along hyperbolis. That's a linear map. That's what the standard automorphism does. And then you can embed these trajectories into a flow and that's local flow given by this system of differential equations. That's easy. So what I'm gonna do, I'm going to perturb the flow. And a perturbation of the flow is by multiplying by this function psi, which will depend only on the squares of S1 and S2. So it's a function of one variable and which graph is here and which satisfies the conditions I wrote before. And so if you look at the trajectories of this flow, so it's again a local flow which is only defined in a small neighborhood of the origin. So what happened is that you have many more points along the trajectory, if you take time one map of the flows and you would have many more points on the hyperbola than it was before. And the closer the trajectory starts to zero than more points you see. In particular, if you start on the stable separatrix that the trajectory will slow converge to zero with a rate which is not exponential. So that's basic idea. So for that I can say that I will consider the time one map of the flow denoted by G and then I consider the map capital G which is my hyperbolic total automorphism outside the small neighborhood and inside is the time one map of this flow. And this will produce a map on the torus which is C infinity everywhere except at the origin because my function psi is not C infinity at the origin. So I lost differentiability at C infinity at this point, at this moment, I will recover it later but at this moment the map is almost C infinity but it's not exactly. So what I wanna say is that because of this requirement it turns out that the map which I just constructed preserves a measure which is absolutely continuous with respect to area. And it's given by the density. This is explicit formula for the density. It's a simple calculation. K naught is a normalizing factor. And this density is blowing up at zero. So it goes to infinity at zero. But it's integrable. It's integrable because of this fact. So it's finite measure absolutely continuous with respect to area but it has singularity at zero. So in order to correct this construction and produce a really C infinity map of the torus I need one more perturbation. And that one more perturbation is the following. Since if you look at what happened in the origin then the area, the new absolutely continuous invariant measure blows up at zero meaning that it gives a lot of mass close to zero. So what I, the idea is to push this mass away from zero. And this, so instead of going along the hyperbole I just push it in a symmetric way in a nice symmetric way outside of zero. This is an explicit formula that produces this coordinate change. This can be viewed as just a coordinate change. And then after doing that we obtain a map which is conjugation of my map G by this coordinate change. And the resulting map will be a C infinity diffeomorphism. Area preserving because this map fee will move this absolutely continuous invariant measure into real area. And this is just a matter of computation. This is explicit formula. It's not difficult to compute that the resulting measure is the area. So the map we obtained is called the Catoch map. That's the definition of the map. And then the question is you have an area preserving C infinity diffeomorphism of the torus. What properties does it have? So that's basically the question. Well, here are the properties. First of all the map F is topologically conjugate to unperturbed linear automorphism. So there is a homeomorphism H which conjugates the Catoch map and the linear map. But the conjugacy is continuous but not herder continuous. And that's crucial for us. Because in the study of thermodynamics we normally like to deal with the herder continuous functions. If you take a herder continuous function on the torus and push it back by this homeomorphism if it were herder continuous you obtain the herder continuous function. You can use this herder continuous function and the linear map A develop the whole business of thermodynamics. It's easy. It's a hyperbolic map. Everybody knows that. Push it back by homeomorphism and get an equilibrium measure whatever you want. Unfortunately just doesn't work this way for the simple reason that the conjugacy is not herder continuous. Start from a herder continuous function you push it back with that non-herder continuous function for the map A. It's known that if you have non-herder continuous function for a uniformly hyperbolic map even linear hyperbolic map then in general you may not have unique equilibrium measure. So since can go wrong if the function is not herder continuous. And therefore it's a question what to do. I mean, it's a nice state fact but the question is how to use it. It cannot be used directly. The second property is that the top map lies on the boundary of a north of diffeomorphism of the torus. So here I would like to say that if this is a space of all C2 or C infinity diffeomorphism on the torus, then you pick the matrix A there is a small neighborhood of this matrix which consists of a north of diffeomorphism, right? Because any small perturbation of A is still in a north of diffeomorphism. So what you do is just consider the biggest possible connected component of A which consists of a north of diffeomorphism. And you end up with I don't know some open set. That's an open set of course. So what one can do what this statement two says that you can start from here and draw in one curve in this space which goes through almost of diffeomorphism. And in the end when it reaches the boundary, so this is A and this is the top map F. It's a simple exercise and I leave it for you because this is not just a conference, it's a school, so you're students so I can give you some problems or exercises. So this is a simple exercise to construct such a curve starting from A and end up with F. You just work with a function psi and you can see how you should vary the function psi to obtain such a curve. But this can't be done. In fact, what happened that if you, here you have all a north of, at this particular curve you end up with a top and you know that it's non-uniformly hyperbolic. A good question is, is that true that any diffeomorphism on the boundary of a north of has non-zero loponoff exponents? It's not a north of because it's lies on the boundary. Otherwise if it would be a north of you can extend it, you can build another ball around it. So it's on the boundaries and it's not a north of. The question is, does it have non-zero loponoff exponent or whether it has all non-zero loponoff? I mean almost everywhere. And I'm talking about diffeomorphism that preserve area. So it's not just finding diffeomorphism, but area preserving. So that's an interesting question. I don't have an answer to that. Now another question is, can you extend this curve beyond a north of, beyond the boundary? Well, for example, you can make a perturbation that does not leave the point zero fixed. It can move it somewhere and do something. The point zero for the cutoff map is a neutral fixed point. The derivative at this point, the differential is identity. So you can always blow it up to produce an elliptic island. And if you produce an elliptic island, that's a coexistence of behavior which is that I mentioned before. So you would have ergodic components with non-zero loponoff exponent outside and an elliptic island inside. Now, this is a picture that I have in mind, but I don't have a theory. So everyone is welcome to prove that. Whether this curve can be extended. And I don't tell you in which direction should it go. It may go this direction, this direction, who knows which direction. But one can produce a direction along which you obtain exactly one island, elliptic island, and the rest will be an ergodic component with no zero loponoff exponent. That's a great problem about the cutoff map, which is open. Now, there are results by Przhtitsky and Leverani who obtained such curves, starting from some other north of map, producing a curve that actually goes through. And here on a small part of this, you can see one elliptic island. So you can go to papers of Przhtitsky and Leverani and find this result. This is a different curve. I believe the result is the same, but I don't know. Then the map is, the next step is, the map is ergodic. In fact, it's isomorphic to Bernoulli. It's topologically transitive, it's mixing, it has very nice ergodic properties. And this is again a nice exercise for you guys to prove that it is ergodic. It doesn't involve any sophisticated argument, you just say, let me have an invariant set of positive area. I wanna show that it has full measure, yes. I'm sorry, can you say louder? Not necessarily, not necessarily. I'm just saying whichever way you can do it. If you can do it this way, fine. I don't think you can do that, but whichever way. And the property that I'm most interested in is that this map has non-zero Lyapunov exponent in almost everywhere in the neighborhood. One thing I wanna say is that, of course, if you look at this, at the cutoff map, so this is a torus, this is a fixed point, this is a neighborhood where we do the perturbation and the trajectory is like that. This point is neutral, the differential is one, it has zero exponents. So you have at least one point with zero exponent. In fact, if you take any point that starts on the stable separatrix, you get zero exponents because the slow down is such that it comes very slowly to zero, it approaches zero with a very small speed and therefore it's zero. And then you push this stable separatrix away and you get a dense set which consists of points with zero exponents. So you have plenty points with zero exponents. However, one can prove that this set has zero measure. So there's no, I mean, there are other points except for the separatrix, there are other points with zero exponents. So this does not exhaust the whole set with zero exponent but whatever this set is, it has zero area. So it's a system with non-zero exponents. Well, other properties is that since it's a system with non-zero Lyapunov exponent, almost everywhere you can construct stable and unstable distributions. But the non-uniform hyperbalistic gives them at almost every point. What this statement says is that we can extend it to the whole area, including zero, by the way, to obtain a continuous distributions. And that's a non-trivial statement at all. And then those distributions can be integrated. Well, there are one-dimensional distributions that can only be integrated. The question is whether we can uniquely integrate and that's the claim. So in fact, you can do that. And those affiliations that you obtain are images under the conjugacy of the standard stable and stable affiliation for the linear map. So the picture is pretty nice in this sense. Now, the reason for Katoch to introduce this map was not just to build this example. He wanted to build C infinity diffeomorphism with non-zero exponent on any surface. And for him, this map that we now call Katoch map was a starting point. So the next step, you start from the torus, but next step is to build such a map on the sphere. And to do this on the sphere, you change the original map. You just take the fourth power of that. There is reason for number four. The matrix you obtain will be this one. And the reason for four is that now this matrix has four fixed points. If you have four fixed points, what you do, you perform the slow-down procedure around each of them in the way I just described. Outside of the small neighborhood of these four points, it's a linear map. Inside, it's those slow-down procedures. This way, you get a map. It has non-zero random of exponents, et cetera, et cetera, still on the torus. But now you use a standard topological procedure which says that you consider the involution map which is given by this formula on the torus. Remember, the torus is just a square. So it's a global coordinate system. Then this involution has these points as a fixed point. It commutes with the Katoch map and the factor space is a sphere which allows you now to bring the Katoch map to the sphere. So now you have a map with non-zero random of exponent on the sphere. Well, you go to the next step, which is punch the sphere in one of these fixed point, unfold it to a disk, and now you have a diffeomorphism with non-zero random of exponent on a disk. It has an advantage that on the boundary of the disk because you punch it at fixed point, the map is identity. So let's remember that. Now you take any surface. You know very well that you can cut the surface so that you get a polygon. Topologically, polygon and the disk are the same thing. So you can move one into another. That's a well-known topological procedure. What was not known before the Katoch work is that using this procedure, you can carry over the diffeomorphism on the disk to a diffeomorphism, not a homeomorphism, but a diffeomorphism on the surface. And to be able to do that, you need that the map on the boundary is identity. So on those cuts, the map will be identity. But you also need the property that when you come close to the boundary of the disk, the derivatives come very fast become close to identity. I'm sorry, to zeros. This can be ensured if you choose a function psi here, very, very flat. So it goes like that. It has to be extremely flat. If the function psi is extremely flat, then the map on the disk will have derivative approaching the boundary of the disk very quickly. And it will be very flat in the neighborhood of the boundary of the disk. And a diffeomorphism with these properties can be carried over to any surface. And this way you get a diffeomorphism on any surface with nonzero lapon of exponent. Well, that was the first step in trying to create, to build maps with nonzero lapon of exponent on any manifold. So Catoch did it for surfaces. Then there were a number of intermediate results. I mean, the problem that actually was a motivated problem for Catoch was not just to build a system with nonzero lapon of exponent, but the method is used now non-uniform hyperbalist theory to claim that this map is Bernoulli. So like immediate corollary on any surface, there is a Bernoulli diffeomorphism. That was an important problem in classical ergodic theory. So the question is, can you extend it from two dimensions to any dimensions? And there was a paper by Brinn, Catoch, and Feldman where they showed that you can do it. On any manifold, you can build a diffeomorphism which is Bernoulli. But that diffeomorphism that they constructed had one zero exponent. So the question then, can you change the construction? So that you have nonzero exponent and Bernoulli. And that was done in my paper with Dima Dogapiat. So now it's completely settled. Any manifold of dimension bigger than two has a C infinity volume preserving diffeomorphism with nonzero lapon of exponent which is Bernoulli. So from that point of view, this problem is solved. The question is, how many we have? What's happened when we perturb them and then we go back to those problems that I just posed. So that was one direction to go when you study Catoch map. The next direction to go is to build thermodynamic. Well, I just wrote that there have been a number of talks where people talk about thermodynamic equilibrium measures, et cetera, et cetera. So I'm not going to repeat that. Just tell you that this is the notion of the equilibrium measure. And what I'm actually interested in is what I call a geometric T potential. So when you have a system with a hyperbolic system, you can consider unstable direction. You take Jacobian along this direction, log of this, multiply by a parameter T, you get a family of potentials. And the question is not just any potential, but that potential, does it have a unique equilibrium measure? And if it does, what ergodic properties this measure has? And that's what I call thermodynamic for this particular talk. Now, if the map is uniformly hyperbolic, for example, if it has uniformly hyperbolic attractor or a hyperbolic XOM-A map or whatever, then these subspaces are further continuous. Therefore, this function is further continuous. And there is a well-known result that any further continuous function for hyperbolic map has unique equilibrium measure. And this measure is fantastic. Bernoulli, it has all the fantastic ergodic properties, exponential decay of correlation, central limit theorem. You name the property, and it has it. So that's the best case scenario. Now you go away from that. Before we go away, I also would like to consider this function. It's called the pressure function. For that particular potential, you have this pressure function. It's easier to draw the graph of this function, which is just like that. It's a real analytic, strictly convex, et cetera, et cetera. This value at 0 is the topological entropy of the map. And the corresponding measure is a measure of maximal entropy. And this number is less or equal than 1. In the case of hyperbolic attractor, it's 1, and the corresponding equilibrium measure is a famous scenario in measure. In the case when it's less than 1, you still have a nice equilibrium measure, and it has certain meanings, certain relations to dimension theory, et cetera, et cetera. So all those measures are kind of interesting. That's why the geometric potential is so good. OK. However, I would like to stress that once we move to non-uniformly hyperbolic case, this dependence is no longer herder. So you lose the property of this potential to be herder. Now, if we use a result by Omri Sarek that Yuri talked about and consider a countable marker, let's say we are on a surface, and you consider a mark of partition on a big countable, you can lift this potential to this mark of partition, and what is important is that it is herder continuous in the metric of this countable shift, subshift. And if this is the case, then you are in a reasonably good shape. You can still get a lot of information about equilibrium measure. So what I'm trying to say is that this potential does not have to be herder continuous, but it's lift to symbolic space should be herder continuous. And that's one can prove in many cases. And that actually works. This idea actually can be implemented in some cases. But in the cases that I'm going to consider, unfortunately, it doesn't work this way. So we have to do something else. Anyway, so if you move from uniformly hyperbolic case that there are plenty of results on existence and uniqueness, et cetera, I just list some of them. In the one-dimensional maps, the thermodynamics was built, in the case of unimodal, multimodal maps. And this is a list of people who worked on this. The lists are not, I mean, many names are missing. So don't blame me if you know of some work which is not mentioned here. I just cannot list everyone. Then there is another good class of one-dimensional maps with some different fixed point, another collection of people who proved existence and uniqueness and studied the property of equilibrium measures. Then there is a polynomial and rational maps which somehow can be viewed as like one-dimensional maps. It's kind of the same story. And the general picture in all these examples is that this function is renal and analytic. Everything is fine. But it's only defined on a certain interval, from T0 to T1. One can prove that on this interval, everything is fine, equilibrium measure exists, and unique. But there might be two points where phase transition occurs. Uniqueness is lost. Very often this point T1 is just one, not necessarily. And T0, it's an interesting question where T0 should appear. I will comment on that if I have time a little later. And then there is a work on non-uniform hyperbolic expanding maps or maps with dominated splitting where one direction either stable or unstable is non-uniform hyperbolic. And again, there is a list of people that I have here. List is again not complete. And there are different results in this direction. Some claim existence and uniqueness, some claim just uniqueness provided it exists, et cetera. I'm not going to, I mean, this is not a talk about surveying all of those results. Just mentioned that there have been a lot of activities. However, if you go to all of them, you either have low dimension, like one dimensions, or you have some uniform hyperbolicity, maybe weaker than the standard one, still built in. The talk map is a case where no uniform hyperbolicity whatsoever is present. So it does not feed in any of these categories. And the question is how to study this. OK, so for my purpose, my function psi will be in a particular form. So this function is not too flat. It's a polynomial function in this sense. So I cannot use it to build a diffeomorphism on any manifold. But I'm fine because I'm only interested in what's happening on the torus. And for the torus, that works fine. Now, my alpha is less than 1, which means that the corresponding talk map is area preserving. It's interesting question whether the results I'm going to explain will work when alpha is bigger than 1 than it does not have. It has an infinite, absolutely continuous invariant measure. However, you can still ask whether the geometric potential still has unique equilibrium measures. That would be an extremely interesting question. I don't have an answer to that. If you ask me, I would say bet that it's true. But I don't know. I may lose. So it's very well put to happen. So here is a theorem. There is a theorem describing cyber dynamics for the talk map. So the theorem says that for arbitrary large number t0, so I can choose this t0 as far to minus infinity as I want. But it's still finite. I cannot push it to minus infinity. There exists r0, which is the size of slowdown domain, such that for every t in this interval, so one critical point is 1, the other one is very far to minus infinity. What happened is that there exists unique equilibrium ergodic measure mu t associated with the geometric potential. So the equilibrium measure exists and unique. Now, because my maps is c infinity, you should understand that by result of new house, it always has, for any continuous function, it always has an equilibrium measure. Since if I go backward, since those subspaces are continuous, they're not herder continuous, but they're continuous. This function is continuous, which means that there is always an equilibrium measure. So in fact, the whole statement is not about existence, but about uniqueness. That's what make it hard. Existence is essentially for free because it's c infinity. Then what we go next is this measure has nice properties. It has exponential decay of correlation, satisfy central limit theorem for the class of nice test functions. The pressure function is real analytic on that interval. At the point 1, so here is 1, and at this point, you have phase transition. You have two equilibrium measures. One is area. Another one is the direct measure at 0, which is clearly an equilibrium measure in this case. And for t bigger than 1, that direct measure is the unique equilibrium measure for the map. So the pressure function is not differentiable here. It just continues this way. So that's the graph. The pressure function goes to minus infinity. I mean, it's like that. But there is some point so I can prove existence and uniqueness on that interval. And it can be arbitrary large. But I don't know what happened after this point. I know they exist. I don't know. What does this point do? Either it's not uniqueness, which is broken, or you still have unique equilibrium measure, but you lose the property that it has exponential decay of correlations. Maybe it does not. Or in fact, you can choose t0 to be minus infinity. It's just the method I use that does not allow me to push it to minus infinity. Maybe you can. That are the questions I don't know how to answer, but would be very happy to know. So that's the situation for the Katalk map. Let me mention that there are other good examples, which are similar to the Katalk map. And the result for those maps are similar. And the techniques that is used is similar, but not the same. And certain things are different and may not be easy to establish. So the first example is that you start not from linear analysis of diffeomorphism, but you start with diffeomorphism, which has a uniformly hyperbolic attractor. An example of such attractor is male-villiam solenoid. I guess everybody in this room should know what the male-villiam solenoid is. It's a classical example of an attractor, but you can't pick any attractor. It is required that the unstable direction is one, but the stable direction can have any dimension. So this is actually a multi-dimensional case version of the Katalk map. Now, which measure does it preserve? There is no area involved because it's an attractor, or volume, or whatever. So there is a result by von Klemenhagen-Demodal which claims that for that map, you do slow down in a small neighborhood exactly in the way I described. You obtain a non-uniformly hyperbolic map attractor, and the result by von Dema and myself claims that you have an SRB measure. So this is a substitute for the area. So now we are back in the game. You have a non-uniformly hyperbolic map with an SRB measure. You wonder, can you build a thermodynamics? And you can obtain the very same result which I showed on the previous slide. And this is a wonderful paper by Agnieszka Zelerovich, who is present in this room. And it's actually a technically very hard paper, very hard to prove result. Just think a little bit. The matter is that this attractor in the stable direction is a fractal set. So it's not as nice as Katok where kind of everything is essentially on the plane, et cetera, et cetera. This is a hard object to deal with. So you have to reproduce. I mean, you follow the general scheme, but the road is much more difficult and much more bumpy than in the case of the Katok map. And the second example is deals with hand non-map. So we consider the classical hand non-map. Now, what's known that if the parameter B is very, very small, then there exists a uniquely defined parameter A star, such that the non-wandering set has a uniformly hyperbolic horseshoe for A bigger than A star. But at the moment of A star, you have a tangency. It's the first moment where hyperbolic horseshoe becomes non-uniformly hyperbolic. So you get tangency of stable and unstable. So that's called the first bifurcation parameter. And at this point, so the question is you look at the hand non-map at this bifurcation parameter. You can prove existence of that SRB measure. The question is, what about the thermodynamics? Well, the answer is that you can prove the very same result I claimed before. It's the same picture for the pressure function. And you can use similar methods that I will explain in a few minutes. And it's a result by Takahashi and safety. The problem, the real problem here is if you go from that parameter, which is very exceptional parameter, it's only one. It can be identified, but it's only one. So the question is, can we go to this positive measure set of parameters of Benedict's young for which, by their result, you have an SRB measure? But that's a positive set in the space of parameter. And do thermodynamics for those parameters. That's a huge, interesting, unknown, I mean, unsolved problem. If I have time, I can tell you what distinguish that particular parameter from all others. OK, so now what I'm going to do is to say a few. How much time I have? Quarter of an hour. OK, I just want to say a few words about the proof. So how do I want to prove that? So remember that I have a torus, my origin. This is a linear map A. And then I have another torus where I have the katoch map F and again the origin. And I have a conjugacy H from one to another. And this is a wonderful loss of map. So it has a mark of partition, finite mark of partition. So this is a slow down area where everything happens. Nothing happens outside. So I choose a mark of partition whose elements are very small. And I choose one of them which is far away from the slow down domain. Far away means something. Doesn't matter. It's not so important. Then what I'm going to do, so I call this element P tilde. And what I'm going to do is this is a rectangle. It's an element of the mark of partition. This is clearly just a rectangle. It's a product of stable lines and unstable lines. You can see that the coordinate system is built by eigen direction. So it's just vertical and horizontal lines. Nothing more than that. So what I'm going to do next is I'm going to consider the first return time to this domain. So for me now, this domain is just inducing domain. I want to induce on that domain. And if I do that, I'm looking at, so let me draw it a little bigger, I'm looking at the level set of the return function. So the set of points which return back at the same time. And what I claim is that if you have such a point, then the whole stable leaf, which is just a vertical line, so say this is a stable leaf, will return to the same set at the same time. So the level sets consist of some collection of stable leaves. So in very time, you get one such small rectangle. Then you get another one, third one. And you have countably many of them. Because the inducing time goes to infinity. The first return time goes to infinity. They do not cover, those sets do not cover the whole rectangle because there may be some points which never return. For example, a point on a separate X may go to 0 and never return. Except this one, is there another point which does not return? That's I don't know. The matter is almost every point is dense, but not every point. And there are some exceptional points which are kind of hard to find. So I don't know exactly. I believe it's true, but maybe I'm wrong. That's I don't know. But whatever it is, it's a set of 0 may. That's I know for sure. And that's what matters to me. And also there are some points which lies on the boundary of Markov partitions, which I also consider kind of a bad point. So if I take those which do not return and those which falls on the boundary of Markov partition, I get some set which I call a bad set. And what I claim that this set has 0 measure on any stable leaf. Now what happens next is that if I consider the image of the stable set by the Kotok map in the appropriate power, each stable set is associated to return time, then I obtain a set which I call a u-set. We just have this form, which is a rectangle which consists of unstable leaves. And it ends exactly on the boundary of the Markov partition. So you have a property that each stable leaf moves into the unstable strip, moves into the unstable strip. This is called a branch. So every and those stable leaves are disjoint. And the unstable leaves are disjoint because it's the first return time. So it's a very nice and simple picture. So then my next step is that I have a conjugacy, which means that I can move this picture by conjugacy here. I get an element which is no longer kind of a real rectangle, it's kind of a curved rectangle, which can still be partitioned by stable strips whose image are unstable strips. They're pretty much the same picture. The return time is the same for the linear map and for the nonlinear map. And in fact, combinatorics of the picture for the linear map is exactly the same as for the nonlinear one. So that's where I use conjugacy. That's important. Now the next step is that the new set that I obtain is what I just mentioned, is built by stable leaves. Now stable leaves for the cutoff map because the conjugacy moves stable leaves for the linear to the stable leaves of the cutoff. So the picture is just easy to see. So we move next. So what I do now is that I consider the set lambda, which is the intersection of all stable leaves with all unstable leaves. This is the set that has full area. But it's not the whole. There are some bad sets of zero inside. So lambda is kind of a set, which is called a set with hyperbolic product structure. And it has certain properties. I want to list those properties. The first one is that for every unstable leaf, the intersection of what's left not covered with this unstable leaf has zero leaf volume. Meaning that it actually lengths along the unstable curve. So if we choose any unstable curve over here, this is my gamma u, and measure the lengths of those points which are covered by this stable leaf, then it has full measure inside the leaf. So that's just by construction. It's easy to prove. The next property is that it's called the set. So if I take the inducing time ti, and now it's not tx, but ti, because it's the same on each stable strip. So I number those strips. It's countably many strips. I number them by i's. And I have for each strip, I have a number, real number, ti. It's actually an integer, right? Where they return back. And they all return back at the same time. So if I measure, if I sum the return time and measure of those which return at this time, then the sum is finite. That's a very crucial property, which I comment on a little bit later. The next one is invariance. You take a stable leaf. You apply the map induced map, the map, the katoch map, and the corresponding part. I'm sorry, it should be capital F. Then it lies inside of this table. So there is a contraction along stable, uniform contraction along stable. And then you apply it here. You get an expansion along unstable. And you have the mark of property. And the mark of property follows from the fact that that was an element of the mark of partition. So I started from a mark of partition. The mark of property is automatic. OK, there are two other properties. Now I consider that what is called induced map. This is just the map on the katoch map raised to the corresponding power. So it's a map, so you take a point x. You take a point x. It goes somewhere. So let's look at the trajectory when you start. Well, it travels. It can go this way, and it can enter, then it can go around, can enter again, whatever. Eventually it comes back. So you just consider the return map. This is what I call capital F. And this map, curled capital F. And this map is uniformly hyperbolic. It's a map from this piece of the element of the mark of partition on itself. This map is uniformly hyperbolic. It contracts uniformly along the stable leaves. It expands uniformly along the unstable curves. And it has bounded uniform bounded distortion. Now how do I know that? The map is non-uniformly hyperbolic. So you have a non-uniformly hyperbolic map, and you found the domain such that the return time to that domain is perfectly uniformly hyperbolic. That's a big deal. So proving these two properties is what constitutes the whole proof of the theorem for the cut-off. My technical part of the proof. Because you have to know exactly how it goes through the slowdown domain. You have to obtain some very nice estimates on how trajectory goes through the slowdown domain. So in particular, what you should know is you can construct continuous families of stable and unstable cones, invariant cones. Katok, in his original paper, said that if you just take a cone around horizontal and vertical direction of 45 degree angle, then it's invariant. And it's an easy calculation. Not sufficient for our purposes. You need to show, on the other hand, if the angle is too small, then it's simply not true. The statement is not true. So there is kind of a threshold, which is less than a 45 degree angle, but it's not too small. And the idea is to find this threshold. It's a part of the business, and it's technically challenging question. Then the next one is to obtain a polynomial bound on the rate of contraction and expansion of vectors which go through the slowdown domain. If you have a slowdown domain, and you take, say, horizontal vector, before you do slowdown, it will exit perfectly horizontal, would have a nice, I mean, the length will be, will expand for the linear map as usual. But if you slow down it here, so you have many more points to go, then this vector will not return vertical. It will return like this. It will be a little bit bigger. The question is how much bigger? And you have to control all of that. You have to control the angle. It will get exit. You have to control how much it will expand versus the time it spends. Because if you go this way, the time you spend is substantially longer. So there will be difference between angle and the length. And you have to control all of that. And if you do, then you can prove this property's y4 and y5. That's what it requires. So I'm not going to discuss that because it's a technical issue. But it occupies like 80% of the arguments or whatever. Now, once this is done, we are close to the proof of the result. Namely, I say that if you have a set lambda with hyperbolic product structure, just think about this Markov rectangle with stable and unstable curves, which satisfies these properties, y1, y5. And again, I just repeat that it means that if you take a stable leaf here, unstable leaf here, the stable sets fill in the full measure. Then you have the induced map is uniformly hyperbolic. There is Markov property invariance, et cetera, et cetera. If this all holds, then this diffeomorphism is called young diffeomorphism. It was introduced, this class of system was introduced by Young in 1998 paper. And what she says that such map has a tower. You can easily build a tower. Now, tower construction, the narcotic theory, is a well-known old object. But that is special tower. That's why we call it Young Tower. What it has, it has a base, which is lambda. And then it has height, which is the induced time. And in our case, the induced time is first return time. So what we built, we built a first return time Young Tower for the Catoch map. Now, what good does it do for us? Here is the main result. So I consider a young diffeomorphism and I consider the geometric potential. That's the class of potential size I'm interested in. I look at the set SN of those stable sets for which the induced time is n. What I mean by that is the following. Let me draw this picture once again. I told you that when you have this rectangle, you look at the time when it returns, say, time n. You want time x to be n. And you look at those points which travel, travel, and return back at time n. What you're going to see is the following. You're going to see a strip, as I explained. But you may find another strip and a third one. So the question is, how many such strips I have at a given time n? Now, the number grows with n. And I want to control how does it grow. And for that, I introduce this number SN, which tells me how many different strips you have at time n. Then there is another condition, which calls the arithmetic condition. You look at the set of all values of tau i. This is a subset among the integers, positive integers. You look at the greatest common denominator, and it has to be 1. So then I say the tau are satisfied the arithmetic condition. OK, why not? Here is the theory. If I have a young tau, just a young tau, satisfies properties y1 by 5, then there exists an equilibrium measure, mu1 for the potential phi1, which is an SRB measure. That's a result by Young. That's what she proved in her 97. And then another paper, 98, she discussed exactly the existence of SRB measures. Next is that assumes that this number SN grows exponentially with some constant, with some exponent h, which is less than minus integral of that potential phi1 with respect to the measure mu1. This is the Laponoff exponent of the SRB measure. Nothing else but that. SN is the number of stable strips, which returns back at time n. Then there exists a number T0, such that there exists a measure muT, unique equilibrium measure for the potential muT. So you have this picture, starting from 1 till some T0, unique equilibrium measure. But that requires this assumption. And in fact, this measure is Bernoulli. You can prove Bernoulli property. Then, if there is metric condition holds, that's why I needed it, the measure muT has exponential decay of correlations, satisfies central limit theorem for the class of these potentials. So you have a very nice, ergodic properties. I'd like to tell you that this statement for SRB measures again was proved by Lyssen-Young. But for equilibrium measure, this is a new result. And the pressure function is real analytic on this interval T, not T. So that's the general result, which was proved by Sam Sainty, Kershenko and myself. The paper appeared last year. And this is the major result that we will use to prove everything for the cut-off map. So what needs to be done is a simple fact. To prove the main result is I need to show this estimate, SN. I need to show that how I satisfy the arithmetic condition. And I need to show that I can push T0 arbitrary far. And I need to show that the measure I obtain, I obtain unique equilibrium measure on the tower. I want to show that that's unique equilibrium measurement on all measures, not just the ones on the towers. Because the tower occupies some part of the space, but not necessarily the whole space. So there may be another invariant measure. So maybe they give more, also maybe equilibrium measure. So I claim that, no, this doesn't happen. So that's what needs to be shown. And then very briefly, do I have like two, three minutes? OK, thank you. I mean, I'm almost done at this point. So I just want to comment on how one can prove this exponential bound on the number. And the idea is the following. If you look, the matter is that here is the cut-off map. And the inducing time for the cut-off map is the same as for the linear map. So if I want to prove anything about the number, I can prove it for the linear map. And for the linear map, I have symbolic dynamics. So what I'm looking at is the number. I mean, if I use Markov partitions, and I'm looking at the number of words, symbolic words, which starts with p tilde with this element, ends with p tilde, and there is no p tilde in between. So I'm looking. So each such word will give me one element which returns a time m. So how many such elements I have is the number of those words. And then there is a well-known result in symbolic dynamics that those numbers grow not faster than topological entropy. So the number of those words s n does not grow with e h n with h less than topological entropy of a. But a is a linear map. For the linear map, the topological entropy is the metric entropy of measure one for a. Because in this case, the area, the kind of SRB measure, the one that comes here at t equal to 1, this area is the same as measure of maximal entropy. So the entropy's coincide. So now when I make a perturbation, if the size of the perturbation is small, this one becomes smaller than the topological entropy. So when I put f here, and I look at the h top of f, then it will be bigger than h mu 1 for f. But the difference between them can be made arbitrarily small if the size of perturbation is small. So there is a distance between h and this number, call it epsilon. I can make a perturbation so small so this distance is epsilon over 2. And therefore, what I would have is h less than this number. And that's verification of the condition. Then one more minute. How to verify the arithmetic condition? Well, again, the arithmetic condition is about the height. That's a common denominator is 1. But the inducing function for a katoch map and for the linear map are the same. So I can verify it for a. How do I verify it for a? I say that I know that a is Bernoulli, which means that every power is ergodic. And then it's simple argument. You can find it in Lays and Young paper, which claims that if you have this property for a, then the arithmetic condition holds for a, and hence for the katoch map. So that's the verification of the two conditions. And the last piece I want to show is this is a formula that can be obtained by calculation for the number t0, the one that bounds the interval. And oops, I'm sorry. This is the topological entropy. This is the metric entropy. This is the metric matter. And this is the number log k, which is computed by a specific formula. And then you can show that if the size of the calculation of the slowdown goes to 0, then this difference goes to 0. And therefore t0 goes to minus infinity. And that's it. Thank you very much.