 recording. Okay, so welcome everybody back. So we have the second and final lecture of today. And it is a great pleasure to have Yakov Pesin from Penn State University. And Yakov will talk about Sinaihuel Bowen measures for surface bifemorphisms. Please, Yasha. Yeah. Thank you, Yuri. First of all, I would like to send organizers for this great opportunity. There are so much missing conferences these days. Of course, we all would like to be to meet in person. But even this opportunity is a fantastic one. And I especially like the whole idea of that, that preceded by school. So a lot of participants get, first of all, involved in the topic. And so it made my life much easier because I didn't have to explain some of the notions that I believe everybody, everybody's that I will be using, but I believe everybody by now knows. So the title of my talk is Sinaihuel Bowen measures for surface bifemorphisms. This is a joint work with one climate hugger and Stefano Luza. I would I mean, I would like to start by just stating the result as it is. So we, the setting is that we have a difemorphism F of class C1 plus alpha on a compact orientable surface M. So it's a two-dimensional case only. And I claim that this F is that the map F admits Sinaihuel Bowen measure if and only if the three conditions holds. The first condition is that this map has a nice domain. I call it gamma PQ and I will explain what this PQ means and gamma means and what the nice domain means. Then it has also a regular set lambda L. And again, it's not clear why L is here, what L means, and I will explain all of that. And the third condition is known empty in the back, strongly recurrent subset A in here. That's the third condition. And again, I will explain what it is. So if these three conditions hold, then I claim F has a Sinaihuel Bowen measure. And in fact, if a map does have such a measure, then these three conditions hold. So it's necessary and sufficient conditions for existence of such a measure. I immediately would like to mention result by Friedrich Raderiges-Hertz, Hamer Raderiges-Hertz, Ravugurus, and Alitak Hibik, which claims that a topologically transitive surface diffeuemorphism may have no more than one SRB measure. So it means that if you add to the three conditions that I just mentioned and other conditions that F is topologically transitive, then you claim that there exists a unique SRB measure for the surface diffeuemorphism. So my goal now is explain all the three conditions. And here we go to the notion of nice domain. So a nice domain is a topological disc, and you can see the picture. I denoted by gamma PQ of small size r. How small the size r will be determined later. So we just fix small r and we claim that a topological disc is a nice domain if it's formed by pieces of stable and unstable manifold curves actually in our case of hyperbolic fixed points. So what I claim is that there are two points P and Q, which are hyperbolic. Naturally, they have stable and unstable separatrices. And those separatrices, we just take the small pieces of this separatrices local intersect forming what one can call a rectangle or a domain. And intersection of course should be transversal. So that's the notion of nice domain. So why it is nice because if you now take the iterates of stable and unstable boundary of this domain gamma PQ under multiples of some time t. So the time t is defined by taking just the period of P period of Q and for example, multiplying that. But you can take any other number which is multiple of that. And that's number t. So in the whole talk, when I say I consider a time i or time n, it's multiple of t. I do not consider any other time. So for me, the time is t times either the set of natural numbers or the set of integers. And the niceness property means that if you apply this multiples of t to the stable and unstable local boundaries, they do not intersect the interior of this nice domain. So that's a very crucial property, which is uniquely two dimensional, which we will use. And later on, I will point pin point the place where that this particular property is used. And there's only one place where it is used. So and that's the only condition that requires two dimensions. So we hope that eventually one can get rid of this condition and then the result will be multi-dimensional. But we don't know how to do that. So the second requirement is regular sets. And this is essentially non-uniform paper balicity, just stated in a way which is in the way which I will be using later. So it's called chi-epsilon hyperbalicity, meaning that there is a decomposition, the splitting of the tangent bundle at every point x on the set lambda, which we call regular set, which we call hyperbolic set. And there is a measurable function c and q, which are absolute temperature. And that's the property which is written here is exactly the property of being temperature. And then we say that, and then you have this uniform contraction and expansion, I'm sorry, not non-uniform contraction and expansion, the standard contraction along the stable subspaces and when you go backward along unstable and similarly for when you go backward for stable and backward for unstable. So and then in the essence of this non-uniform hyperbolic condition is that I will define this for given set lambda, given number lambda, I define the set lambda l, which is what I call regular set. So on this set, the constants c, x and k, x are bounded from above and respectively from below, which means that the conditions, the uniform hyperbolicity estimates becomes uniform on that set. That of course has a lot of correlates in particular on the hyperbolic set, you can construct a stable, local stable and unstable curves. On a regular set, they become of uniform size and they depend continuously on the point, et cetera, et cetera. There are plenty of good results when you see it on a regular set. The only problem with distinguished non-uniform hyperbolicity for uniform is that regular sets are not invariant. However, they are nested. So when l grows, we get bigger set and they exhaust the whole set lambda. So that's basically the major properties of the regular sets. The next step is pre-card. So this is the third condition. So we have set lambda, we have local, as I said, we have local stable and unstable set, curves whose sets are uniform on the regular set. And then I pick a subset in lambda l. I say that this subset is recurrent if points in this set returns backward and forward iterates to the set A every point, which means that every point returns infinitely many times. And when I say iterates again, I take iterates from multiples of t. And which is a major property is the back strongly currents. That means that you first of all, it's recurrent. And second of all, there is a point x in A and a stable leaf through this point, such that which has positive one dimension on the back measure. So what happened is that you have a point x, you draw the unstable local leaf and there are a set of positive measures of one dimension on the back measure, such that every point on the set belongs to the set A. So the intersection of A with this is a set of positive measures. Why we just take unstable leaf is that you can draw stable leaves. You use absolute continuity to obtain a set of positive measures. And if point A returns to A, then the whole stable leaf returns very close to this area. And that's exactly what we need to use for our construction. This will be kind of an important part of our construction. Okay, so that's the three properties under which we require for our result. I would say a few words about SRB measures, because I want to define it in a way maybe a little bit different from what you might have learned before, which is that we take a set A and we call it fat if the union of stable leaves through the points in A have positive area. And this can be done in general, it can be volume. And then we say that mu is an SRB measure. If it is hyperbolic and for every set in the set X, oops, I didn't define the set X, this is X. Or you can say, okay, you can say that the set X, which is fat has a full measure. So in other words, my X is A. So there is a fat set X, which has full measure. Now, that definition is adjusted to the two dimensional case. Now, it works fine in the two dimensional case. In the multi dimensional case, there are some differences. You need to do some work to show that this is equivalent to an SRB measure. But in the two dimensional case, it's essentially the same definition as the classical definition of SRB measure, which is that if you take conditional measures generated by mu then one unstable leaves that they are absolutely continuous to leave volume, which in our cases one dimensional, the back measure or just the legs. So this is the definition of SRB measure, which is a little easier to use in our construction of SRB measure technically. There's no principle difference, of course. I would like to say that nice domain are, of course, an important part of our approval. But it sounds like artificial condition. But an argument is that it is actually not. And the idea is that if you are given by the theorem, that if you have a C1 plus alpha surface diffeomorphism and mu is an any invariant hyperbolic measure, non-atomic, then given number R, you can construct a nice domain of size R. And in fact, given a number L, you can construct a set A of positive measure, which is recurrent. And this is true for any hyperbolic measure. But if my measure is SRB, then it is strong, the back strongly recurrent. So what it means that the conditions that I stated in the theorem and the main theorem are necessary conditions for existence of SRB measures in the two dimensional case, of course. So nice, this is necessary to construct SRB measures and the back strongly recurrent is also necessary to construct SRB measures. And again, I want to stress that in this statement, I deal with what is called hyperbolic SRB measures. There is an ocean of SRB measures, which uses this absolute continuity property of local stable manifolds, which is not necessarily hyperbolic. So it allows zero exponent outside of the unstable leaves. But in my cases, all SRB measures are hyperbolic. The measures I'm going to build will be hyperbolic SRB measures. So that's important to stress. I would like to say some few words about construction, some history and some interesting problems related to SRB measures. First of all, as you all know, the SRB measures were constructed for uniformly hyperbolic situation by scenario and bonds in the 1970s. And then, so for the uniformly hyperbolic case, the works of scenario and the bond essentially answered any question, almost any question I would say about SRB measures. Certainly existence, uniqueness, and robotic properties up to Bernoulli. Bernoulli was established by by Bob. However, in the techniques, the use was partitions of finite subject of finite time. That was a great achievement of the time. It brought not just existence, it was much more than existence of SRB measure, because it brought the whole idea of symbolic representations of the systems, which opened up a lot of possibilities. Then the next step was constructing the SRB measures for partially hyperbolic system. But I'm going to skip this part. It's a huge area with a lot of work in this area and a lot of interesting results, ideas, et cetera. But I'm not going to talk about this, because I will not be mentioning partially hyperbolic system for a simple reason. In two-dimensional case, there are no partially hyperbolic different morphisms. So from uniform hyperbolic, I will jump immediately to non-uniform hyperbolic case. And here, there is a conjecture which was posed by Viana at his ICM address in 1998, which it was at the very end of his address, where he says that if you have a C1 plus alpha diffeomorphism with non-zero level of exponent with respect to volume, so volume is not necessarily an invariant measure. You don't have an invariant measure, you want to build it. And so you just say, I have a set of non-zero level of exponent positive volume, or you can even assume that it's full volume. Then this system admits an SRB measure. That's an interesting statement. And I will comment on this statement a little bit. First of all, I would like to say that the requirements that differ morphisms of the plus C1 plus alpha is important, because it kind of hints that we want to use non-uniform hyperbolicity. We talk about non-zero level of exponent in this conjecture, but I would like to say that if you just have non-zero level of exponent, that fact does not provide you immediately, at least, with any techniques of non-uniform hyperbolicity. The point is that you need to know more. You need to have, I mean, one way to explain this is to say that we need what is called Lyapunov-Pyrrhon regular points. The points for which accelerated theorem, multiplicative algorithm theorem holds. Without this, it doesn't look like you can use the power of non-uniform hyperbolicity. At least I don't know anything about how to use non-zero Lyapunov exponent without having sufficiently many regular points, oscillators or Lyapunov-Pyrrhon regular points. Here's what I wanted to say about the conjecture. Whether it's true or not, it stimulated a great body of work on constructing SRB measures for various systems. I'm going to mention two of them. One is my paper with von Kliemann-Haver and D. Madolpa. In which we constructed SRB measures, but we use a new notion of hyperbolicity called effective hyperbolicity. What that actually is, it's a version of non-uniform hyperbolicity, which is adjusted to study dissipative systems. Namely, you have an attractor and in the neighborhood of the attractor, you assume that there is a positive volume of points which are effectively hyperbolic. Again, the effective hyperbolicity is somewhat similar to non-uniform hyperbolicity, but it's not the same. It's a new notion of hyperbolicity. The conclusion is that if you have positive volume of points which are effectively hyperbolic, you have an SRB measure. The second result, which is much closer to this talk, is a result of Snir Benavadi, who constructed SRB measures under conditions which are essentially the ones I just mentioned, this condition of hyperbolicity, fatness, and strong recurrence. They stated maybe a little bit different way, but essentially they are the same. So, what is great about this work is that it works in any dimension. The result by Benavadi does not use niceness property. What it uses instead is a countable mark of partitions for systems which are non-uniform hyperbolic. The construction of this countable mark of partitions goes back to early work of Omri Sarif, who did it in the two-dimensional case, and then Snir Benavadi extended it to any dimensions. And the beauty of having countable mark of partitions is that it allows you again to obtain a symbolic representation of the system using mark of shift on a countable set of states. This is a great technique and it had many other important results coming from it, but one of them is the construction of SRB measures. Now, the difference between the result by Snir and our result is that we do not use mark of partition. Instead, we use a very geometric approach, which would lead us to a different symbolic representation, which is a representation by young towers, and that's what I'm going to explain. And at this point of time, young towers have certain advantage in the sense of implications to non-uniformly hyperbolic case than the mark of shift on a countable set of states. For example, in terms of study correlation decays and central limit theorems and stuff like that. I believe that the two representations are somehow equivalent and whatever you can get with one, you can get with another, but it's not the state of the art at this point. So, let me now also mention that as I said, I do not see how Vianne conjecture can be proven in general without attributing to one or another type of non-uniform hyperbolicity. There is also another point, which in my view may prevent Vianne conjecture to be true as stated, is the notion of Lebesque strong recurrence. In our paper and in the work of Snir, the Lebesque strong recurrence is explicitly required. So you have a set of points, which return to itself in a certain way. Particularly, they will have a return with positive frequency, which is nowhere to find in the statement of Vianne conjecture. In my paper with von and Dima, it's not explicitly required, but effective hyperbolicity is stated in a way, and that's where it kind of differs from non-uniform hyperbolicity, is that it essentially produces recurrence as a local stable lift starting with certain size, may go down, but the size can go down, but it eventually must recover. That is a correlating of effective hyperbolicity. So, you do have recurrence, in fact, strong recurrence in the case of effective hyperbolicity. So, any of the three results which build SRB measures in non-uniform hyperbolic case do require strong recurrence property. And I would like to mention a result by my former, my student Alansari, which says that, in fact, if you have a set of regular points of Lyapunov-Peron regular points or oscillations of regular points, points comes from oscillations theory of full measure, and if you have a subset which is back strongly recurrent, then it means that the measure is actually a smooth measure. So, you don't get genuine SRB, you get that the volume itself or a measure equivalent to the volume is invariant. So, you cannot get much if you have these two properties, except the just invariant volume. So, that actually shows the intrigue kind of way the conjecture was stated. It's extremely interesting, very stimulating, but nobody knows whether it's true or not, and there is no counter-example, which would be nice to have one. I believe that one exists, but I don't know anything. So, let me from now on move to the proof of the main theory. And again, I remind you that what we have is a non-uniformly hyperbolic set, a nice domain, regular set, subset A, which is strongly recurrent, and what you want to do is to prove that the map F admits an SRB measure. So, how are we going to do about that? First of all, there are a few steps that I'm going to make, essentially four steps. The first one is the notion of almost recurrent. That's a new notion, new object that had not been before in non-uniform hyperbolic STC. Before I state it, I need the notion of a full-length stable and full-length unstable curves. So, what it is, is that you have a nice domain gamma PQ, you have a point X here, and so this point you draw a local unstable leaf, local stable leaf. What you want to say is that if the size of the domain is small, and that's where I use a small R, so I fix L, which is the level of regular set of order L, the sizes of local stable and unstable manifolds here are fixed, and if R is small, that this local leaf is long. So, what I do, I just cut it so that its end points lie on that unstable boundary of nice domain. And similar is for the unstable. If you have a point Y and you draw unstable leaf, which is a full length, so it goes from one point on the stable boundary to another point on the stable boundary. And then there is a notion of almost return. So, you take a point X in A, this is my point, and it returns to a nice domain, so it's under the time I, I call it FIX. What happens is that if it returns to A, then this is what I would call true return, that's the best situation. It is back to nice domain, but it may not be returning to A, so it will be somewhere in this domain. An almost domain means that it's close enough to the set A in the sense that if you take the image of the stable local manifold, that, oops, I'm sorry, this is where it is, this is an image. And then there is a point Y in A such that you draw a full length unstable terminated intersect. So, if you have this situation, then you say my return is almost return. You are very close to actual true return. So, what I'd like to, you want to understand that when you take a point X in A and you return to the nice domain, there are three possibilities. One is a true return, second one is an almost return. And this sort, when the point comes somewhere and the stable leaf is like this, and it doesn't intersect any of the unstable leaves through the points in A. That's, you can say that this image of the stable leaf falls into a gap. So, true returns and almost returns are not gaps. Otherwise, it's a gap. So, that's the notion of an almost return. And once you have an almost return, you can kind of start playing the game. Namely, before I do this, I want to say that local stable and unstable manifold exists only through the point on a regular set. We need it, we need to extend this regular set. For that, we don't know how to do that because we don't have stable and unstable leaf outside of regular set. So, but what we can do is we can deal with admissible manifolds. Admissible manifolds, as I believe most of you should know, everyone knows, is almost local leaves. So, what you have, you have a stable leaf through good point, regular point, and there is a cone which we build in the tension space that you can build at every regular point. And the cone depends continuously on the point. So, we can draw curves which, if you leave them by exponential map, they will lie in a cone. So, they're close to the genuine unstable and or respectively unstable curve. But they're not stable and unstable. So, they are stable or unstable if you move them by finite time. Their behavior is pretty much close to the actual local stable manifolds. But if the time is infinite, they do not survive in that sense. So, so, however, I'm going to use them so I will extend the collection of local stable and unstable manifold to allow local admissible curves. And I'm going to use them to construct stable and respectively unstable strip. So, that is an actual two-dimensional domain which is bounded by pieces of stable and unstable curves through periodic points P and Q and some local admissible stable or unstable manifolds. If I have those domains, I call them stable and unstable strip. And here's maybe the main property that maybe the main notion that the notion of hyperbolic branch. So, what it is, you have a stable strip here. And it maps into an unstable strip. So, you have stable strip and you have an un... So, let me look at this picture. So, you have a stable strip and unstable strip. So, what I'm saying is that you take a stable strip and you apply some power of the map. And it goes, the stable strip shrinks in the stable direction, expand in the unstable direction. And if the time is chosen appropriately, the image will be this unstable strip. It may, may not happen. You don't know if such a time exists or not. But what I'm saying is that if it happens, then I call the pair a hyperbolic branch. And the time is an intrinsic part of this hyperbolic branch. So, it's two strips and the time that makes a hyperbolic branch. And here is the main result, which says that if you take a point x and it returns to a, almost returns to a under time, which, and again, I stress it, the time I consider a multiple of the capital T, the capital T is needed to use nice property. So, if it returns back to a, almost returns to a at some time i, which is multiple of t, then there exists a hyperbolic branch, which is absolutely non-trivial result. And it's a major part of our construction. I would like to make a few results. First of all, once you have a point in a non-uniformly hyperbolic set, there is a so-called regular neighborhood around this point. It has a property that's another weight of describing regular sets is that in the small neighborhood around the point x, whose size depends on x, the map fi is uniformly hyperbolic from an x to the image, to the regular neighborhood at the point fi x. So, that's an important property of regular neighborhood. Of course, when x runs through the regular set, the size of regular neighborhood is fixed. And this is very important. So, we have two numbers now, c and lambda. c stands for the rate of contraction when you go from one regular neighborhood to another. And c is a constant in the hyperbolicity estimates. And if x lives on this regular set, then those lambda and c are constant. They do not depend on the point. Now, the second comment I want to make, this regular neighborhood are heavily used in the construction of hyperbolic branches. Namely, the hyperbolic branch are first constructed in regular neighborhood, not in the nice rectangle, but in the object which is bigger. Regular neighborhood contains our nice domain. They're much bigger. So, we use this regular neighborhood to construct hyperbolic branches within this regular neighborhood. So, what we'd like to do is to simply restrict them to a nice domain. If that's exactly where the nice property is necessary. For example, if you have a regular neighborhood, this is a regular neighborhood which contains our nice domain, and you have a hyperbolic branch and you want to restrict it, so if you start from x, you don't have the picture like that, because it has to contain the starting point which is inside nice domain. So, the picture that you would like to have is something like that, and then it maps into something like that. However, what can happen is that you start from this point and the stable strip is nice, but unstable strip is something like that. It contains a boundary. I'm sorry, the picture may be not so nice. I will show it a little bit better. So, this is a nice domain. You take a point, you have a stable strip, and the unstable strip is like this, so that part of the boundary is inside. That you don't want, because you cannot then restrict on the nice domain. And the nicest property guarantees that it doesn't happen. So, then you can restrict, and that's how you get a branch property, hyperbolic branch property. That is the only place where the niceness property of the domain is used. So, somehow one can avoid that, then you can do it in the multi-dimensional case. We don't know. It's pretty much the dimensional argument, the second part of the proof, but the first part of the proof is not, and it's a new reason. Now, another property of hyperbolic branches is what we call concatenation property, which means that you have one hyperbolic branch, F, I, C1, S, C1, U, and the second one, which has time J. And then, so you move from this one to this one, and from this one to this one. So, what you would like to do is you would like to move from, apply time I plus J, so you go from this dark rectangle in through this one to the dark rectangle here. So, the corresponding parts, the corresponding strips are given by these two formulas. What this new strip that you obtain is called the concatenation of the original branches. Now, if you just look at what, I mean, if your two branches are C-lander branches, so you have constant C and lambda, then a priori, the branch which you obtain as a concatenation branch, would have constant C squared lambda. So, the constant will grow as you concatenate branches. If you do it multiple, if finally many times, you get a huge constant in front. It turns out that you can construct hyperbolic branches, C-lander hyperbolic branches, such that the result in concatenation branch is C-lander, no matter how many, but finally many concatenation branches you will use. So, that's another one important result in constructing hyperbolic branches. So, the first limit claims that you can construct one, the second claim that you can construct in a way that it keeps those branches to be C-lander hyperbolic branch. So, here I also would like to say that, yeah, I already mentioned that, so I will skip that. So, the next step is the notion of a saturated rectangle. So, remember what we have. We have a set subset A which has, this is regular, lies inside of regular set, has the back strong recurrent property, etc., etc. What we would like to do is we would like to extend it to build a rectangle. A rectangle is a set which has hyperbolic local product structure. It means that you take a point X in gamma, then it is an intersection of local stable and local unstable curves through some two points of in gamma. So, every point in gamma is an intersection of local stable and unstable curves through points in gamma. So, that's the notion of rectangle. The difference between this rectangle and kind of a well-known rectangle on the plane is at first of all, the boundaries are kind of curves. But more important is that it's, as I said, it's maybe like a counter like set. It doesn't feel any domain. So, what we want to do, we want to construct a rectangle which contains A, contains in the domain, nice domain, and it is saturated. Now, what it is, the notion of saturated saturation is another new important element of the proof. What it means that, I mean, there are several meanings of this property, and it will be used in several ways. But first of all, it is a maximal invariant set for the dynamics within this nice domain, which is generated, being maximum, maximal in a sense that it's generated by hyperbolic branches associated to almost rich ops. Now, what it means, it means that you, it means that I mean, I draw a pair, an analogy with the standard horse. And the standard horse, you have two rectangles, which are basic rectangles, which is what the construction starts with. And then the construction goes by building the maximal invariant set, which consists of points which remains in these two strips, in forward and backward times, infinite forward and backward times. So the difference between the, so the construction of saturated rectangle, the idea of saturation is pretty similar to that one. But the key difference is that in the course you say, you can have at most finitely many branches, which are pairwise dejoints of stable and unstable strips. In our case, every point can belong to infinitely many hyperbolic branches. So in course setting, there's one branch, one of the two strips to which it belongs, in kind of an original trips, when you start the whole construction. In our settings, this is, you may have infinitely many such branches, when you start the construction. So in a sense, the construction of horses is also construction by hyperbolic branches. The difference is that those hyperbolic branches are associated with true returns. We do not consider any almost returns. And as I said, you may have infinitely many branches which would contain a given point. So that makes a huge difference in our construction, but the idea is exactly to construct the maximal invariant subset. So to do that, what I would like to do is I consider all possible hyperbolic branches, which comes out of almost return. So you start this A, you consider all possible almost return to each you construct a hyperbolic branch, and you get a collection of hyperbolic branches. Now, since F is a diffeomorphism of compact manifold, its derivative is bounded, which means that for a given time I, you can have only finitely many such branches, which means that the collection of all hyperbolic branches that I just described can be indexed by two indices, i and j, where i is a return time for almost return, and j goes from one to some finite number and j. Okay. And so once we choose this collection, this collection of indexes, you have the collection of the corresponding branches, which are the branches which you obtain by using almost returns. So this is kind of a symbolic way using hyperbolic branches to describe the dynamics of points which belong to this hyperbolic set. So the saturated rectangle consists of all points X for which there is a sequence of hyperbolic branches, again hyperbolic branches which are obtained from almost return, such that the orbit falls into stable strip under iteration of the corresponding return times. So that's, again, that's pretty much similar to construction of horseshoes, but of course it's much more elaborate, because we are in a completely different setting. So this way we are constructed saturated rectangle, and it has the following important properties, that you take a point X in A and consider, and consider it's almost return, then it's actual true return to gamma, because points in A, A bill is a part of gamma, so you take a point in A, which belongs to gamma, then almost return of this point is actually a true return to gamma. Second, because A is Libert strong vertical, gamma is also Libert strong vertical. And third is that if you have a point X in gamma, then you can construct two sequences, one going forward, another one going backward, such that if you take images of X under the finitely many iterations by returns associated with hyperbolic branches forward, you fall into the corresponding stable strip, and similarly for when you go backward. So these two sequences, one goes forward and one goes backward, can be viewed as a symbolic representation built by hyperbolic branches. So this is a different type of symbolic representation, it's not mark of shifts, it is not young towers, it's a different type of symbolic representation in this case. And we use it to carry out our proof. So the next step is young towers. So once you have a saturated rectangle, what we do, we just induce on this saturated rectangle. So we build young tower, and it will be the first t return time young tower. So the time will be first return, but it must be multiple of t. Such that the base of the tower or inducing domain is my saturated rectangle gamma, and the high time is the first return to gamma. That gives you a tower. Now of course, it's not young tower yet. I mean, it's easy to see that since the base is a rectangle, it has hyperbolic product structure, which is a requirement in construction of young tower. But you have to verify other properties, which is a mark of property of returns, property of bounded distortion, and the most important one, integrability of the return time. That's the three properties that you need to check. But in order to do that, first you have to define the partition element of the base. So the young tower, as you know, is constructed when you have the base, it partitioned into what is called S sets, and then above S sets you have returns. So that's kind of an abstract picture of the young tower, and I'm pretty much sure everybody knows. So this will be the height. Under this time, the piece returns back to the base of the tower. So how do we define this partition? The elements which will have the same inducing type. Well, you just take two hyperbolic branches. So what do you know? You take a point X, and you have many hyperbolic branches which contain X. So you take two of them. One is a time i, and another one is a time i prime, and it has stable and unstable strips, Cs hat, C u hat. Hat stands for the strips being inside of the nice domain, and C prime, s and u. Now, what one can prove that they are either disjoint or nested. So the picture is that you have a rectangle, you have a nice, you have a hyperbolic branch, and then the new branch, the second one will be, and you have a point X. So it must be, if both of them contain some point X, they say there is a point X in this one and in the stable one, then they must be nested. So one of them will be a subset in another one, or otherwise they are completely disjoint in, there are no point in the stable which will belong to the stable of another branch. So that means that you can choose a sub-collection i prime, so you can order all the branches by this nasty property. It's a partial order because not every two branches are nested, which means that you have partial order in the set of indices i, a, which we can denote the set i prime, a, which consists of those, which is a maximal set with respect to this partial order. So you get these those sets. So in other words, once you have a point X, you choose the maximal branch, the branch for which the stable strip is the biggest possible, and if you do that, they will be disjoint and that gives you the assets, which partition the base of the tower. So now you have all elements of the young tower. You have a base which is, which has direct product structure. You have assets and respect to your assets, which assets you obtain by taking images of assets under the corresponding time, because they form hyperbolic branch. And all you need to do is to verify market property and verify bounded distortion and verify integrability. So that's the three properties you need to to verify. Well, the the market property is immediate from immediately followed from hyperbolic branch property. That's that's almost immediate. The verification of the bounded distortion required some estimates and it follows from the fact that within the hyperbolic branch property, when you consider the map FI that moves CS into CU, this map is uniformly hyperbolic. And that is the reason why we can prove the bounded distortion property when you start from this assets and go back to to the base of the tower. The property that needs to be, the third property is the integrability of the induced and that's where we use the fact that we have a Lebesque strong recurrence. And that's the only place where a Lebesque strong recurrence is used. So I told you where the niceness is used, here where the Lebesque strong recurrence is used. The basic idea of how to prove it was actually goes back to a result by Pinheiro. He actually introduced us to this idea. That's why they're very simple to him. The idea is that you take the induced time, you take a point x and you look at the induced time at x. You have a function, measurable function on your base and you apply and consider a big of averages of this function. The big of averages of you want to show that big of averages converge to something finite because the system, the map is eroded that it will converge to the integral of the induced function. So you use just big of eroded theorem to conclude the integral is finite. The idea is that you can tie the big of averages induced time induced time function to Lebesque strong recurrence in a way that Lebesque strong recurrence gives you a positive frequency of return. And you can connect the big of a big of a going which average to the positivity to the frequency of return to the tower. And Lebesque strong recurrence property gives you positivity of the return, which means that the big of averages are finite. So the connection is that the big of averages, the limit of big of averages is equal to the inverse of the frequency of return. So if the frequency of return is zero, then the big of average converts to infinity. If the frequency of return is positive, the big of averages converts to a finite map. So that's basically the relation between integrability of the return of the induced function and Lebesque strong recurrence. So that completes the proof of the theory, because the final result of the distance of SRB measures immediately follow from the famous result of the lesson Young, which says that if you have a young tower, then you have an SRB. And that's the end of my talk. Thank you very much. Thank you, Yasha. So let us all thank Yasha. Do we have questions or comments for Yasha? Pablo, Pablo Carrasco raised his hand, his virtual hand. So please, Pablo, we can't hear you. What about now? Good. Yeah. All right. Hi. I have two questions, which are essentially the same. Probably the first is easier. Have you guys thought if this type of technology of hyperbolic bands or hyperbolic slices will work with for local and local diffeomorphisms or surfaces? It's very well, maybe. I mean, the matter is that what I described to you is kind of very, very overview of the proof. There are a lot of technical details behind the scene. And if you go to local diffeomorphism, then there may be some some things. I mean, on the first glance, I would say, yes, it may work. But one has to check all the technical details. There are too many of them to allow me to say definitely yes. Okay. That's a very interesting question, Pablo, actually. I think that's a very interesting question. Yeah. I at least have not thought about it. Yeah. Nice. Nice. Please go ahead. No, I think Pablo, we can't hear you well. Sorry. I'm thinking because if you have a local endomorphism, you can leave to the inverse limit, and it's going to look locally, but foliated by two-dimensional leaves, and you're going to have a separating properties. So you can define two-dimensional rectangles. I don't know if anything is known for SRBs, for endomorphisms, for example, I think that's not much. Sorry, Pablo, are you talking about endomorphism, which are non-uniformity? Yes, in the tutorials. Non-uniformly expanding or also with contacting that? Expanding and contracting. And I know if something can be said for this type of map, in particular, the existence of XRBs or something like this, but it's just curiosity. What about partially hyperbolic two-dimensional center? Do you think that you can reduce this technology to prove the existence of SRBs for that type of map? I don't know. I mean, this is argument, it's so far two-dimensional, and it's essential. So I don't know if it can be extended to higher dimension, even assuming that it's a map of partially hyperbolic, that's hard to say. Someone else wants to ask questions or make comments? Sorry, just a comment. Maybe it's relevant. I think I saw a paper by Pavel Gora and co-authors on SRB measures for example of endomorphism, which is essentially what you described. I mean, I think the idea is that, I mean, it's for just the family of examples. It's not a general result, but maybe it's helpful. Yeah, I'm sorry, I'm not aware of this paper. It's called the singular SRB measures for a non-one-to-one map of a unit square. Oh, I see. Okay, thank you. I would have to take a look at it, but I can't comment. Someone else? I'm asking because I actually have a bunch of questions. Okay, so let me ask my questions. One nice thing is that you mentioned these three kind of ingredients that are used in it, which are, as far as I understand, hyperbolicity and some sort of regularity like temperedness and recurrence. And these three things were mentioned explicitly in the mini-course as being main ingredients either to construct Markov partitions or to construct young towers. And actually, your topic is a topic that has been successfully treated in both methods. So you presented the young towers construction that uses young towers, and there is also this work of SNE that you mentioned. And on the direction of SNE, I remember that he proves that the SRB measure exists under recurrence and under strong recurrence, then you get finiteness. Yes. So my question is, if instead of only asking Lebesgue strong recurrence, you asked for Lebesgue recurrence, can you build a sort of infinite SRB measure? I believe so. We have not done that. We just didn't think about that, but probably yes. And because, as I said, the strong recurrence properties used only in the very end when you prove that the induced time of the tower is integrable. And if it's not integrable, then you get an infinite SRB. So it should be true. Well, Yasha, to be honest, I would be more affirmative than that. It's definitely true because we have a first return young tower. And so there's no problems. It definitely becomes a sigma finite invariant measure in every case. And we only use the strong recurrence for the integrabilities. So as Yasha said, we didn't think about stating that explicitly. I just didn't want to be absolutely certain without having things written. Yes, I agree. My final question regards lemma too. I got curious to know how you are able to, instead of having c square, you can still remain with the same c. Oh, yeah. That's a good question. And I mean, it's a technical fact, but the basic idea is that, I mean, how would you do it in the uniform hyperbolic case? You adapt the metric such that c is one. Okay. And then, of course, one square is one and you get what you need. Roughly speaking, that's the same idea here. But technically, it's more complicated. But that's all I can say at the moment. But yes, the idea is rough. Okay. Yeah, maybe, I mean, maybe if I, if you don't mind, if I just had something, you know, I mean, for people who are a little bit familiar with the Lyapunov charts, then you know that in the Lyapunov metric, you are uniformly hyperbolic. And actually, the constant comes when you transfer from the Lyapunov metric to the regular metric on the manifold, right? So you just avoid doing that, right? So you stay, you don't need to go up and then down each time. You just stay up on the Lyapunov level until you want to, until you, until you get to where you want to go, basically. Yeah, yeah. That's a good, yeah, yeah, sure. Okay. So do we have any other question or comment for Yasha? Okay, so if not, let's thank Yasha again for his wonderful talk. Thank you guys.