 Well, thank you so much. First I'd like to thank the organizers of this conference, limitation and for the opportunity to speak at such a wonderful place. Thank you so much. So my talk is devoted to a certain symbolic dynamical system, which is called verticotomophism. It can be regarded as a symbolic analogue for many dynamical systems of parabolic type. For example, for interlock exchange maps and translation flows, which we'll discuss a little bit later. But let me start with the definitions. So I consider a graph gamma with two m vertices that I arranged in two levels. There are m vertices on the top level and m vertices at the bottom. Here it is. We, well, all edges go from the top level to the bottom one. And we assume that for each vertex from the top level there exists an outcoming edge, as well as for the vertex from the bottom level that exists, an incoming one. And we also permit the multiple edges. So this graph gamma can be represented by the incidence matrix. That is, we enumerate the edges at the top and in the bottom. And we just count the number of edges that start from i. And let me now consider a sequence of such graphs, gamma m. I put graph gamma m plus 1 over the graph gamma m. And consider the graded graph, which is the union of these graphs gamma m. OK, and I consider the set of all paths in this big graded graph. The set I'll call the Mac of Compactum. So each point in the Mac of Compactum is just the sequence of edges. So on this set of paths we have an equivalent relation. We say that two paths are equivalent. If there exists a level that these two paths are the same starting this level and above. And let us denote gamma infinity plus the equivalence class of a path x. So it is something like that. So these equivalence classes form the vertical variation, the name vertical. I won't explain why it's called so. But this is just a foliage defined in this way. In a similar way, we can define the horizontal affiliation, which is if we identify the paths that are the same below. So it is very nice structure, but I want some dynamics in it. So to provide some dynamics, I first order the paths. I provide an ordering on the set of all the paths. So I start with the ordering on each vertex v. Well, I start with an ordering for the sets of all edges that go out from every vertex v. And thus I obtain a linear ordering on the set of all paths in this tells set. Well, now I want to provide some dynamics on this. I consider a Markov compactum that corresponds to the graph, which is infinite from one side, which is the unit of graphs gamma 1, gamma 2, and so on. Well, since I have the ordering, I define the natural map, which switches the path x to the path tx, because it's successor with respect to this ordering on the set of edges. So this map is called the Veshka automorphism. Veshka automorphisms are the symbolic model for, for instance, for internal exchange maps. So let me consider a surface and a flow in it. I have already partitioned my surface into the union of the rectangles, like that. Now I want to, well, and I consider the directional flow. The first term map of this flow is the internal exchange map that permeates the intervals here. So I want to identify the paths in this big graph with the points on the interval. So there is an initial map on these surfaces, which is called the Vich induction. Informally, it cuts off the part of the rightmost rectangle and puts it somewhere above. So how do we give the correspondence between the paths and the points on the interval? We say that two paths are equivalent to this relation. If, after the iterations of this Vich induction, we fall into the same rectangle. For example, points t and tx are equivalent. Well, each leaf in this graph corresponds to the orbit of y under the, and it is x. Well, OK, of y under the internal exchange map. Clear? So here's the picture of how to describe, how to build one level of this graph after applying one step of the Vich induction. But, naturally, we want to encode the orbits of the flow on the surface. We want to have some suspension structure on our Markov compactum. For this purpose, we consider the infinite Markov compactum. That is, the Markov compactum that corresponds to the graph that is infinite from both sides. In this case, the telecommunications classes are not discrete. So we do not have any successors. We do not have photocatomaphism. But we can define a flow on the fallation formed by these sets. Very informally, you switch the arrows very far below, and then on high and high levels. And then, and so this is how you obtain the flow. Well, there is another example of it. You switch from the number 0.23 to 0.24. When you continuously write, you can think of the levels of this graph as of the digits. And switching the arrows is just switching the digits. While going to the number 0.24. But more formally, such a symbolic flow is defined as follows. We move along the leaves of these fallations with the unit speed. How to define the unit speed? One has to provide some sort of a back measure. Well, more geometrically, this leaf corresponds to the orbit of the flow in this surface. So how to define the positive measure given a sequence of incidentous matrices of these graphs? We shall assume that we've seen for a while, but then we'll show that for a typical MacCuff-Compactum this assumption holds that there exists a sequence of vectors with positive coordinates such that they have such an equivalence property. And the first property and the second property can be described as follows, as follows that for small arcs of the flow, the measure is exponentially decaying. So how given these sequence of vectors, we can define a measure. So for any vertex, I consider a canonically chosen path that goes through this vertex. And I set the measure on the set of all paths coincide with this path that I am. Since this level m, or more geometrically, you define measure on the set of this arc that are the sides of the rectangles of this MacCuffian partition. So this is how we define the measure. And indeed, if the number of edges grows sub-exponentially as n goes to infinity, we can extend this measure to any arc of the flow. Well, thus we obtain the flow, which can be viewed as a suspension flow over the virtual cartomorphism. Namely, we restrict ourselves onto the sub-graph, which is obtained by deleting all the negative sub-graphs. And actually, the first system map is just switching the arrows, which is indeed the virtual cartomorphism. OK. Well, now we want to obtain the answers for the following questions. Whether the flow ht is a gothic? With respect to the measure, we should preserve. Preserve the product of the libc measures on the stable, and not stable, but vertical and horizontal foliage. And there are two other questions. Precise asymptotics, doing all the precise asymptotics of the gothic integral. And last question, when this integral is uniformly bounded in x and g? To give an answer to this question, I provide, following the books of Alexander Boffatov, the construction of finite-layered measures. You might have heard of it if you attended to Professor Klamenko's lectures. So these are the same measures as the libc measure that we defined before, but we refuse from the property of the positivity of these measures. Actually, these measures are the same. They are given by sequence of, doesn't work, of these equivalence matrices. So by the definition, finite-layered measures are the subspace in Rm, where m is the number of vertices at each level, just by the definition that we identified these measures with such coordinates. So but we can say something more. To say something more, I want to obtain a sort of the renormalization map. Namely, it is just the typological mark of shift. How does it x on the mark of compactum x? It just pulls it down for one level. So the shift of the mark of compactum, the n sub-graph of the shifted mark of compactum is the n plus 1s of the previous. Well, so this map admits a sort of a cycle that is just the product of these insidious matrices. But these sub-graphs, gamma 1, gamma 2, gamma 3, and so on. So let us provide some additional assumptions on these products of matrices, namely, I assume that there exists an ergodic shift of merit probability measure on the set of all mark of compactum, such that with the positive probability that exists, a sub-graph, gamma 0, with insidious matrix that has all the positive coordinates, together with the properties of invertibility and the integrability of this co-cycle, we realize that we can apply the oscillators theorem to this co-cycle. Namely, we can decompose the space rm where the product of matrices acts into the sum of the stable and the unstable subspace, where the growth of vectors in ECS, the stable subspace, is at most sub-exponential. And the growth of the expanding vectors is given precisely. So with us, we can identify the set of finelative measures with the unstable subspace of this co-cycle. This is why does it hold? Because we have said by definition that it decays exponentially for small arcs. Thus, we obtain such a notification. So I can consider the measures on the transferral co-coliation and this space is isomorphic, because this space B minus is isomorphic to the stable subspace of the transpose co-cycle. We are almost ready to write a statement about the precise and totics of this integral. Well, for this purpose, let me consider a functional, which is the product of the positive measure on the vertical variation and an arbitrary finelative measure on the horizontal variation. Second, well, these functionals are invariant under the action of the flow. And due to the pairing between the measures on two variations, we can obtain the following asymptotics of the ebotic integral for Lipschitz functions. So what is the first term in this decomposition? It's just the positive Lebesgue measure. So we get the ebotic theorem. So the answer to this question for a typical Markov compactor is positive. So it's ebotic. Moreover, we can give a precise and totic of this integral. OK, but what about the third question? Well, the third question, well, we want our integrals to be uniformly bounded. Of course, if all finelative measures are zero for our Markov compactor, well, on the contrary, if they are not zero, we cannot hope for this. OK, let us assume that all finelative measures are zero. This is equivalent to the statement that all these distributions are equal to zero. How to kill this term O t times epsilon? So we must smoothen somehow our function F. So we consider the space of continuous functions on x that have Lipschitz derivative along the horizontal relation. You'll ask me, is it possible that you can differentiate my function in this symbolic space? But I can say that we consider the derivative as just the limit. And this is precisely defined. The argument is that we differentiate over along the filiation. And along the filiation, the derivative can be defined. So given these two assumptions, we can state that these integrals are uniformly bounded. Well, due to Gorsuch's theorem, the previous result implies that if we consider such an equation, then it has a valid solution. So what does this ecological equation mean? You can ask yourself what suspension flows are equivalent to each other. Well, the suspension flows, when you answer this question, you should write this equation. So how much I have? Six minutes, yes? Well, I must go to the proof. Well, for the proof, I'll switch to the geometric language if you won't mind. So this theorem could be reformulated in terms of torsion flows, and finally, and interrelation maps. And before it was this type of theorem was proved for these dynamical systems by Mami Vesaiyikos and Forni. So what do we do? We draw an arc of the flow. We decompose it into the union of arcs that are the arcs of the rectangles of this partition, arcs of the type gamma n plus, whatever it means. And we can think of this arc as the sum of the union of two parts, the small part and the big part. For the small part, we get a geometric series, because the number of edges and thus the rectangles grows sub-exponentially, while the heights, or while these lengths decay exponentially. So we get a geometric series. The problem is to prove that the integral along this long Markovian arc is uniformly bounded. So the main argument is that we can approximate the integral over this long arc gamma by the finally additive measure on the arc gamma. And do it uniformly. So if we can do that, assuming that all the distributions are 0, we just arrived at the statement of the theorem. But why can we do that? I'll give just two claims for that. And then we will end. Everyone will be happy. OK. So the first claim is the following. Let us consider a long Markov rectangle. And I consider two closed arcs, gamma n and gamma n prime. It turns out that under our assumptions, the difference between two integrals along these arcs decays exponentially, since the n is the number of steps of reservoir induction. It decays exponentially, thus, in the similar way that we did for the definition of finally additive measures, I just, for every Markov rectangle in this series of partitions, I just choose one of the arcs and define a vector whose coordinate is just the integral along this arc. This sequence of vectors is not a covariant, but it is almost a covariant in a sense that the difference, the norm of this difference decays exponentially. And here the final claim comes that if we have such a sequence of vectors, then we can choose the one from the unstable subspace of the cycle, which is the product of matrices, such that we can approximate the sequence uniformly. So this is rather a technical demo. He is sort of a proof, but I think I don't have much time to discuss this. So thank you so much.