 So, the title of my talk is limit theorems for interval exchange transformations, okay? So, I think I don't need to remind what the interval exchange transformation is, but I want to recall what is a limit theorem in this context. So, if you have a dynamical system in this setting, say you have the map on an interval, and this interval is endowed with the Lebesgue measure, which is invariant in this setting, then you can take any function and consider, so now you look at this as a random variable, so x is chosen randomly with respect to this mu, and so you are interested in this distribution. And since we have the transformation, this is natural to consider the averages with respect to this map. So, we take Sn of pi, x, which is, for convenience, I will not divide by n here, just to make other formulas looks nicer. And so we are interested in the distribution of the normalized averages. So, we take a, which is average, divided by a square root of this variation. So, we are interested in what is the distribution of what is the limit of this distribution. For the, in the situation when your system is chaotic, say hyperbolic, you have more or less the same central limit theorem as for Bernoulli as, just for Bernoulli process, but here we have no such simple asymptotics, there are no limit for the distribution, but we have more delicate setting, and I will describe what we have. So, the base for this case is a case which of the continuous system, which is a special flow with this map and the base, which is the translation flow. So, now we have a flat surface, and think about it as a bunch of rectangles, and so we have the interval exchange map that maps this interval, so it's interchanged this intervals by some way, say, here it's one, two, three, and you have a point goes up with the unit speed, and then when it reaches the top of this interval, it goes to the bottom with respect to this interval exchange map. So, go here, then this point maps out there, then goes back, goes up again, and so on. So, this is a translation flow on this flat surface. Indeed, there is a method how to stitch together the side parts of this rectangle. So, now we have defined how to glue top boundary to the bottom boundary, but what about the sides? So, they can be glued to some nice surface with some singularities, but this is not the topic of my talk. Note that on the same surface, you have another flow, which is a horizontal. So, you just go to the right with the unit speed. Then, this system obviously also has the invariant measure, which is the back measure. And we will assume further than the total area of the surface is one. We can consider the same averages for this situation. So, we take S pi, which is an integral, the Wodic integral. And again, we normalize it like that, and we're interested in asymptotics. The theorem by Buffett of states that there is a limit along some subsequences. So, Buffett of theorem works not for all translation but for generic ones. I will discuss this a bit later. For generic translation flow, the distribution of this, all right, 5T, which is a normalized. This guy has some limit for some subsequence. And the reason for this is that, so how to describe this limit? The reason is that when you take the surface, you have a Tehmuller flow, which is, so we basically shrink the surface in one direction, stretches in another direction. And the other thing you can do with the surface is that you can change your section, your base interval. There is a culture of the induction, which can make your interval shorter or longer. And so you can think about this, about the set of all translation flows factorized by this result induction. And on this factor, on this model of spaces here, Tehmuller flows, flow X. And so you can, now if you apply Tehmuller flow to the surface, your ergodic integral change the time, which is this T. So this T is multiplied or divided by E to the S. So you can think about this, not as you make longer and longer interval, but as you take just interval of length one, so you have T equals to one, but you change the surface. And therefore, you can look now what is about, now how to describe which sequences can be taken here. You just take those subsequences, such that, well, the logarithm of your initial surface, flow. And so it takes such time, such that this has some limit. And then along this subsequence, you have the asymptotic, so you have Tk equals to the E to the SM. So when your surface tends to some fixed surface, you have this asymptotics in distribution. Note also that indeed there is another technical detail is that you should say that phi is not an arbitrary function, but for phi outside of some subspace of co-dimension one. This is what we've had of theorem states. Now we take, so and the limit of this, limit of this, so the limit of this distributions can be also quite explicitly defined. You take, so-called, finally additive functionals, which is essentially goes like this. So you have the Razi induction, so we have the following set of matrices. So the heights of the intervals, so h are vectors of the length. Equal to the number of the rectangles. So the height of the rectangles on the next step are some linear combinations called the heights on the previous step. And so obviously this set of equations, so here n belongs to z, have an n dimensional set of solutions. And among the solutions is one, which is where all vectors are positive, there are heights of these rectangles, but indeed you can also take other solutions. They're not positive, so your vectors have some negative components, but nevertheless if you take any such solution, so we take a set of vectors satisfying this, you can measure any, you can describe another length, so another functional to any vertical segment. Take, say this segment, it has a length which is a zero, so it is first component of h zero, if this is the partition defined by h zero, which is number to zero. And if you have another segment, you have, you can do like this, so first put some elements of the partition number zero or maybe even less, then have some elements of the first partitions and of the second and so on. And what you need is to have some, that this set, this series is summable, that is converges to some numbers, so you can define your measure for any segment. And this can be done if you have this n, tens as n goes to minus infinity to zero exponentially, with then you have that this series convergent, everything is okay. And indeed, what is the basis for this result by Buffetter is that when you look at this integral, that this integral behaves essentially like this, like this, find the editor functional. So I'll write this as phi plus and here is a segment from x. Now that this, this is also written as false, it's just a definition. So this is just a length in this strange sense of this vector plus some plus this error term, which is large of t to the epsilon. And so when you normalize, you have that this is the most important part and indeed when you divide by this variation, first you subtract the average, then you divide by the variation and so indeed you should consider the most powerful part of these asymptotics. And indeed, when you look at this exponential, about this, all the subspace of all solutions of this equation, which satisfy this condition. Then in this setting, we have a sequence of subspaces, a flag of subspaces, and so the first subspace is, which is just a length. The second one is the next, an asymptotic sense so on. So, and so on. So indeed what we have in this setting is more or less this guy, because this one is killed by subtracting average and the next guys have this is the lower order. So this distribution is in fact, the distribution of this phi two plus of x t or indeed of x one for this. So we can say that the limit is the distribution and so what is my result? I try to do almost the same things but for interval exchange transformations. What are the main troubles here? So there are two troubles. First of all, there are no, so here you have tecmoiler flow which works quite nicely. So if you take tecmoiler flow, your average by the time t goes to the average by the time t multiplied by e to the s. So it just stretches or shrinks your time uniformly. And if you perform resin induction in this situation, you have not the average by same time, time million. But for some random time because the resin induction yields you not the iterate by the first return map. And so instead of this, you have more complicated thing which is sums with various number of elements. And another thing is that you have no, well, so the resin induction is non invertible so it needs some more care with respect to that case. So my theorem is that again for the same holes. So if you take phi and k, fusion tends to some limit which is indeed exactly of phi two plus. So they have the, so the set of all these distributions is exactly the same as here. And the only difference is that the first need to describe what and k is. So here we have just the technical flow and at that moment this technical flow moves your surface close to this m. And here the situation is a bit more complicated because there are no initial surface. So here is the setting. And one more thing I want to discuss before going to prove is that what the word generic in this works states. Indeed, this generic means of the set of measure one by some measure. And this measure should also have generosity for some properties. For example, so indeed there are two properties for this measure that should, first of all the uniquely ergodic translation for it should be generic. And the second property is that when you look at this equation, you can think about it as about a cycle for this system. And so you have to require that this cycle satisfies the conditions of a slated theorem. That integral of the logarithm of its norm of its inverse should be integrable with respect to this measure. And if this is okay then for almost all translation flows for almost all m with respect to this measure mu. Okay, this mu is not that new. Point we have what is written there. So let's say that the set of all this, so let omega or say omega zero be the set of all such translation flows such that this holds. Then indeed what is written here is that generic I t is a set of all translation flows such that there exists vector h such that t with this rule function with this. So when you consider the special flow like the drawn there, then this belongs to omega zero. This is a new set of omega zero hat and I say that indeed for belongs to this omega zero. And the proof goes, so now I define it in all details. So the proof is goes as follows. So first of all, as it was written here, you have to fix any hate such that this flow is such that for this translation flow you can apply it with that of theorem. And if you apply this to the theorem, you can see that when you define new functions as by the following equation. So you take your function which is take any point, then you define your measure, your function along all this segment to be equal to the value of pi of pi on its bottom point divided by the height of this integral of this rectangle. So if you do this like that, you can see that this average integral from zero to some time. Just an ergodic integral, but with the not defined time to end. And this time can be defined by the following property which is exactly the same formula, but for function pi which is identically equal one. Then you know that this time does not depend on function. It is indeed the time you need. So if you take any point, you just take this point and then you integrate or until you go up and large rectangle. So this time is doesn't depend on phi. So here we have this. And then you can decompose both these formulas by this bouffet of approximation lemma by this composition. And again, the first term is given. The second one is the one in question. And the remaining is negligible. And here you have, you need to perform the same trick as when you say prove the differentiability of the implicit function. You here you have some equation and you have to prove that indeed that this is n plus something of the order of phi two plus of xn. And this can be done by several steps of estimation, better and better. And finally you get that this tow of an axis and minus something multiplied by this phi two of xn. And then you substitute it here and then you have that here is two terms. And again, you have that the term, the first term is just average of phi multiplied by the time n. And the second term is phi two. So we have the same asymptotics. But this is not the end of the game because indeed what you prove if you're just looking at this approximations is that the distribution of, well, of this phi nk tends to the distribution of this phi two plus of one x where x is distributed, say x hat where x is uniformly on I, on the interval. So you have uniformly distribution on the bottom line of this interval. But fortunately, when you can first, you can average by some fixed time, say up. So you have not this integral but the integral this plus some constant. And this constant should be taken uniformly random on some large interval. In this setting, you have almost by this first step you have almost uniform distribution of your starting point on this flat surface. And so you just, and on the terms of estimate this first part of the integral is negligible. It's just a constant. And so combining these two ways of estimation you obtain this result. This has a limit. Thank you.