 I want to thank the organizers first for inviting me. It was a long time, first time came here in 91. Actually, I just spoke with a gentleman who's helping us, and he's working here from 1990. So always I was coming, he was there. I mean, it was extremely helpful all the time. And it was always wonderful to be here. It was many exciting results and talks were announced in this room. So it's really nice. The two characters which I'll discuss will be renormalization and rigidity, and they're related. And since here was 10 lectures, I think, of Marko Martens who introduced renormalization in some details, it will help me, but I also plan to define most of object I'll need. So it is related to what Marko Martens discussed, and it is also related to what Stefano Marbi discussed. And I would remind you that Stefano discussed interval exchange maps, and he started with the simplest exchange of two intervals, which is a circle rotation. And so the usual picture here is that if you consider unit circle, and which is represented by this zero one interval, and then the map is given by a graph, and it consists of two pieces. It just, the first piece is a linear piece with slope one, it started point row, and then starting from point one minus row, is another piece, which is exactly matching this point. And this represents a rotation. So this map T row is just a map which takes x to the x plus row, model of one. It's a rotation by angle row. And most of interesting things from a gothic and dynamic point of view corresponds to irrational row, but rationales are also important. And then next step what Marbi was discussing, and he discussed what is called generalized interval exchange. She said, why should we consider this linear map for slope one? Why don't we consider non-linear maps instead, which are monotone, but follow the same pattern? So we just replace this by something like this, and like something like that, and just two independent pieces of monotone maps will be still matching positions of end points, this point and this point, this point and that, but otherwise it's two basically arbitrary monotone functions, smooth ones. So two branches, I would like them to be smooth, and right smoothness for me is something like c to plus alpha because I would need some distortion arguments, so lowest smoothness is possible, but it's always tricky. That's a good setting for the thing. So both pieces of smooth can be more smooth than that, of course, the smoother the better. But I'm not putting any condition of a type of, Stefano Marbi putting that, derivatives are matching and derivative are equal to one, it's just two pieces, right? And what I'm getting here is a map with breaks. It's a circle map, homomorphism circle, but it has two break points with breaks. So one of the breaks, it's points where first derivative of a map has jumped discontinuity. So obviously at this point, derivative from this side and from that side are different, not necessarily the same, and at this point derivative on this side and that side is different, not necessarily the same. And actually these two break points, so the point zero and point one minus r and point one minus row is a trajectory of the same, line of the same trajectory. So if we take point one minus row and map it by the map T, which is now non-linear map, they will get to point zero, the image will be point zero. And so they belong to the same trajectory and actually these two break points, the two break points of this picture, two breaks is equivalent, and I ask you to believe me, you can conjugate it to a system which have just one break point. It's equivalent to one break point. And so we come to a class of maps which I would discuss first and for which we have complete understanding of why now it took 25 years kind of to achieve this and it was many papers with medical collaborators and I would list them, Elena Wul and Dima Khmelev, Alexei Teplinsky and Sasha Korshich. And so I'll formulate results later on and I would say who's results are there. But this is maps with one break point. So that's a class of maps I want to look at. So it's a maps, map T belongs to C2 plus alpha outside of unique point, point of break. And the point of break derivative at this point from the left is not equal to the, is not equal to derivative from the right. Minus and plus will be here. And both of them are positive. So there is no critical point. And if you think about what are the, so you have singularity of this break singularity at one point and the parameter which characterizes the singularity, I'm sure Marko discussed critical circle maps or critical maps, whatever. So there is a order of critical point which characterizes the singularity. So here parameter which characterizes singularity is the ratio of these two numbers. So if we consider the ratio of these two numbers, this is a parameter and this is a positive number. So there is no harm to take square root of it. It will be convenient for me. This C is a parameter which characterizes the strength of the break. And the aim of the rigidity theory is to show that if you have two maps with the same type of singularity, so if you have T1 and T2 and they have the same, both with one break and the size of the break is the same, so they have the same type of singularity and they have the same combinatorial structure, meaning that rotation number is the same and irrational. Sometimes, but most of the time you have to put some different time condition plus different time condition, condition. So the equivalent topologically, and the local structure of singularity is the same, then you want to conclude that the equivalent in a very strong sense, much stronger than topological, you kind of upgrade this fact that the topological equivalent to smooth equivalence. So you want to conclude that this information implies that after change of variables, giving by phi, T1 and T2 are conjugated by phi, where phi is either C1 or preferably C1 plus epsilon or maybe C2 or maybe C3, but at least C1 is where rigidity starts. That means that metric properties are the same. So that's the aim, that's rigidity, to upgrade topological equivalence, which is given by D'Andrea theorem to higher level. And the other tool here to achieve such things is renormalization. And the scheme of renormalization for circle maps is the following, it's always scheme of renormalization is the same. So you have kind of system of intervals, a system of sets, subsets, which are getting smaller and smaller. And instead of original dynamical system, given by T1 or T2, you can see the first return maps on this increasingly smaller domains. And so you can see the high iterates of your map, but you can see it in the coordinate system where this decreasing intervals become of size one. And then your hope is that if you do this procedure, then of course you'll forget details of this T1 and T2, but you will reveal universal features of the thing. So what you'll get will not depend on where to start it from, but you'll see some universal feature. That's the idea of renormalization method. So in first circle maps, the intervals which people normally consider are the following. So here is the circle. You take any point, we call it mark point X naught, it's at some point on a circle, in its circle. And then you form trajectory of this point by the map T. So this is T, the homomorphism. I can define it on a level of homo, but it will be smooth later on. And we'll think that rotation number is irrational. So a rational rotation number would have continued fraction expansion, give you partial quotients, and then I will also have approximations, given by truncated continued fractions. So Qn denominators of continued fractions of this Pn over Qn will be important here. So basically intervals which we want to consider will be the following. X Qn minus one, it's a trajectory of this point. So from X i, which is equal to T i X naught. And when i is equal to Qn minus one, you get this point, we should come close to X naught. And then you'll have X Qn, size is not very important here, X Qn on the other side. And this will be sequences, next will be Qn plus one, Qn plus two, and they will be oscillating coming closer and closer to X naught. But so I stop at this level, I can see the interval i n, which is interval between X Qn minus one and X Qn, and I can see the first return map in this interval. And so I define this first return map and this first return map will have two pieces, one corresponding, if I start here, that I will have to iterate. So if I start in this one, I would have to iterate my map Qn times to come back inside. And so if I consider T Qn or on this piece, I will have to iterate Qn minus one time because after Qn minus one, I come inside, that's point X Qn plus Qn minus one. And I would rescale also coordinate system. So how you rescale it, I would think that this is kind of unit size. You can do it in different ways, but I'll just take a fine change of variables so that the coordinate of this point zero, let me put it in another color, this is zero, z coordinate of this point. And say at this point, that coordinate is minus one. This fixes the size, after that my coordinate system giving by z. So call this coordinate z instead of x. It's a renormalized coordinates, it depends on n, but I don't indicate this here. And then I can define these two maps, which will be Fn on z, which is equal to An on z, going from z to x, then applying T to the Qn, and then going back to z coordinate. So that's a first renormalized map and there is another renormalized map which is defined here, Qn z, which is the same coordinate system, An and minus one, but iterated Qn minus one, rather than Qn. So this pair of maps, Fnz and Gnz is really first return map in the renormalized coordinate system. And that is a sequence of renormations which we want to study in the limit when n goes to infinity and look what's going on with the things. So that's how renormations are defined. Notice that this is z as a coordinate, the name of the coordinate here, and if I fix zero minus one, I cannot fix this one, it will be some number. I will denote it An, so this is renormalized coordinate of this point, is An, and it's basically a ratio of this interval to that interval, to the length of this interval, that renormalized coordinate of this point is something which I cannot also control at this point, it's denoted minus bn because it's negative. Now also notice that I draw picture this way, but actually which one will be left and which will be right, and then whether n is even or odd, so it switches orientation, but it's not essential for me. So you want to study the thing for different maps. And I will be studying it later on for the map with parameter c fixed, but I will start, we can just start discussion with c equal to one. So c equal to one is a case when there is no break, there's the same numbers. So it's diffeomorphism case, and this was well studied and that is a realm of linearization theory discussed by Marmy. So this is Hermann theory, and many people contributed here. So basically result is that if map is smooth enough and c2 plus alpha is enough for that, and rho is what typical number, full measure set of numbers, actually defantite properties which should depend on alpha, then you can smoothly conjugate two maps. However, in this case, these two maps, there is a distinguisher presented for diffeomorphism, it's just linear diffeomorphism with slope one, so then you conjugate giving t to this linear one and that linearization theory. And this can be done. Now what does it mean? If you start with linear map with slope one, so if you start with this one giving by this slope one, of course slope will not change, you iterate how many times you iterate, it will be still slope one. So basically these two things for linear map will be linear function of z. And for linear map, you can easily calculate that it will be z plus an, and this one will be z. So this one is taking zero to minus one, so it's z minus one. So that's what you'll get if you do this integration scheme for linear map with slope one. And what you want to show is that if you do it for any diffeomorphism, any not close to linear, but any one with relation number one, this thing will converge to that one. It will look the same. Now an you can calculate here, it's a mathematical quantity because it's a linear case and an will be given by truncated continuous fraction starting with k and plus one and so on. So it's a number which is well defined to depend on your continuous fraction. So what you want really to show the derivatives of this maps are close to one. And that would indicate everywhere that derivative is close to one, then it's a linear function. So the aim in this theory is to show that derivative of fn prime is close to one when n goes to infinity. And the derivative of this map of course doesn't change because it's a fine, it's the same as derivative of tqn. So you want to say that tqn, if you differentiate it at every point in a circle, then it's close to one. If you manage to do this, you're done. I mean, then you get certain rigidity result and randomization converges and you get certain rigidity result. You have to work on it. You have to use the other condition, but basically it's well defined. Well, that's everything. Now, it's possible to prove it's not very hard, but what I want to say is that imagine that you do the same scheme but you use two smooth pieces but they have break point. Now, of course, when you move, in case of break, so if you can take any point x naught, for break, you take break point as a starting point of your procedure and then when you move this interval many, many times until it come back, it never touches this point. So you never really encounter this break thing. And this interval, it goes, it never encounter the thing. It never cover this point until this time. So you kind of iterate two smooth pieces all the time of small intervals. And then there is a kind of very conceptual effect that when you iterate too many times smooth functions on small intervals, then the resulting map is going to be not linear. It's going to be fracture linear. So there is a very conceptual effect which is easy to prove that F, N and G actually converges to the space of fracture linear maps rather than linear maps. Now, linear maps is particular class of fracture linear maps but very particular. Now, and in case of the second difference of break, you really will have fracture linear map which are never linear. And this thing is governed by the following actually effect. There is a way to understand if you have several break points, for example, when you can have linear things and when you will not have linear, when you will have fracture linear thing. The effect is very simple. You have to take integral of the second derivative of your map divided by first derivative of your map which is positive and integrate it over circle. And if this integral is zero, then you will have linear case or piecewise linear case if you have many breaks. But if the integral is non-zero, then you will have fracture linear. And that's very easy to understand. And maybe I'll not at this point discuss it but it's a very simple fact. And so what it gives to you that immediately just on this conceptual level, you reduce your problem from infinite dimensional one because T can be anything. It's a functional space of maps of a circle to actually finite dimensional one with very few parameters characterizing it. This is a linear map and this is fracture linear map. So this one characterized by three parameters. This characterized by less parameters. So actually if you calculate the parameters correctly and you think about it a little bit, it turns out that in fracture linear case, these parameters A, N and B, N completely characterize the maps which you'll encounter. So the whole thing will depend just on two parameters, A, N and B, N. So what it says that starting with infinite dimensional space you converge to a plane depending on two parameters and renormalization will be effectively renormalization and renormalization transformations, transformation which take FN, GN to FN plus one, GN plus one will be effectively two-dimensional transformation. And then you have to study it. I'll be more specific about this fracture linear map in a little bit later but at this point I want to formulate first theorem. And this theorem is a final theorem for the maps with one break and it was published last year. It's a theorem which we proved with Sasha Koshich and the theorem says the following. That renormalization converge. If you start convention from this one and that one they converge. Now converge is a bad word. Then they converge to something. They approach each other but people say converges are renormalization anyway. And they converge with exponential rate and this exponential rate is universal rate. It doesn't depend on T1 and T2. It depends only on number C and nothing else. Of course it also depends on smoothness but if I fix alpha in class of smoothness then what theorem says that they exist. Lambda depending on C which is less than one such that for any two T1, T2 with the same size of break. It's not less, it's not equal to one. And with the same irrational rotation number. And here you don't need to put a different condition. For convergence relation you don't need it. Only for rigidity you need it. Then fn1z minus fn2z say in C two topology on minus one zero that's where they define. The z goes from minus one to zero. Norm is less or equal than constant which depend on T1 and T2 times lambda to the end. So they exponentially converge to each other with rate lambda C which is less than one. And that's a final result on the level of renormalization theory. So they, any two renormalization approach each other. And again I want to emphasize that it's the fact which we see in all renormalization schemes that to get convergence relation you don't need to put different conditions. Different conditions need only later to deduce from this that there is a rigidity. And there is a rigidity result which follows from this. And this is theorem which says I formulated the following way. There exists a set of full Lebesgue measure of rotation numbers such that if, say set A, if rho belongs to A then T1 and T2 are C1 conjugate. So which means that this T2 is equal to phi. How did I put it? Phi minus one and phi. I think I put it phi minus one, phi minus one, T1 phi. Where phi is C1 on a second. Now you can say much more about this set of rotation numbers. Actually it's more than full Lebesgue measure. So what we allow is that integers and contained fraction expression can grow even exponentially fast within. So basically it's a little bit funny situation and I'll say about this few words. There is even odd phenomena here and for even positions it depends on whether C is greater than one or less than one but either on even odd positions you don't need to put any restrictions okay no different condition but the other positions you have to put conditions and these conditions you can allow exponential growth. And so they have the same rotation number all. Thank you. And so that is a result of Sasha Koshich and Elio Matzea and T2 will be published soon. Now Sasha Koshich has also results showing that this C1 cannot be much improved in general. So for many rotation numbers you can improve it. Say for bounded type it can be improved to C1 plus something and that was known for some time. Sasha proved that you cannot prove this result for C1 plus epsilon for any epsilon. So basically the theorem of Sasha and the paper was as far as I understand was just accepted to communications in the following one that so they exist a set called A1 of rotation numbers of rotation numbers of full measure. So there is a set of full measure of rotation numbers such that for any row from this set and for any epsilon positive they exist T1, T2 with the same rotation number being equal to row for which conjugacy is not C1 plus or epsilon for which phi does not belong to C1 plus epsilon. So you cannot really extend it to larger class but of course for bounded type you can do it. Yeah there are contra examples where it's not C1, it's not full measure set. So for better rotation numbers we can construct examples where it's not C1 and actually we are proving this out but there is still certain regularity and the result is that conjugacy would not be arbitrary bad. I mean for diffeomorphism conjugacy can be arbitrary bad if rotation number is real bad. Here it will not be C1, it will not be Lipschitz we can construct examples but it still will be herder with exponent close to one. So for any epsilon positive it will be herder one minus epsilon herder. But I'm not going into it, it's a little bit of delicate. Say it again. No it's not even C1, it's not even Lipschitz. And no, so it really can be singular with conjugacy. Okay, it's a good question because for maps with a critical point you actually can prove that for all rational rotation numbers C1 conjugacy holds. But for this singularity you don't have this robust rigidity, you have some traces of this robust rigidity but not here. Anyway, so let me explain what's going on here and why this theorem was so difficult to prove and it took so long time and what is the main point here? Say it again. No, I cannot have, we can see them without critical points. Yeah, it's without critical points, that's essential. No critical points, I should put it here. So C prime is strictly positive. All right, meaning that at the break point it's also both of them are positive. All right, so there are two phenomena here. One is convergence to the space of fractional maps which I said depend on two parameters and I want to show that the map is concrete. I would just write some formula, don't be scared. I mean, they're not very scary formula but there's concrete formula. So let me draw it here and because of this plus minus say changing of rotation symmetry you actually, what the picture will draw will have two parts. So I would introduce a class of fractional maps which is invariant class for transmission transformation and then I'll apply, describe in this temporary. So V F A V C, C is a parameter here on Z. That is A plus C Z divided by one minus V Z. And then G, this is the same F and G but they're capital now, you'll see connection in a moment. That will be equal to A Z minus C divided by AC plus Z times one plus V minus C. Now parameters here are not A and B as I described here geometrical parameters A and B and parameters are A and V and connection between V and B is the following. So A is the same as A and N, the same meaning and V has the following meaning, it's C minus A and minus B and divided by B and N. It's more convenient to use this coordinate rather than B coordinate but otherwise it's the same thing. All right, so this is invariant set for renormalization. So renormalization acts on those maps and the resulting map will be the same class. So how does, you do the following. You take images of point minus one, so zero, if you plug zero here, so no, I mean, zero goes to E. This map, zero goes to minus one. So what you do, you take point minus one and start to iterate it by this map. So it will be F on minus one and then next one maybe, second iterate of minus one until you cross from one side to another side of zero. You start at minus one and then you do monotonically and here say case iterate of minus one is negative and case plus one iterate of minus one is positive. At this moment you stop and then you calculate new F and G. So if you start it with F and G, so the new one will be the following. So new F will be case iterate of F and iterate it with G. So you start it, maybe I'll start with G and then iterate F K times and then new G will be just old F. But you also have to change coordinate system because your coordinate system will be based on this point being minus one and now that point will be minus one. So actually you have to calculate it to your, the new coordinate system will be zero. We'll stay there and you'll have minus a n. So if you take a new Z, I would have to multiply by a n and this is minus one over a n and this is minus one over a n, a n minus a n Z. So that's how renunciation acts on this family F and G and of course effectively it just acts on parameters a and v. So you just calculate new parameters a and v. So you start it with a v which is denoted here a n v n and then after you apply renunciation transformation you get new parameter a prime v prime or a tilde v tilde which actually corresponds to a n plus one v n plus one. So this is two dimensional map but of course it's a lot of discontinuities because the number of times you iterate depend on what you started with on your rotation number. Nevertheless, you can study this dynamical system to the measure dynamic and it habits very strong hyperbolic properties. So that's the key to study this renunciation. Hyperbolicity of this information. So what you can prove and this is some work and this basically was done in our paper with tip Linsky. You can prove the following picture. I delete Hermann path that so the play of parameter a v this is C less than one and this is C greater than one. There is certain domain. This is point C minus one. This is C minus one over two. This is C and there is certain domain where all dynamics happen. So this is a tractor. Everything end up in this domain and there is similar domain here. C minus one will be negative. C will be less than one. There is certain domain here and so you'll have curves corresponding to smooth curves corresponding to a rational rotation numbers. So a and parameters a and v corresponding to maps with the rational rotation numbers and they will be stable manifolds for renunciation. So if points belong to the thing there is universal convergence, uniform convergence of them with exponential rate. So if they lie on the same stable manifold so there is also unstable manifolds along which you go away. So there is a whole true structure with uniform expansion and contraction rate here and here. So what happened point from here maps here and then goes back. So it's kind of period two thing. So there is a structure of stable and stable manifolds and actually there's a counter set here and here invariant and universal counter set which depend only on parameter C and points underization converges to corresponding points of a counter set. So it's as good as you want to prove. I mean there is a horseshoe, there is hyperbolic horseshoe, uniformly hyperbolic horseshoe and that's what's going on here. And you can prove that everything is nice on this picture. There is also as I said, conceptual exponential converges to this picture starting from any map is a break. Now you want to combine these two together and that's very hard. Because here you have convergence only when rotation number is the same but when you converge to this plane it's very difficult to control rotation number. And that's why the results which were available before this paper with Scottish were only for a strictly class of rotation numbers for which there is no difficulty which I'll describe now which is called unbounded geometry. So, what I discussed now is very different from the case of difomorphism and from the case of critical circle map. You can see this phenomenon only here. So this just describes why deophantine conditions are needed in this case in case of break singularity and other things. All right, so what means deophantine condition? It means that certain condition of a growth of this integer numbers can that they cannot grow too much. When they grow too fast, you get some problem. So what kind of problem you'll see? And problem is the following that if you look at the map of fn and fn is defined between minus one and zero this map it doesn't cross diagonal it doesn't have fixed point otherwise it's not a map it's some monotone function here. And say typically say for something like this for example and then what we do we iterate point minus one until we cross zero and if this graph is very close to the thing then you will iterate many many times until you go and cross zero. And even if you have control of how good is this function here on step n if the number iterates a huge you're losing the control or you may lose this control. Now it turns out that behavior here is also well controlled because it's always parabolic singularities almost touching almost tangency here you can control it. So this is well-defined thing it's not a problem I wanted to discuss. This is about it but pretty much bounded geometry anyway. The difficulty in break case is that sometimes graph doesn't look like this. Again this is even odd situation and for even or odd cases depending on whether C is greater than one or less than one you will not face this one you will face this one in half of the cases even or odd. But the other one you will see the following situation that your graph would look the following way. It will be function which comes very close to the diagonal not in the middle but at the end point so that trajectory of the break point is very close to be periodic and what will happen that the graph will have such a form. So it's not zero here but derivative here is close to C this number C I assume that C is greater than one and here to one over C so overall product is equal to one it's kind of similar to this derivative to one but it is something with derivative greater than one and this is less and you have huge number of iterates but these iterates when you start to iterate say you go from here you get a lot of very short intervals so the intervals here they start to accumulate they're very short and there are many of them and the size of them will be over what of E or say one over C in power KN plus one over two so KN plus one is the number of iterates we will see in the next step half of them here half of them there and these intervals are getting very small and those intervals are also getting very small because one over C is here so you get very small intervals accumulating here and there and then boundary they are really very very small and you have to deal with this situation now the difficulty is that if K is much larger than N say N squared can be anything then this is much smaller than exponential but all control is all exponentially good so we cannot get anything better than exponential and convergence to over generations but here we are facing numbers which are much smaller than that and you have to control this and apparently it can be done and that is done in a paper with Sasha Koshchich so that you can deal with the difficulty and prove that still the integration will converge even in this situation even with this very unbounded geometry all right now let me just use last five minutes to describe important feature which I didn't discuss yet it's important feature which allows you to prove hyperbolicity of this picture now it's kind of complicated system how you can in principle prove hyperbolicity so it's two dimensional map you want to prove that there is one stable one unstable direction now it turns out that this map this relation map which takes A, V and transform it for giving C and transform it to A tilde, V tilde and actually to new C which will be one over C it's kind of switching from C to one over C this map have a symmetry and we can... I will write formula for the symmetry it's set on a formula level you can see the symmetry and it should have geometrical interpretation I always say it has geometrical interpretation and not know this geometrical interpretation so we see the symmetry it's time reversal symmetry but we don't know really what is behind it so the symmetry is the following it's concrete involution which take A and V into C minus one minus V A, V minus V over C and it turns out that inverse relation map which you can define is conjugate by this involution with a direct relation map so this is time reversal symmetry which indication of certain usually indication of certain symplectic structure Hamiltonian structure in two dimensional case it's kind of exact that if you have the symmetry then you will have symplectic structure but in multi-dimensional case it's not exactly one to one anyway if you have the symmetry what it gives to you if you know that there is unstable direction this gives you that there exists a stable direction so if you know that for any combinatorics row there is unstable direction then it shows that it also has stable direction for any combinatorics row just by symmetry because stable because of the symmetry now and where instability comes from instability comes from rotation number because there is always instability in rotation number you have to fix rotation number in order to see stable manifold but along the direction of increased rotation number there is unstable direction and that's relatively easy to prove so if you prove that then you can prove hyperbolicity of this picture it's a very good conceptual explanation however in the paper we prove it the other way we prove that there is stability and then we get instability the other way which is not the ideal way but it's possible to do also anyway the symmetry is very important part of the game and now I want to say that the direction which we are moving now is the direction very closely related to what gain that is Cosmarby but it's again another setting not linearizable interval exchanges but interval exchanges which are essentially fractional linear and we are staying on a level of maps of a circle so it corresponds to interval exchanges with a genus 1 as discussed by Marmin so that means that you are having circle maps circle maps are important because in this case you have Dandrua theory so you know when maps are topologically conjugate to each other but what will happen in this case we want to consider many break points find that number of break points so you have two maps T1 and T2 and both have K break points so let me call it Y Y1, YK break points for T1 and say Z1, ZK I'm sorry for that it's not coordinate break points for T2 and they have certain sizes of breaks C1, CK and those also have C1, CK so we say that two maps are break equivalent if they exist topological conjugacy, Homo Phi which conjugate them that's not big deal because the rational rotation number conjugated with the same rational rotation number but we want this conjugacy to put break point into break point so this conjugacy is such that topological conjugacy such that that I goes to YI by this conjugacy so they conjugated point with the corresponding breaks so on the break level on singularity level they are the same if there is a conjugacy the rotation number is the same and so the conjecture yet it's still conjecture it's still work in progress but we have certain progress in understanding the thing is that if you have the thing then again there is rigidity that you can upgrade this conjugacy to be C1 conjugacy and I put it as a conjecture under some different conditions conditions Phi is C1 but second part of conjecture that without any different condition the renunciation will converge the renunciation constructed for one and for another converge exponentially fast and then you can ask what are the renunciations and we are having renunciations scheme here which is different from Razi induction which was discussed by Marmi it's another induction scheme for this kind of interval exchange information it is not interval exchange information but what we call interval exchange scheme so our first return map is defined on small intervals which surround the break points Y1, Yk so each of the break points is surrounded by small intervals and these small intervals are actually changing by the maps one interval exchange where they are connected they are disconnected but they are completely going one to another and there are a finite number of combinatorial schemes of these things and what is important that we can see similar symmetry acting in this renunciation so the coordinate system is important because then you can see symmetry and this symmetry we can see and this is much less trivial than here one combinatorics basically or two C goes to one over C here there are many combinatorics and symmetry is taking renunciation map with one combinatorics connected with another combinatorics but still is symmetry so still if you can prove that there are K and stable direction you get stable direction for free and that's the line of research which we are pursuing right now and I think it's very promising anyway there is an interesting topic of extending of interval exchanges in Russia induction and Zorich Konsevich co-cycles from realm of maps with slope one which is interval exchanges to realm of either piecewise linear maps with different slopes or fractional linear maps and these two worlds exist and they have nice hyperbolic properties and nice top here thank you