 good morning. So let's continue this renormalization of Hennon maps. So let's recall briefly what we discussed yesterday. So a Hennon map is something of the form where this f is some unimodal map and this epsilon is something very tiny. So that is the first coordinate and the second coordinate is our simple x. Let's make a picture of what it means. So we have the square, unit square, and so the image of the square is something like that. So it's because the bit of this strip is epsilon. It's a very tiny, it's a perturbation of some unimodal map. But what is going to be crucial, and we discussed this already yesterday, if you take a vertical big line of length one, then that will be strongly contracted and will be mapped horizontally. So there is a very strong contraction and the size of this is like the epsilon dy, which is much smaller than one. So there is very strong vertical contraction. And just a reminder, the Jacobian of the map is exactly this, the epsilon dy. The Jacobian is also going to play an important role. And so because of the shape of the map and because the Jacobian is very small, vertical lines are going to be strongly contracted and put horizontal. So these are our maps. And then we wanted to renormalize them and let's recall the definition of renormalization and of a renormalizable map. So the situation of a renormalizable map is something like this. There exists a little tilted rectangle with this horizontal top and bottom. And that will be mapped to something, a little bend. And then it will be mapped back inside, something like that. So this curve is going here and this curve is going here. And the left, so this curve, they become vertical. And then in one step they become horizontal. So now you see the trick. If you take the vertical foliation here, it will be mapped horizontal. So if you pull it back, you will get some foliation here. And these foliation, as we discussed yesterday, they are exactly the ones. If you take one of these leaves, they are exactly the ones which are going to be mapped by the second iterate to something horizontal. And so this fact that there is this foliation forced us to use a nonlinear rescaling to define the renormalization. So there is some map C1, which is a different morphism, and we will come back to the precise shape of that one. And we will have here, after rescaling, we will get our renormalization. And the essence of this diffeomorphism is that it straightens the foliation to become vertical. And so psi, which goes in this direction, straightens the foliation. And then the renormalization is defined as you do psi 1v, then you do the second iterate, and you come back. And so this is the renormalization operator on our nonmaps. And then yesterday renormalization becomes very useful if this operator is itself hyperbolic. And indeed it is hyperbolic. And let me summarize the statement from yesterday by your picture. So what's going to happen is there is the space of unimodal maps. Have you observed yesterday because of the Jacobian is very small, that the limits of this renormalization has to be unimodal maps. So in the space of unimodal maps, there exists, have you saw that before, there is the unimodal fixed point. Now we go, and also what we have is there is some stable manifold. This is all in the unimodal world. So code remains one stable manifold in the space of unimodal maps. And there exists some unstable manifold in the unimodal space. So and because of this superfast convergence to the unimodal maps, because of the Jacobian, this hyperbolic picture extends, and it is really not difficult to prove, extends into the space of unknown maps. So the stable manifold of this fixed point becomes the stable manifold of f star. It's just another notation is this map. It's the same map. And so this becomes the stable manifold. And the original unstable manifold, maybe I should draw it a little bit like this, is still the unstable manifold of the unknown fixed point, which is a unimodal map. And so we get exactly the same picture. And then if, okay, from this picture, a consequence will be, again, universal parameter universality. I remember we had this, this, this pair doubling cascade going up to the boundary of chaos. And from this hyperbolic picture, you will see again, that the bifurcation moments go to the boundary of chaos exactly with the rate 4.6, which is the unstable eigenvalue at the fixed point. And so again, this is a boring picture. We, we, we get exactly boring, but it explains why people in, who do physics and engineering, when they see a cascade, they see, they see the one-dimensional numerics. And so we have, we explain something. Oh, so, so this unimodal map is actually a quadratic like map. So we, we work with holomorphic unimodal maps. And deception is also something holomorphic. C0, C0. But then you get everything because everything is holomorphic. Yeah, so if you get C0 convergence in a holomorphic context, you get convergence in any topology. And so this is like perfect. It's perfect. Okay? So now let's, let's, let's go to the dynamics. Maybe let's, let's recall something. If you take the Hennon family, you remember, that was this one. And so, and that has a parameter space by A and by the Jacobian. So this two-dimensional parameter space will, will live somewhere here. So somewhere here, and you will see an intersection in the Hennon family with the stable manifold. And so somewhere here is, is some little curve, an analytic curve. And here you will have the maps, which are infinitely renormalizable. And at our infinite renormalizable maps, we are going to look at. But you should think about these are maps in some codemains and one subspace. So let's look at the dynamics of infinitely renormalizable maps. Not necessarily Hennon maps, but Hennon like maps. So let's, let's extend this, this picture again. And so what you see is, and so we have our map and we have the boxes. It's our exchange by the map. Then we have a picture of the renormalization, which is obtained by some non-affine rescaling. And then we also used the map C1C, which is just the rescaling, and then brings it to, to the other box. So F of psi of V. So this box we will call BV. And this box we will call BC. And then this process continues. So we look again, inside here, we will have again two boxes, which are exchanged by the renormalization. We, we renormalize, we rescale. This is our diffeomorphism. And we get the, the second renormalization. Okay. So and then again, you see a box here and a box here, which are exchanged. And then this process continues like that. And so, and now from this picture, this picture is sort of the key picture in the whole analysis. So it will come back all the time and everything is happening in this picture. And the analysis is really mainly going to sit in, in these maps. So but now let's, let's look at the dynamics. Let's go slow and let's not go too fast. And so, and so these two, two pieces, of course, we are going to introduce cycles. So the first one is, are, are these two pieces? But then in these two pieces, we will see copies of those guys. So those guys, they will land somewhere inside here. And then they are mapped somewhere here. So these four pieces will be the next cycle. But then you take, go to the next renormalization. You have two pieces here. They go inside here. They go inside here. Those guys, they go inside here. Those guys go inside here and inside here. And you see, as in the, in the one dimensional picture that you get the sequence of cycles. And so here there are two pieces. Here there are four pieces. And here there are eight pieces. And you see a doubles each time. And then, of course, you take the intersection. So C is the intersection of all those cycles. And the theorem is that this is an attractor. So if f is infinitely renormalizable, and then you can do this construction. And then Cf is a counter set. And for every n, there exists exactly one periodic orbit in the n-cycle of period 2 to the n. And if you take a point which is not, these, these periodic points, they are of saddle type. So they have stable manifolds and unstable manifolds. And if you take a point which is not, sorry, in the stable manifold of all those periodic orbits, then the limit set is exactly this, this counter set. And so our counter set, which, which is living somewhere here, is the attractor of the system. So this is actually sort of an old theorem by Gambeau-Dôme, von String and Trécerre. In a slightly different context, but essentially it's the same. So in, in, in the context we are discussing, this is work by the Cavallion, Nisalubics and myself. Okay, so let's, let's start to look more carefully at this counter set. And so what happens is, and this is again by André de Cavallion, Nisalubics and myself. And the statement is that from now on we will always be, everything will be about an infinite renormalizable map with a, with a tiny, tiny bit. So there exists exactly one invariant measure on, on this counter set. This, secondly, there is a conjugation from the, the counter set we are studying here to the counter set of the one dimension of X-poly. So topologically, and you, and you can see it in the picture, this counter set is just an adding machine, like in the one dimensional case. So the topology of the attract is completely clear at this moment. And then there are this measure that has what they call characteristic exponents. So one of them is zero and the other one is log of BF. So what it means is, I have to tell you what this log, what this BF is with this zero characteristic exponent corresponds to the directions. And in this counter set there will be certain directions. And you should think about them as sort of tangent spaces to the counter set. And in these tangent spaces to the counter set, the dynamics is not expanding at all, neither contracting. And so there is, there is a stable, stable direction which corresponds to the tangent space. The other direction corresponds to the contraction. So let me tell you about this BF. BF is the average Jacobian. And so BF is e to the power, the integral of the log of the Jacobian with respect to this measure. And I put a box about it, around it, because this number, it's just a Jacobian, just the average Jacobian, is going to play a major role in geometrical sense. This is an ergodic theoretical number, but this average Jacobian is going to play a major role in geometrical sense. This is already a little surprising. Ergodic theory related to measure, to geometry. But this is more surprising is that it is going to be, interact with the topology. So this ergodic theoretical number is actually going to be a topological invariant. So this, this is going to be a crucial number. Everything is, is sort of in this number, all the complications and you will see the complications or the surprises or are related to this average Jacobian. So this is a crucial number. And the first observation is that this number controls the contraction, the contracting directions. So now let me show you, and yesterday you saw already the, the stable directions. So let me, let me recall. So the characteristic exponent zero corresponds to the stable directions in, in our counter set. But it also, absolutely, absolutely. Oh, oh, sorry. Yeah. Yeah. A neutral direction. Sorry. Absolutely. Yeah. Absolutely. Absolutely. So neutral direction. That's okay. No, you're absolutely right. The direction in which sort of distances don't change. The other direction will correspond to the contraction direction. So you should think about that. So there is all the tangent lines. You should think about them as also as tangent lines to this counter set. And of course it takes some work to make sense out of that. But think about it like that. There is this counter set lying there and it has tangent lines. And in the case, for an example, when the B is zero, and then we just have a one dimensional system and the image is just the curve of, of, of, of our unimodal map. And this counter set is, is lying in this most curve. And so, and then if you take a point, then this neutral direction is really the tangent line to this most curve. And so this, this identification works perfectly in the degenerate, in the degenerate case. And so now we, we plot all these tangent lines and you see they are going to form the envelope of the unimodal part of the map. So, and now I would like to, so this is the picture. I can show you the picture for real. And so this is the real picture. So you see, you see sort of an envelope of the unimodal envelope of some parabola. So that is the picture when we are in the one dimensional case. So now let's go just a little up to point one. So we turn on the two dimensional aspect just a little bit and then you get, you get this. So you see this is just like in the top of the page you see point one. And so we just, have we just turned the B on a little bit. If you take hundreds you would get quite a similar picture. And so what happens is, apparently, if you look at this top in the thing in the, in the top there it's like a sort of a star. And that means that this counter set starts to twiggle a lot. So and, and this, so it becomes wild. And so it's not, it doesn't look at all that it is contained in some smooth curve. All these tens, tens of lines they start to form stars. And these stars are sort of all over the place. There's a star there and there are many stars. And so now if you go up a little bit more, I think to point four. So if you make everything a little bit more two dimensional, you get, oh, you get this. And so this is for when the B is point four. And you see it starts to wind more and more and more. And this counter set gets crazier and crazier from a geometrical point of view. And even if you go up more, it starts to get really wild. Okay? Excuse me? No. So this theorem, everything we do is perturbative. So our B is small. So this is a calculation. Sarma, Sarma made these pictures. We go there. That will be the next subject. And you're, of course, like if you look at, if you look at this picture, this is for like a little bit and you see stars already. So I, at this moment, I wouldn't bet that for hundreds you don't see stars. So this picture, the message is going to be that if you perturb the one dimensional and non-map, only a little bit, a billion, a trillion, the curve will break. And then there's a theorem about that. So all these ones, so this curve, if you are looking in the curve of infinite renormalizable maps, so that's the only place where we are looking. Yeah. Yeah. Yeah. And between now that this curve exists. And then we can do this experiment. It is an involved experiment. Of course, Sarma had to work hard to do that. But now there is a theorem. And so let's, let's explain these pictures with a theorem. This is, again, with the same group of people. And again, we look at infinite renormalizable maps with a small Jacobian. And that statement is that there is no continuous invariant line field on our, our guy. And you maybe don't get scared of what this is. This is a line field. It means that at each point of our contest set, we attach a line. And you see a picture here. So this is a line field. And the, the statement says that if you take the line field coming from neutral, neutral, neutral directions like this picture, this is not continuous. And so this theorem, there's still this complication going on. And so a consequence of that, of course, is that if there is no invariant line field, it also means, it also means that a consequence, our contest set is not contained in a smooth curve. And so this answers Stefano's question. And you can not, there is this contest set lying here. And if you take B at trillions, and for B at zero, it is clearly contained in a smooth curve. But now if you take B at billions, then this curve breaks down and, and, and you cannot find a smooth curve. So the stars is really representing what is going on. There's a lot of twinkling going on. And that's it. Yeah, this is, this is, this, this picture is the picture of the, of the oscillating line field. Absolutely, absolutely. And it is measurable. But it is not continuous. It is really measurable. We, we come to that. It's, it's, there is something, but it is, I need some time to explain what is going on there. Because there is at least one point where that happens. That has to happen. For example, where we are zooming in with our renormalization, will be a place where the stable direction and the, the, where that contest set is going to be tangent to the stable direction. Exactly there, for example. But then it is all over the place. Okay? So, so from this we know that we really left one of the dynamics. And we are really doing two dimensional dynamics. It's really, it really involves the plane. Okay. So that looks all, all very scary and you cannot deal with that. But let's start this a little, a little good news. And that is by Misha Lubic and myself. That is, these counter sets are contained in a rectifiable curve. And a rectifiable curve is something with, with a finite length. On each scale you have some finite length. But that can be very wiggly. But you can, there's a curve with a finite length, but where this thing lies in. So there's something that's not completely crazy. Okay. So now let's, so this is a little good news. Let me tell you more good news. And let's, let's give you two universality results. And, and it is a little sort of, it's going to be technical. It is going to be a formula. But this time there is sort of fun in the formula. If you look at, at the end renormalization of system app, it will look like something like this. Of course in the bottom we have an X. And then we have something unimodal. This is something unimodal. And then there is something, our average Jacobian starts to show up. And in a very precise way. So it's BF to the power 2 to the N, like no constants, B to the power 2 to the N. And then there is some analytic function which doesn't depend on anything. It's a universal analytic function. Then there is a Y. And then there is an exponential small error. And so all this is universal. So, so what we see is that this FN converges exponentially to our unimodal fixed point. This function A is something analytic and universal. And universal in the sense it doesn't depend on the map you start with. Okay. I just remarked you see the convergence of the unimodal part is exponential to the fixed point. So this is clearly the main contribution to the renormalization. Because this is something super exponentially small. So we have a very small part of something which is exponentially close to something you understand well. And then there is a super exponential perturbation. So that cannot play a role, you would say. So everything is going to happen inside here. Okay. So that is one universality theorem. There is another one. This is all sort of the good news. And again the same group. So remember we had our picture, C1. And we had the box here. And then we had the new one and we had the box here. And this is C2V. But then this one lands up somewhere here. And then there will be another box. And we zoom. And so you will see a sequence of boxes which is shrinking down to a point here. Let's see. This is the box. BNBVN. And this is the one B. I didn't see a box, a sequence of boxes shrinking down. This is the place. Let me not put symbols. So this is the place where renormalization zooms in. And so our renormalizations, which are described in this very precise way, tell us something about like high intervals F to the 2 to the N on this little piece. But now this is the problem. So the renormalizations we understand very well. But of course these renormalizations are rescaling of this very high iterate by a composition of diffeomorphisms. So you know very well the behavior of the renormalization operator. But if you start to compose all these diffeomorphisms it might become a mess. So it might be that all this precise information in the end is telling us nothing about what is going on here. So this is the danger of using diffeomorphic rescaling and not affine rescaling. This is a delicate point. And you might say, yeah, because the picture blows up because these rescaling are not affine and they start to do crazy things. And it means that renormalization has nothing to do with what is going on here. The contrary is true. And it is going to be again a formula. So let's take our map F and let's take our end renormalization and let's take the diffeomorphism here which is used to rescale what you see, what F2 to the end is doing there. It's like the composition of all our diffeomorphisms. And then this rescaling has again a very precise form. It's sigma squared to the end. It's minus sigma to the end. Sorry, goes the other way around. Maybe it doesn't matter. It doesn't matter. It's one. It's some number. I will tell you what they are. Times sigma squared to the end times minus sigma to the end. And then there is some diffeomorphic part and there is just a Y. And then an exponential error. So times one plus something exponential. So there is this formula describing how this rescaling happens. So let me tell you what these numbers are. So the sigma, what is the sigma? Now if you take the union modal fixed point and then we are looking at the domain of renormalization and that has length sigma. So it's what they call the scaling ratio. So this number T, so in particular it doesn't depend on F. It's a universal thing. This T depends on F. You see this matrix without this will be identity with this little T that corresponds to a little tilt. So this T is corresponding to some tilt. And it is proportional to our average Jacobian. Okay. And then what else? I have to tell you about this function here. So this function S and X is going to be this universal again. It is a universal again. It is some analytic function just depending on X, not depending on our original map. Plus some number which does depend on F times Y squared. It corresponds to some bending. And then plus some exponential small error. So what it means is that we had to consider the possibility that all this composition of this non-affine rescalings is something which is going to blow up. But apparently that's not the case at all. It's going to convert to some universal diffeomorphism. So the scaling is not, is not that fine, but it is going to have a universal shape. Not depending on the map at all except for this little tilt. So it's all very precise information and universal information. So let me tell you a bit more precise. What happens in this? Let me show you what this picture is. And you, you will see what happens. So that picture, because of this tilt, of this scaling here and because of this tilt, the little picture is going to look something like that. Where this is of the size sigma square to the n. And this is of the size sigma to the n. And then there is a tilt here which is of the order of b. So this picture is, is where your renormalization lives. So it's just a little tilt, a little tilted. And now, you know, in these boxes, there will be two copies of the next renormalization level. And so, but you will see is, if you, I tell you, and believe me, it's a simple, a very simple exercise, if you have this condition that like the tilt times how sigma to the n, if that is proportional to sigma square to the n. So if you have this condition, this is the crucial condition, what will happen is that in this box, there are two boxes of the next scale. And it will look something like this. If you have this condition, they will be above each other, under this condition. The two boxes inside, they used to be, like, they used to be, I don't know, so they used to be something nice like that, two pieces next to each other, a little bit above each other. But if you have this relation between, between the Jacobian and the scaling, they will come exactly above each other, each other. This is something very simple, but it has a consequence. Because remember, when we introduced the non-maps, the thing was that vertical lines are strongly contracted and then put somewhere. So now they are above each other. So if you now map these things, let me take the formula away because that's sort of, we don't need it anymore. So if you now just apply your map, and so this picture, remember, is sitting, is like this little, little, little box here, and now we just apply our map and we will see something here. So what you see there is that these two pieces are above each other, but they will be strongly contracted. So what you will see is something like that. So they are very close compared to their size. So and this is, this is a bad picture. And like if you think one dimensional, one dimensionally, this is a 2D picture. So in 1D, and you would expect that it would be something next, nicely next to each other. In 1D, the pieces are always nicely next to each other, and the distance is comparable to the pieces. But now you see that the distance is actually much smaller compared to the size of the things. And so here you see the cause of that effect that there are no a priori bounds. And the geometry degenerates. So let me write down more precise statements about that. And now comes the bad news. So there is a theorem by Peter Hazard, Michel Lubits and myself. It says the following. There exists some set A in the interval. It's a set. No dynamics. There's some set and the Lebesgue measure is one. And now the following happens. If your Jacobian of your map is in this set, so it's purely a number theoretical condition, there is this set of numbers. If your Jacobian falls in this set, then unbounded geometry. And that means that you can go, you can repeat this picture in smaller and smaller scale and the pictures are getting closer and closer together and bad stuff happens. It's not at all what you expect it to be if you think about the one dimensional case. It is sort of interesting that it is purely a number theoretical condition. Now let's select this. You know, so let's think one dimensional. And so then we have, we are building up our counter set. Pieces and pieces and pieces and pieces and it continues forever. But now if you zoom in here and then what you will see is some nice little interval and there will be nice two pieces inside. So wherever you zoom in, you will see that all the pieces which are lying there, they are comparable size. So nothing goes wrong. But now if you go, so this is in Bandy, so let me make this picture better. So now let's go to our hand non-map. And now this, here we have our one dimensional counter set, which has a topological copy as inherent attractor of the hand non-map. So this is our counter set. So now if you would take the same spot here and you would zoom out, then it will happen that it doesn't look at all like this. So there will be pieces which have a certain size that is the equivalent of this one and there will be a piece which is the equivalent of this one which is really close. So the picture, the conjugation should bring this picture to that picture. These are the topological equivalent places. But the geometry here is very nice, just nice three pieces, nothing weird. But in the hand non- context, these pieces are screwed up. They are sort of, they are folded. No? Yeah, that is the next theorem. So in the hand non-family, like in this family, B is just BF. It's the Jacobian. So it is just, if you move along this parameter, then for almost every value of B, you will find the unbounded geometry. But the curious thing is that this condition is just number theory. It has nothing to do with the fact that this is a curve in the hand non- family. There is number theory here. We don't know. We don't know. That's no progression. So we don't know. Everything we know is for tiny Jacobian. Not like very, it's like the small. Okay? So let me answer Stefano's question, which is coming after this one. And so the theorem is again the same group. If you have two maps which have different Jacobians, like take one here and take one here, maybe very close, then you see that there is a conjugation of the cantor set from one to the cantor set of the other. And now, hey, you're right. This is at most C1 half to the one plus log BF1 divided by log BF2. And this is always smaller than one. And so in particular, if you have, here we have our fixed point. This is our cantor set. And here we have our map F, which is a certain positive Jacobian. And it has this cantor set. So here we have our conjugation. And now, in this case, B is zero. So it becomes your C1 half. This is at most. No, no, no. So this map is at most C1 half. So at most C1 half. So you turn on your, you have your very nice universal cantor set. You turn on the B, a ridiculous small amount. And the conjugation goes from C1 plus holder, what we were expected in the Vandermeister theory, suddenly to C and half. So this picture is representing something. The cantor set really is deformed tremendously. The conjugation jumps from C1 plus something into C half. It's really a dramatic change. And this, like B is zero to B positive. Okay, so it seems that everything breaks down. The whole theory breaks down. And at some point we told Feingenbaum about this result. And he said, this is not true. This cannot be true. Because universality and rigidity, that is our, the soul of our belief, of our renormalization, this cannot be true. So he did the measurement. He went to the computer and he started to, to measure the structure here. And we told him where he should look. And he found bad spots. But he also found good spots. So he found many good spots, but he also found bad spots. So let me try to make a theorem about that. And now the probability stuff comes in. And so if you just think about rigidity as we thought before, that the conjugation has to be differentiable on the attractor. And then you see that that notion of rigidity breaks down immediately. And it goes from something differentiable to something that is at most older. But there's something good happening. So it has to do with the scaling structure. So let's take again our unimodal fixed point. And let's take a little piece here out of the end, the end cycle. And we can blow this up. So we look at our microscope. And we see that this little piece is essentially a straight line. And inside here there will be two other pieces. So there will be a, let's call this eye. And this we call IC. And this we call IV. And as always, pieces have two pieces inside. And then we look at the scaling number of IC is the length of the C interval divided by the length of C. It's like the scaling ratio which we discussed when we discussed the fundamental map. Just the ratio of the C interval is in the other one. And then there's also a scaling ratio for the other side which is just the relative size. And you know, let me put a star here. Stars refer to things which are universal. And so these scaling ratios are the universal scaling ratios. And these are numbers, you remember, in the set of, the counter set of scaling ratios of the fixed point. So these are universal numbers which you can measure. They are there. It's a rich set, but they are there. So now let's look at the same spot. Let's use our conjugation. And let's look at the same spot in our Hennon counter set. So here we have our Hennon map and we have our counter set. And like this little blob is going with the conjugation to some little blob here. Something complicated. And let's zoom this out. You cannot expect that it looks as nice as this. We know that sometimes it really goes bad. So let's measure how bad it is. How far is it away from this picture? So inside here you will see like the two blobs which is like the C1, the equivalent of the C1 and the equivalent of the V1. I draw them sort of nice. We don't know how they are. And let's measure how far they are from this. So what we can do is to do that, we can draw around it some rectangle. And that rectangle will have a certain length and it will have a certain thickness. I'm trying to make a one-dimensional picture, but think about these very small but maybe it's not at all. And then you can cut off a little box measuring the size of this guy. And that we call sigma C times L. It's like the relative size of the length of this in the long box. And the same you can look at sigma V times L. So you measure three numbers which sort of describe the geometry of this piece. So you get an H which measures how thick the thing is. You measure a sigma which measures how long this guy is and you measure how long this guy relatively is. And now you say, here we have our Psi. This is the definition. So epsilon is epsilon standard if the picture differ by epsilon. And so if H is smaller than epsilon and if the scaling ratios which you measure in the hand-on are epsilon close to what they are supposed to be. If there would be rigidity. And this should be smaller than epsilon. And now you also measure the other one. And you also want this to be smaller than epsilon. So if epsilon is very tiny, it means that this picture really looks like that. So standard pieces are like one-dimensional pieces. And now there is a theorem about this, about how many there are. And there the probabilistic aspect of universality comes in. So this theorem is called probabilistic universality. So what you do is you collect all the pieces where epsilon i in the end cycle which are epsilon standard. So these are all the good pieces, the epsilon good pieces on the end construction of the counter set. And now the statement is that there exists some number theta smaller than one which is a universal number. And what we know about this is that if you take the measure of all the good pieces and good pieces which are exponentially good, so you take all the pieces at the end level of the cycle which are really exponentially close to what you want it to be. But it will be in the one-dimensional context. And now you take the measure of those guys. You count them. And that is larger than one minus theta to the end. And so we know that there are bad pieces which is indicated by all these stars. We know there are bad pieces but the bad pieces only form an exponential small fraction of all of them. So if you go deep enough, then most of the pieces will just look one-dimensional. So from a probabilistic point of view, the counter set which is reformed, these are really bad pieces but from a probabilistic point of view, they didn't reform. So that is the other theorem to finish. And that answers Stefano's first question. It is also by Michel Loupitz and myself. That is called probabilistic rigidity. So now we are going to speak about the quality of this conjugation. So we have our universal counter set and we have our real counter set. And there is a conjugation. And we would like to know the quality of that. So we know that it is only C and half. So we cannot just say it is good, something good. But what we know is that there exists a sequence of sets in the counter set of our map and what we know is that the measure goes to 1. So you start to fill up the counter set. Sort of you start to take, there are places where the stars are and you throw the stars out. And you go to find, that is what you do. You throw out the bad stuff on each scale. And then what we know is if you look at the conjugation from this Xn to its image, in the unimodal world, this is C1 plus beta. And again, there exists a universal holder constant. And on this subset, the conjugation is differentiable. And so rigidity breaks down. Have you proved that if you perturb, if you take your unimodal map and you push it a little bit into the unknown world, the conjugation breaks down and becomes C1 half. But from a probabilistic point of view, everything is fine and it continues to be C1 plus beta. So rigidity survives in a probabilistic sense. Let's go home. Excuse me? So the exponent is fixed, but the holder norm grows. And I think it grows exponentially. I'm not 100% sure. I think it grows exponentially. Yeah, but you know, there is something. So I don't know the answer, but we have the technique to calculate this norm. We know very precisely how these things look like. And a remark, I remember that there was a thing about you have to choose the Jacobian according to a number of theoretical conditions of the Jacobian. And that will make the geometry unbounded. So these sets, they have a topological construction. And the way they are constructed depends only on this average Jacobian. So if you take two maps with the same average Jacobian, the conjugation will identify these sets. So these are not arbitrary sets. There is a lot of structure here. And they are topologically defined by, and that is the strange things, they are topologically, it's all contradictory. They are topologically defined by a measure theoretical invariant. It's crazy. But that's how it works.