 Good morning. So I know you will find this boring, but let me make the little diagram again, OK? It makes me happy. So I was thinking we have our space, and we have our map, and we take a U, and we take the first return map, and that gives us our renormalization operator. And then there was this thing that renormalization, which we saw, connects topology, and we started to see the connection with measure theory yesterday. And today we will finish the measure theory part, and we will start to look at the difficult part of renormalization. And if time allows, we will look at the bifurcation part. And if time allows, we will also look at the geometry, but maybe not today. OK, so let's go back to the measure theory. So the theorem says, which we were discussing yesterday, so if you have a diffio, let's say there's a diffio, which is infinitely renormalizable, then there exists a unique ergodic in variant measure. So that means that circle diffeomorphisms, infinitely renormalizable circle diffeomorphisms, as they are called uniquely ergodic. OK, so let's do the proof of that. Is the machine working? So let's do the proof. And probably by the exercise yesterday, you know the proof by now. So let's look at the end renormalization of F. And you know that there's a little map like that. That is something like that. It's an Ln and an Rn. It's like a little map. But it also comes with a partition and it comes with a tower. So this is like Ln. And let's say this is like Rn. And there are all the floors. And there are all the floors here. So this tower also represents the dynamical partition of the circle. And so this is just a cut-off version of the circle. But now you see that this interval here is Un. So we can represent this tower even more schematically by putting a dot here, which represents the basis of the building. And then this tower, we can represent as some loop in the directed graph. So this is like the left loop. And then for the same thing, we can take Rn and we can represent all these intervals. Also, by dots, on my loop. So here we have the directed graph we were discussing yesterday. So now let's look at the next renormalization. So you know how you do that. And you have to cut, say, somewhere here. And you have to, I know, it's better, it's better. So we start over with the whole thing. So let's make a picture of the next renormalization. So you know that it was like cutting and reorganizing. So you cut here and you have to take this part of the tower and you have to put it on top of this one. So what you get is you get something thinner here and you get something here with this other thing on top. So this is Ln plus 1 and this is Rn plus 1. And now this is again a tower, et cetera. And we can represent this again by the left tower. We can represent again by the left loop. And the right tower we can represent also by a loop. But now let's take, so this is the right loop. And now you see that this tower represents the next partition. But we know that partition is a refinement of the previous partition. And so there should be an inclusion map here. So let's figure out what it is. So we have to write down our winding matrix. So let's say this is left, right, left, right. So let's take the left loop here so that the left loop corresponds to this loop. Now what you see is the left loop starts out with the old left loop. So the left loop will go first around the left loop here. So it hits the left loop. And then, if you caught here, you have to go to this part. The this part corresponds to this part. So that corresponds to the right loop. So what this left loop does, so this one, what it does is it starts to go out through here. And then it goes around this one and you're back. And so the left loop uses once the left loop and it uses once the right loop. So now let's take the right loop so that is this one. So now the right loop is just a part of the old right loop. And so it just goes around here. So what you see is the right loop doesn't use the old left loop but it uses once the right loop. You see? So this was a year you see. This is the plus case. And here you see the corresponding plus the corresponding binding matrix. So you can do the same thing if the critical, the cutting point was here and you would get, you would get the left renormalization thing. And the corresponding matrix would be 1, 0, 1, 1. Yeah. And so and now what we did is we looked at, so now we have a, let's say this loop comes Xn and this loop is the next one. Yeah? Oh, it's the same one. Yeah, thank you. Sorry, thanks. It's like this one. So now we see our loop and we have our Xsn, our partitions and we have the inclusion given by this winding matrix. And so we have our Xsn here and we have our inclusions and this converges the inverse limit of these things, which is X. Of course, you also see that from here, or from here, let's forget about it. So these are our things. This is just a representation of our partitions. And now we started to look, we started to look at the corresponding spaces of invariant measures on this scale. So what is the invariant measure possible in this one? Now you have to put some bait here and one here and one here and you can choose anything here, anything here, et cetera. So the possible invariant measures up to this scale are characterized by two numbers. This is m1 and this is m2. So this is our measure space. And on the next scale, we have our same measure space and then we understood yesterday that the inclusion from this partition in the next one induces a map between the measures, which is just the plus or minus matrix. And so we will get a sequence like that from R2 between R2 and plus or minus matrices. And this will convert to the inverse limit of all these measure spaces. And that is the space of invariant measures of our circle diffeomorphisms. So these are the invariant measures, our original map. And so now we see what happens. Like if you have your little cone here, that is going to be mapped maybe in this one. And now you see what happens is, and we have to understand what if you take the positive quadrant and you have to project it down, and you have to take the intersection. So yesterday we played this in the exercises. And you can see that these projections will actually intersect into a single line. And let's skip the details. We can discuss this in the exercises. And the fact that you get the intersection of the cones in a single line corresponds to say that the dimension of this, the dimension of M is 1. And it's going a bit fast. But yesterday in the exercises you saw with those. And that proves that our circle diffeomorphism has unique invariant measures. OK? And so we are done with this part. OK. So now let's go to the difficult part. So somehow what we are doing in this course is showing using a renormalization machine. And if you know enough about this renormalization machine, you can detect all parts of dynamics you want to understand. And we are sort of doing the fun part. We are doing the topology. We constructed math. We are doing the topology. We constructed measures. We will do today by vacation. But the hard part is renormalization. So now we have to go back to do the by vacation thing. We have to understand a bit better what this renormalization operator does. Yeah. Now you know, so renormalization is not very useful on a topological level if your map is not infinite renormalizable. And that means when it has rational rotation numbers. So in the boring case, rational rotation numbers, renormalization is not so useful. Also, if your maps are not smooth enough, know that it's always the same. That is purely topology. Yeah. So boring topology is an obstruction for renormalization to be useful. Another obstruction is, remember that we used C2 maps. So low smoothness is also an obstruction for renormalization to be useful. But you can also say low smoothness. Nature is not low smoothness. So, and so far I think that that's our sort of the main obstructions as far as we understand. Okay? Yeah. So topologically it's small. If you take a family of circle diffeomorphisms, there will be an open and dense set where the rotation, if you take a generic family of circle diffeomorphisms, there is an open and dense set where the rotation number is rational. So the complement is something like a counter type of set or like, so topologically it's not a lot, but all the interesting stuff is happening there. Okay, so do you have another question? So let's go to discuss the difficult part of renormalization. Don't get nervous. So we are going to speak about renormalization. And so renormalization is an operator on the space of systems. So that's a dynamical system. So we can ask ourselves whether renormalization as a dynamical system is hyperbolic or not. Whether it's simple or not. Hyperbolic systems are the simple ones. So that's the question. R is hyper. Let's first look at, in our circle case, it is easy of the rigid theorem to assess the following. Let A be the set of rotations. That is rho in the circle. So F rho is just so, okay. So a lemma says, and you can do this yourself, this is a thing, that renormalization takes the rigid rotations and maps it to the rigid rotations. And you can write down a formula. So this is like the rho. And I will make the graph of how renormalization acts on this circle. And it looks something like that. I think this one is like the new rho. R rho is like rho over 1 minus rho. That's in the minus case. And in the plus case, I think R of rho is 2 rho minus 1. So it's an explicit formula. How renormalization acts on the rigid rotations. And this is dynamics we understand very well. Okay. So now let's write down, I write a big star here. Because this is the generally difficult part of renormalization. And the theorem says that the following. Let F be a C3 circle, maybe F and G. These are infinite renormalizable. This A and of G is A and of F. And that means F is topologically equivalent to G. And they are conjugated. What we discussed here. So we look at two maps which have exactly the same topology. So then ren of F and ren of G, they convert to A. So if you start to renormalize some diffeomorphism. And you have little, you can make a little picture here. So you start to look at your renormalizations. You will have some bubbles here initially. But if you go on, these branches here, they will become straight lines. And that means it becomes the rigid rotation. And of course for our other map, you'll be the same. And so apparently the rigid rotations are the attractor, form the attractor of our renormalization operator. Which previous theorem? This is part of the convolutional A. Is it related to the theorem of that? This is part of, so these are the conditions. So if you take two maps, F and G, which have the same rotation number. It's a topologically equivalent. Okay, let me write it differently. You like it like this? So take a circle diffeomorphism, which is infinitely renormalizable. Then the renormalizations become like little rotations. Okay? So that is the first part. And we will come back in how fast this convergence goes. And that's going to be important. So the second part of the theorem. Now take F and G, which are C3 circle diffeos. So the orbit of this accumulates at this. So it gets, it could accumulate at the whole of A. So these are functions. And we proved that they satisfy apiory bounds. So the limit set is some compact set. And that compact set is contained in this one. And remember the action on this thing is given by this map which has entropy 2. So there are many candidates for limit sets. And all of them are possible. Excuse me. If the rotation number is rational, the thing stops. So look at the set of rigid rotations. And so some of what is happening is, here we have our space of C3 diffeos. And then we have our renormalization acting there. And inside here there is a little curve. A circle. These are the rigid rotations. It's a circle. Let me make it a circle. And now if you take some map F here, which is infinitely renormalizable, and you start to renormalize, it will start to run around this thing. This picture. Now, so if you take a rotation that's labeled by the angle. So the set of rotations forms a circle, a set of functions. And so if you look in the space of all diffeomorphisms, then there will be this curve here. And this is the set by this formula. Oh, the dynamics of this one know that the derivative is one here. It's tangent here. Yeah. That's this thing. Yeah. It was an exercise. Yeah. If you look at the definition, it takes a minute. It's easy. Okay? Yeah. Absolutely. Yeah. That is the next statement. It's a rational map. This is the actual formula. It's just you have to take some ratio of intervals. And so if you take a map, if you have a rotation number row, and then it is row over one minus row on this side. And on the other side, it is like two aromas. It really doesn't matter. What is a bit nasty is that the derivative is one here. That's a little bit nasty. So this thing is not hyperbolic. It has this neutral fixed point. And that is, yeah, but we don't do that. Yeah, but we don't do that. We don't do that. No. No, we could do that. But then you have to do a long story. And for today, it's not necessary. It's not necessary. And also, if you do that, the difficulties are going to be hidden. So in this formulation, you will see the difficulties. And the difficulties have something to do with this. And if you do the fast one, you wouldn't see this. And so the beautiful picture is the one, is the Gauss map. But the more honest picture is this one. It reveals that put renormalization is more exposed. OK? So let's go back to how renormalization acts. So we know that if you take an infinite renormalizable guy and you start to renormalize, it will start to accumulate at our little loop of our attractor. So how do you see in that picture something is finite? So inside here, there is the set of all s, where rho is in the circle minus the rational angle. And so if the rotation number is irrational. So and then this set, are the infinite renormalize so on? The actual renormalization of the normal rotation. Yeah. So, but every one of this mapping. Yeah. So it was sloppy. Have you only defined things when you always have plus or minus? So if it is C is equal, you can still see what renormalization is. And then it extends. Let's skip over it. And as a matter of fact, if you do it right, so you know the statement is correct. If you take the irrational rotations in this set, then on that set renormalization is defined and it is defined by these formulas. But now if you take a guy, it will start to have a limit set. And that limit set is going to be compact. And there are infinite renormalizable guys whose limit set contains rational rotations. And so if you want to speak about the attractor, you have to include those stupid rational guys. So this is really the attractor. The statement is correct. And formally the thing is... You see? I'm sloppy here, you know. But allow me to be sloppy. Yeah, no, no. I don't want to scare you with the real picture. And it has something to do with the previous question about the measure of the infinite renormalizable guys. And so apparently if you take a family, the infinite renormalizable guy, they form a counter set. So that means that it looks much more... Let me not make the picture. But the fact that the guys have... If you take a family and you look at the guys which are infinite renormalizable, that is sort of the interior of a counter set. So it's a zero-dimensional set. And that indicates that the picture is more complicated. But let's not go there. Exactly, exactly. Okay? Yeah, absolutely. Okay? Okay, so that is about... So let me... So the statement is correct. If you start to renormalize, you will convert to this set. So the second part of the statement is, if you take two maps, which are C3-circle diffeomorphisms, and they have the same rotation number. That means the rotation number of f is the same as the rotation number of g, and it is part of the irrational numbers. Then the renormalizations of f and the renormalizations of g go to zero. And again, that is in C1 topology. So somebody had a question about shadowing. If you take your map and you look which rotation number it has, you can look at the map here, which has that rotation number. And now, if you start to take the regular rotation with row and f, then they will come together. So for each map, f, there exists. So here you have f and here you have a rigid rotation, and they will come together. And so rigid rotations are the ones which are shadowing. So then there is a third part, and that is a little technical, not only a little. And so we see that if you start to renormalize, you start to go to a rigid rotation, so the branches will become fine. So let's make a technical statement, the third one. If you take the n-trenormalization and you look at the tower, l, n, and r, n, and you look at, say, a certain level here, fi of r, n, and then you can look at the i-thread. So then what happens is that this map, this takes the r, n to its image, is also going to be f-fine. So this map converges uniformly, that means in n and i, to an f-fine map. So not only the return maps are becoming f-fine, but also the tower becomes f-fine. So it is like perfect. So let's use this. So these three ingredients to prove it is the difficult part of renormalization. In this case, we can do this. There are relatively simple tools which allow you to prove that. And for people who are a bit familiar, you can use non-inliarity, which you have seen, and something like Swartz in the relative. Very classical tools, and in the circle case, you can prove that. It evolves, but not very difficult. Doable. In other cases, which we will see later, this is going to be one of the most, I would say, one of the most sophisticated theories in dynamics. There's a star. And so in this course, we are not going to discuss the star. We are only going to discuss the fun of it once you have a good understanding of renormalization. And the main point of this course is that you can get everything out of renormalization in the context we know. So let's do an example. Let's keep this picture. And you know it is like this. If you take your map, it has a rotation number, and that rotation number is sitting somewhere here. And this point will have a limit set under this map. And that limit set is the limit set of your map. And that can be anything. It can be a periodic orbit. It could be a fixed point. It could be the whole thing. That was nice. Okay, so here we have our map, our renormalization on the attractor. And now let's do an example. And let's take a map F for which all A's are one. For all M. The Fiber-Nazi case. Because in that case, the rotation number is one over one, one over one, et cetera, which is like square root of five minus one over two. So that is our little row. And in this case, if you take F row, this is this rotation one, which is the rigid rotation, the rotation, and then if you renormalize twice, you get the same map. And so this Fiber-Nazi rotation corresponds to something like this. It's like the period two point. Difficulties to draw. And so F row is periodic period two point renormalization. And of course, you see, this thing is sitting somewhere here and sitting somewhere here. So you see that the derivative along the attractor is larger than one. C2 plus alpha is okay. Let's take this. Absolutely, absolutely, absolutely, absolutely. Perfect. And we will use that a little later. And so we are looking at our simple example where we have always plus, minus, plus, minus, plus, minus renormalization. And we get the golden rotation number. And that is a period two point of renormalization. And of course, we see that if you take the derivative in this point and say along the attractor, that is larger than one. And so you see it in the picture. The slope here is larger than one. So in the attractor, it is an expanding periodic point. And so it is an expanding period two point of renormalization. So that starts to look hyperbolic. So let me make a picture of how renormalization acts in the neighborhood of this thing. Not even a neighborhood. So let's say here we have our space of C3 diffused and somewhere here we have our little map F-row and somewhere here we have the renormalization of our little map. And now you see that this map is contained in the attractor. So it's contained in this circle. So there's a curve here. And that curve, according to this expansion thing, is the unstable manifold of this thing. So we detected the unstable manifold of renormalization in this point. And for similar, this same curve will go around and there will be the unstable manifold of the renormalization of this thing. So now we have the thing that if you take any map, it will start to convert. If you take any map with the same rotation number, it will start to convert to those two ways. So what you will see is there is also a stable manifold and there will be a stable manifold here. And this is the dynamics. So renormalization has a one-dimensional unstable direction and a co-dimension one stable direction. And in here, so under these combinatorics, this will actually be exponential convergence. So it is really a definite contracting manifold. So this is an hyperbolic picture. That is part of the star. And we could do it if we have one or two days. We could prove this whole picture, a lot of it, but we don't do that. We want you to have the fun of it. So this is the dynamical picture of renormalization and now the fun thing comes. And that is the main point, the main advantage of renormalization. So if you look, maybe I should draw this very big because this is really important. This is really important. So let's take the set of all f in the C3 diffeomorphisms where all the an's of f are one. Or in other words, the rotation number of f is squared to five minus one over two. Let's look at all these maps which you can renormalize in exactly the same way. So have you learned that if you have two maps in that class, they are topologically the same. We constructed the conjugation. So we know that this is a topological class. Topological classes are all the matrices that are topologically equivalent. And so before we proved that that matrix, the same rotation number, are topologically equivalent. So this is exactly a topological class. And now the main point is, I should give it a bit more space because this is important. So this could be a bit smaller. Let's look at the topological class. And now what you see is that the topological class is that. It's the stable manifold of f row. So this is the key point which is a codemain in one, manifold. This is the key thing. This is one of the most important consequences of renormalization. So if you look just at this set, don't even try to think about it just in general. It is a very complicated thing if you don't know anything to deal with it. But apparently we identify it with a stable manifold of an operator, of a hyperbolic one, and there we know that these are many faults. So from this whole renormalization thing, we are done with bifurcation theory. We know that the topological classes are many faults. So done. And in particular, remember in the first day we did this exercise that if you have a circle diffeomorphism, you can push a little bit and then the orbit becomes periodic. So you can change the topology by pushing up a little bit. And then we use the trick where you can just rotate a bit more. But that is a very special trick which only works for rotations. But if you have this picture and you know that the topological class is a codemence of one manifold, and of course it is obvious how you can change the topology, you just push out of the manifold. That's it. That was our difficulties. But if you have this general picture, you can always prove this. So for people who have heard about the C1 closing lemma, and for people who know how difficult it is to prove the C2 closing lemma, you see that this is the way to prove higher smoothness closing lemmas. And so this is a very important consequence. The logical classes have manifold structures and this dynamical meaning. They come from hyperbolic theory. So we are done with the bifurcation picture, we are done with the topology, measure theory we did. So let's go to geometry. Let's do it Monday. The same will be true if you take any bounded sequence. It's still perfect. If it goes up, I don't know what happens. If you have some non-dyed, if you have tyrophantine numbers, it will still be okay. If they become very bad, if you take any, yeah. Okay? So let's discuss Monday like the geometric part.