 OK, good morning. So we continue with our big diagram of the relation between renormalization and topology of maps, measure theory, and geometry, and bifurcation of systems. So last week and yesterday we discussed the relations for circle diffeomorphisms. And today we are going to part two of the course, which is about unimodal maps. And most of it is going to be a little bit historical. It's going to be relatively soft today. OK, so what are we going to look at? We are going to look at a unimodal family, and they have t from the unit interval to itself. And like it's something like this, like a family with sort of unimodal maps which get higher and higher up to the end and say, if the family is labeled by how high you are. So this is just a full family. You move from the bottom to the top. And we are interested in the boundary of chaos. And in this case, the boundary of chaos is reached by a pair of doubling cascades. So what happens is, you have seen this picture, even in the first class of our course. So if you write down here the parameter, so we are going to move up our family, then what you will see is for small parameters values, you will see if you take like this parameter, you will see that the dynamics is controlled by an attracting fixed point. You can easily see this in the picture. So if you take like this one, and you see like there's a fixed point here, and points will converge to that fixed point. And so for very low values of t, you see that the dynamics is controlled by an attracting fixed point. But then what happens is, at a certain parameter value t bang, this attracted fixed point bifurcates in a pair of two points. So you will see that the fork happening. This is something like that. So if you take a parameter value here, the dynamics is controlled by a pair of two attractor. So the original fixed point will continue forever. And then the same thing repeats. And so there will be a little later parameter value t2. And there will be, again, a pair doubling bifurcation. And what you see is if you take a point a little higher than this bifurcation moment, you see that there is a period four attractor. And again, this periodic two point continues. And then this process continues forever. So you will see another bifurcation at the moment t3. And it will continue like that. And you will see the sequence converging to a t infinity. It's like that. And so now what you see is that on this part of the bifurcation pattern, you see that the dynamics is very simple. The attractors, they are periodic. And so this is definitely not chaotic. So we are in the simple dynamics. But then if you are sitting on the other side, what will happen is that on this area, you will see that the number of periodic points of period n is like e to the power, something like that. It grows exponentially. So suddenly when you are beyond this t infinity, you will see an explosion of periodic orbits. And the rate of how many there are is h. And this h is positive. And it is called the entropy. So that looks bad. It looks like chaotic. And indeed, this is the chaotic. So what you see, let's draw the same picture. But let's forget about all the details. And there is this moment t infinity. And here you see no chaos. And here you see chaos. So there is a sharp moment where the behavior changed dramatically. And now the historical part comes, you can call this a phase transition. It's a phase transition. This is terminology from statistical physics, that you have a parameter and you change the parameter. And suddenly, the physics is very different before and after. Exactly, yeah. Exactly. As you see, here the attractor has period two. And here you will see the attractor has period four. So each bifurcation, it doubles. And then after this, you will not have all periods. But there will be many. And if the period grows, it grows exponentially. And so this growth rate starts with zero and then slowly it starts to go up. And so there is an exponential growth of periodic orbits. But it is small, just that. The endopic growths are continuously. Eighth depends on t, yeah. Thanks. That's a good way to say it. Eighths of t. Thanks, yeah. That's the right way to say it. Yeah, when you get to this thing, like the full map, then if you look at how many periodic orbits there are of period n, that is like 2 to the n or something. I never know. I think it's like it grows like 2 to the n. So for this guy here, this is the worst situation. The number of periodic points grows like the end of p is log 2. Ah, yeah, OK. It's log 2. So it grows like 2 to the n. And that is the worst you can get in this type of dynamics. But it starts very slowly. So let's go back to the global picture of what is going on here. So we have our moment t, which is at the accumulation of period doubling. You call this moment, as you'd have said, this is the accumulation of period doubling. And in that moment, you see a dramatic change in behavior from cyclic behavior to chaotic behavior. And let's call this phase transition. And this terminology comes from physics. And I would like to tell the physics story. Because the physics story inspired three people to introduce renormalization into dynamics. So this point of view, that there is a phase transition, is like a crucial historical moment. So let me tell you a little bit about this phase transition stuff. So this is called spontaneous magnetization. So what you do is you take a piece of iron, and just a regular piece of iron. And then you cool it down, cool down. And then you make a diagram. So here you write the temperature. And in this axis, you write the magnetic force. This is just a regular piece of metal. Like something in the chairs. So that means for room temperature, there is absolutely no magnetic force to feel. That is just a piece of iron. But then if you get to what they call a critical temperature, then suddenly the thing spontaneously becomes a magnet. Something like that. And it jumps up, sort of like a root. And then what happens here, and that is sort of amazing and very surprising, that this magnetization at the temperature is proportional to something like this, 0.3, when the temperature is smaller than the critical temperature. So it's not amazing that it behaves like a root. That can happen. But what is amazing, if you take some other material, which has the potential to turn a magnet, and there are many of them, this whole class of these materials, which show the same behavior. You start to cool it down, and suddenly it becomes a magnet. So there are many of those things. And then the amazing thing is that for each material you choose, the critical temperature will change. And the critical temperature depends on the material. But then this number is always the same. So there is some sort of a universal. There is some universality. I should not make it too much a mess. And so what you see is clearly there is a phase transition here, so from not magnet to magnet. And apparently, there is some universality. I should write this big, because this is really important. And unbelievably. So what you see is the critical exponent is independent of the material. And the name had this independence. This independence refers to universality, universal, in parameter, the dependence. So this is the point. This is the point. It's not, yeah, let's keep the, this is the picture. This is the picture. There's something going on there. So there's something very special going on in this phase transition. And let me give you a very rough and maybe a very naive idea of what is going on here. And the idea will lead to renormalization. So the study of this introduced renormalization in physics. So let me give you an idea of what is going on there. So you know that if you have a piece of iron, the iron atoms, they are organized. They are sitting in a grid. So it looks something like this. It is a grid of, and we are very naive today. So let's make it two-dimensional picture. So you see something like this. And in each corner, there is an iron atom sitting. And this is under a super microscope if you look in this piece of metal. And now what happens is that if you, this little atom, has itself a little magnet. So it has a little magnetic direction. That's one ingredient. So there are two parts. Each atom has a spin, a little magnetic direction. And suddenly each atom is shaking. How do you write shaking? It's vibrating. And so if you look just at this one little atom, the guy is sitting there. It has a certain magnetic direction. And the thing is moving. It's like shaking like that. And the shaking, the amount of shaking, or how wild it is shaking, is proportional to the temperature. It's related to the temperature. So the higher the temperature, the thing is shaking like crazy. And you slow the temperature down, then they start to shake less. But then if you look at this neighboring atom, maybe it's spin was like that. And if they are shaking a lot, the spin interactions are very weak. And you can forget about it. But if the shaking gets sort of slow, then they can align. So energetically, less has less energy. So spins bond to a line. But if they are shaking too much, they cannot align. But if you scroll down the temperature, they start to shake less. And suddenly they align. And apparently, if you go to this temperature, the shaking became so weak, it's almost nothing, that all these guys are aligned and all of them are pointing in the same direction. And then the thing becomes a magnet. So low enough temperature. So you get alignment of the spins when the shaking, shaking, or temperature gets low. So that is the picture. So now how do you describe something like this? And now we are getting to the renormalization thing. So the picture is something like this. We have our grid. And here we have our atoms. And they have their spins. And the way they model this is as a random process. And the model is random. And the spins are assigned randomly. But of course, if you remember your probability courses, there is something like an independent random process. And so that is the way they assign the spin here is completely independent of the spin here. And of course, that's not true. So this random assignment has to involve some interaction between the pieces. There has to be some interaction. And so this is not at all an IID process. There is some dependence in it. And that makes it complicated. If there would be no dependence, then you would get sort of a central limit of the theorem behavior, like something like the bell curve. But now the claim is that once you get to this phase transition, there is one universal law which describes this random assignment of spins. And so at phase transition, there is a universal law assigning the spins. And now this universal law, a universal in the sense that if you model iron, you get something. If you model another atom, you would get still the same random process. And this universal law and this law, this universal law, is a fixed point. You will not be surprised. It's a fixed point of renormalization. So let me explain you what this renormalization is. Because this is where historically our story began. So what you can do is take these spins and let's take four of them. And they have randomly assigned their spin. So there is some common distribution of this. So what they do is they take four of them and they replace it by one spin. So they group them together and replace these four spins and the distribution of those four spins to a distribution of a bigger piece. This consists, if you look inside, you would see this picture. And this operation is called renormalization. And this acts on the space of distributions. So there's a law describing this. You block and you will get a new distribution. And renormalization is acting there. And now the claim is that this process has a fixed point. And with whatever material you begin, you will see that the statistics on small scale is controlled by this one and the same universal distribution. OK. So this was the naive and very simplistic description of what is going on in phase transitions. This is not complete. There's one difference. You see, they go from the little scale of atoms to one scale up. So they zoom out. And the physicists, they zoom out. That's the only difference. So somehow, you see, they take this thing and group it. So they zoom out. OK. So this happened in the late 70s. And then something nice happened for us. So there was Coulet, Pierre Coulet, and Charles Trecer, and independently, Feigenbaum. So Feigenbaum was already an established physicist. These were young kids, students, but in physics. And this was in the 70s. So and then this is considered a spectacular story in physics. So they got inspired by how this universality observed in phase transitions. And then they thought, OK, in dynamics, have you also have a phase transition? No chaos. Chaos. So it's a phase transition. And then you are a physicist, and you say, OK, if you have a phase transition, you have to see some universality. Whatever it's going to be, you have to observe at the boundary of chaos at the transition moment some form of universality. So what they did is, so this is a computer, of course. So what they did is they made the picture I erased. So they took, say, the quadratic family as, let's say, Qt. Qt of x is, I think, 40x1 minus x. So a very specific family, the quadratic family. And they drew the picture. So you see the bifurcation, the first bifurcation moment, the second one, the third one. And this continues like this, up to t infinity. So in this family, they calculated this bifurcation moment. And then they observed. And that is the equivalent. Oh, you remember, I erased it, there was the 0.3. So this is what they observed is. The first thing is that this tn converges to the limiting value. That's like the critical temperature where the phase transition happens. And they observed that this rate, this is exponential. And the rate is like 1 over 4.669 and some other numbers. And then they did the same picture for another family. And maybe something like something with sinus, like something like t sinus pi x or something. And for this thing, they made the same picture. And they measured the moments where the bifurcation happens. And they observed that it converts again exponentially with exactly the same rate. So this is what they call parameter universality. And there is some sort of geometric organization in the bifurcation space, in how the parameters move. There's something going on. And that is equivalent to what you should compare that like if you had the magnetization. And here, there was this power x to the 0.3. And so this 0.3, you should compare to that one. It is something which is universal in the parameter dependence. So then they did another measurement. And remember that if you take the attracting fixed point, then at some point, it flips and becomes a pair two attractor. But this orbit continues. So it's like somewhere here, p1. So and then you look at the pair two attractor, and you look at like this one. And when it bifurcates, it still continues. And this you call p2. And these are the periodic points which are survived. And you continue like this. This one continues p3. And so this has period 1. This has period 2. This has period 4. And it continues like that. So this sequence pn converging to some p infinity. This is the positions of the periodic orbits. And what they observed is that this pn converges exponentially fast to p infinity. And the rate is 1 over 2.6, I think 6 squared. And then, so this day first, they did it for the quadratic family. But they were hoping for universality because they believe in physics and they believe. So they did it for the sinus family. They did the same picture. And they calculated the positions of this periodic point. And they saw again that it converges exponentially fast. And the rate is the same number. And so this is phase space universality. Universality of dynamical geometry. And so this starts to smell a little bit. Remember yesterday, we constructed the conjugation between a circle diffeomorphism and a rigid rotation. And it had something to do with the cutting place. Had to be on the same place each time. And that created rigidity. And so the fact that these periodic orbits, we will come to that later, the fact that these periodic orbits are at a very specific place, asymptotically, is an indication that we are going to get rigidity, something equivalent to the Hermann theorem. OK. So these are the observations. And then, the Coulain-Tréssère, they were physicists and quantum physics. So they know they have to look for a fixed point. Somewhere in some space, there exists a fixed point. And that fixed point is describing what is going on here. And we are going to discuss precisely how that works. So for today, what we will do is we will discuss what renormalization is in this context. And then tomorrow, we will see how it connects these dynamics. And so of course, it has something to do with spirit 2. So let me say how it works. So let's be a little bit precise. So our space of maps are unimodal maps. So they look something like that. There's some critical point here. And let's say that this behaves, this is 0, 1. This is 0, 1. And let's say that it behaves as a C3 map. And let's say that at this point, the second derivative is negative. So the critical point is not degenerate. It has a nice sort of quadratic shape on small scale. So this is the space on which we are going to look at renormalization. So inside here, there is a space of renormalizable maps. And let me tell you by this picture what these maps are. I shouldn't screw up. So a renormalizable map. So renormalizable. So it looks something like this. It looks something like that. So this goes here. And then this goes here. And the critical point is here. So you look at this interval and you look at this interval. So what this is, you have here the critical point. Then this point is the critical value, which is value 1. Then this point is the second critical value. And then this is the third critical value. And then this is the fourth critical value. And the map is renormalizable if these four critical values are in this combinatorial situation. So the third here and the fourth between the critical point and the third. So these are renormalizable maps. So what is important about them is that this interval is mapped to that one. So this one here is mapped to that one. You see it happen immediately. And this one is mapped back. So you see here an interval of period 2. So this was our map F. Let's make a picture of the second iterate. So you remember, if you renormalize, you have to choose a domain and you have to look at the first return map. That's always the scheme. But the delicate thing is, which domain do you choose? So what we are going to do here is we are going to choose this interval as our domain of renormalization. So let's call this v. We might use it natively. So there's a u-interval and there's a v-interval. So let me draw them here. Here is u. And here is v. And you can sort of check easily that the picture of the second iterate looks something like that. No, that doesn't look right. No, no, no. That the picture looks something like that. Sorry. You can check it. So now you see that this thing here is the first return map to u. So this is our renormalization. So rescale. So you rescale this one. And you get a new map. And that you call the renormalization of f. And so it is a r of f is a rescaled version of the second iterate restricted to u. So it's all everything we used to do. OK. I'm not sure how much time do we have. You know, let me finish with the difficult picture. So it's about how this operator behaves. And you will not be surprised. So there's a theorem. And it says that r has a unique fixed point. And that is some unimodal map. And it has some properties. It's an hyperbolic fixed point. The unstable manifold as dimension is 1. And the stable manifold has code dimension 1. And what is important, that the stable manifold is exactly the class of infinitely renormalizable maps. And this is a topological class. Because having this order of critical points is a topological condition. And so you know that this is, again, we identify, like in the rotation case the other day, we identify the infinite renormalizable guys, which is a topological class with a stable manifold of renormalization. 15 minutes. Oh, it feels like Lundstein. You know, OK. 15 minutes. Wow. And this insight of Coulain, Treser, and Feigenbaum happened in the late 70s, like 77, 78. And now we have this theorem. It took a long time. The proof of this theorem has a long history. And let me name some players behind this theorem. So of course, Coulain, and Treser, and Feigenbaum, you are behind this picture. Oh, I should make the picture. And so it looks something like what we saw the other time. It is something like that. Stable manifold and the unstable manifold. And so Coulain and Treser, Feigenbaum, are behind conjecturing this picture and using this picture to understand the boundary of chaos. So Bidendurg is quite a list of people who helped establishing this in some little parts, some very large parts. So the first proof of this theorem was by Landford and Epstein. This was a computer-assisted proof. So then Sullivan came in. And he introduced Teichmuller's theory behind this renormalization operator. And he got convergence. And then there was MacMullen, who understood the expansion of renormalization and many other things. Then I helped in constructing fixed points of renormalization and periodic points of renormalization. And then the final of the present picture and a very beautiful picture is obtained by Avila and Lubitsch. So this is the place where you should go for understanding this theorem. No, no, no, no. You can answer that yourself. And you know, that is a very good, that is a very, that's a such to the point question. So let me first answer why it is no. So if you start with a polynomial, then you have to look at the second iterate. So the degree becomes like double. If you have a polynomial, you put in another polynomial. The degree gets much larger. So this fixed point of renormalization is definitely not a polynomial. It is an analytic function. It is analytic. But it is quite a mysterious thing. And tomorrow when we co-discuss the geometry of the attractor of this thing, you will see that it is a very rich thing and something you can clearly not make by hand. It is a very rich object. It's very specific. And that shows that in this context of unimodal maps, the situation is going to be painful. Because remember, when we did the circle diffeomorphisms, like the first observation was that the attractor are rotations. So you know, like the first observation, you know already the limit of renormalization. And these are just affine maps, so something very simple. And the fact that the limits are so simple allows you to develop a renormalization theory for circle diffeomorphisms with elementary tools. So in this context, the limits are complicated object. There are some analytic functions. So this is an indication that this is going to be quite a story. And I think this is, I would say, among the most beautiful pieces in dynamics. So the next part would be to have exactly what we did in circle diffeomorphisms, to use this picture to go through all the parts of dynamics. And you will see again that we can understand everything in terms of renormalization. And the most interesting part, so let's go home. And let's go have lunch, and then tomorrow we start with the blobs. It's come here, so let's go home. Bye.