 Good morning. I would like to start with an administration remark. There will be lecture notes on the course on partial hyperbolicity. It will be on the web page. And there will also be notes about the renormalization course, but they are just handwritten, so don't expect too much. I think you can use the notes for the renormalization to recall the definitions, but it is handwritten. Okay, so let's begin. So we are going to talk about renormalization, low dimensional dynamics. And we would like to discuss renormalization in any dimension, but it will be very low. We will be discussing one and two dimensional dynamics. Okay, so what is this renormalization? We start from scratch, so don't worry. So what is renormalization? It's a very simple idea. So you just take your dynamical system and you have your map. This is our state space and our map. And then what you do is you just choose some subset and then you look at the first return map. So the renormalization of f is the first return map. And this is the little tricky part. A well-chosen u part of x. So what you do is you take a point x in u and you iterate it a couple of times and then it returns. And that is the how you define the renormalization. So you get the renormalization of our f. There will be a map from u to u. But then it was off because you could take this u very tiny. So if you don't rescale it, you might not see anything. So you rescale also. You go back to sort of a unit size. And then here we see really our renormalization. So there is a rescaling involved. And just to get back to unit size. Okay. Okay. So you see this is a very simple idea. But in the cases we understand, so in most of the situations we don't understand at all how this behaves. But in the cases we do understand, something nice happens. So renormalization unifies the main aspects of dynamics. So if you here have renormalization, so if you do dynamics, you would also like to understand the topology of your original dynamical systems. And very often, in the cases we understand, renormalization allows us to give a topological description of the dynamics. This is vague but we will see examples and then this connection will become very precise. Another part which is closely related to the geometry. And that is easier to understand. Because you see this renormalization, it takes like a ball on a small scale and you see what happens. You blow up. So renormalization is like a microscope. And a microscope is perfect to understand geometry. And so the first thing about renormalization is really understanding geometry. So then there is another part is measure theory. And so the existence of relevant inferred measures. And another part which is missing is bifurcations. And so in the cases we understand, renormalization connects all these main branches of dynamics. And that is a nice aspect. And a little bit surprising that this very simple idea of using a microscope sort of starts to connect. You can easily say the main branches of dynamics. We are going to discuss concrete examples. But let me clarify these connections a little bit. So let's say something about topology. So if you do, if you choose a piece where you are going to look in, a smaller piece, you associate to that a type of renormalization. And so this what they call this topological information, the type of renormalization, the type. And there is combinatorial information. Combinatorial information. And for example, it could be that here you have our original system. And it could be that our set U and the return map to U is something very simple. It could be that the set U just goes to another set and then it returns. And so it means that the renormalization, here is the renormalization of F, as it is just the second iterate of the map restricted to this U. Yeah. No, no, no. So no, this is like the simplest case. U goes and it returns in one step and the whole thing. This is the simplest of the simplest situations. And in general, it will not be at all like that. In general, this return map will be something complicated. But you will see examples where this will turn out. So this picture illustrates that a renormalization has a type. And this is topological information that says that the renormalization is built in such a way that you take the piece, you map and you come back. That's a specific type. And clearly that is combinatorial information. So now in the cases, we understand. And so if F and G have the same type, a renormalization type, and lately we will see precisely what that means, then F and G topologically conjugated attractors. And so here we have our map F and here we have our map G. And you can renormalize F and there is a certain type and you can renormalize G with the same type. So then there will be an attractor here of F and there will be an attractor here of G. And then there will be a conjugation between the attractors. And so H of F is, and so apparently the renormalization type implies the topological structure of the attractor. And that is sort of the machinery how this works. If you know the renormalization types, you can figure out what is the topology of the attractor. And again we will see concrete examples of how this works. So let's go to the geometry. Okay, so let's take again, so F and G have the same renormalization type. And then what happens is that if you start to take repeated renormalizations of F and you start to take the corresponding renormalizations of G, then in the C1 topology this converges to zero. And so if you start to mean, what the means is that if you have your map F here and you have your map G and you start to zoom in here, say this is U1, here we have V1 and now we start to zoom in here U2 and we start to zoom in V2. And it continues like this, so you get a very tiny UN. And if you blow this up, you will see the end renormalization of F and you go very deep here and you go to end level and you zoom out, you will see the end renormalization of G and then they are almost the same in the smooth topology. But that means that if you look in the geometry of what is living here and the geometry of what is living here, these are parts of the attractors so we are zooming in to a very tiny part here and we are zooming to a very tiny part here at the corresponding place, then the pictures are the same. And so apparently this convergence of renormalization implies a certain universality of geometry. And so you get here, it's almost the same on small scale, it's universality of geometry. So this is sort of a key ingredient. And maybe not everywhere but this is at least in the place where we zoomed in, we see that we see the same geometry. So it can be, so this is I should have written this larger because this is really a crucial aspect of renormalization. You see convergence of renormalization gives you universality of geometry but sometimes it goes even further. So in the picture we zoomed in on a very tiny scale here and a very tiny scale here and the geometries are the same. So that means that this conjugation is differentiable in this point. So it might happen that it can be, there's a notion of rigidity, that this conjugation from the attractor of F to the attractor of G is differentiable. And so if you start to renormalize, if you have systems which you can't renormalize, then sometimes it happens that the attractors they get a universal geometry everywhere. So these are crucial notions of universality and rigidity when you're talking about renormalization and we will see very concrete examples of this. So let's go to the measure theory. So measure theory. So again in the cases we understand the renormalization type, if you know the renormalization type and it will allow you the construction of invariant measures, of relevant invariant measures. And what we want to have is that if you start to average a function and then that converges to the integral of respect to the function. This is called the physical measures. So in the cases we understand the physical measures can be constructed out of only knowing the renormalization types. And again we will see examples. So this connection is also there and about the bifurcation pattern. That is sort of where everything began. You know, if you have a question or if something is clear, let me know and we can try to make it clear. So what happens is, so clearly this renormalization operator, what it does is it takes a system and it produces a new system. And so what renormalization is, it is sort of made to study the space of systems. Because that's where the renormalization acts. So we have the space of systems and renormalization acts on that space. You take a system, you zoom in, you get a new system. And in the cases we know, in the very simplest cases, you see another dynamical system here. And renormalization by itself is a dynamical system. And in the cases we understand, it is one of the simplest ones, namely hyperbolic ones. And so what you very often see is this picture, you see, the dynamics of renormalization has a fixed point and it has some unstable manifold and it will have some stable manifold. It's a hyperbolic picture and the holy grail of renormalization is to prove this picture, hyperbolicity of renormalization. And then the key thing is that is, again in the cases we know, this stable manifold of this fixed point is the same as the topological class, as a topological class. And that means that if you know the renormalization types of your system, that will define the set of all systems with the same type. It's a topological class. And it turns out that that has a manifold structure. And this is something that's quite the result. So we discussed very briefly how the main questions of dynamics are related, are connected by renormalization. And in the cases we understand renormalization well, this picture is working fine. OK. So the reason why this picture is working fine is that renormalization comes with so-called renormalization phenomena. And we mentioned already universality and rigidity. And so in this phenomena by itself they are interesting and fascinating. And let me give you an example, a simple example of a simple and surprising example of this intrinsic renormalization phenomena. And I'm sure that you have all seen this. But let's speak about period doubling. And let's first speak about a period doubling verification. That's what we have is we have a family. And what we can plot is we can look at that we can plot the parameter. And what happens is at a certain parameter value, before that value the dynamics is controlled by an attracting point. Like if you take this low parameter value, what you see here is something very simple. Ft. And you're just sitting somewhere a fixed point and everything converges here. This is like the simplest type of dynamics you can imagine. But now you change the parameter and you get to this specific value t0. And then what happens is that this periodic point is attracting, stops being attracting and it becomes something different. It's still a fixed point but not attracting anymore. And it splits into a period two orbit. And so in this case what you will see is if you look at this parameter and you look at the picture that there will be like two points which are exchanged and things will converge to this period two attractor. So this is a period doubling verification. And I'm sure that you all have seen this picture in your first courses in dynamics. And then there's something, the period doubling cascade. And so I'm sure you have seen this picture. It's sort of a famous picture. And so now we have our family again. And what happens is that we start with our attracting point and then there is a first moment where we see the attracting point bifurcating the period two attractor and then a little later the period two attractor will split into a period four attractor and then a little later it will bifurcate by a period doubling bifurcation in period eight. And it will continue like that. And it will be a limiting value. And very often this is the boundary of chaos. And so if you push your parameter a little bit further the system turns chaotic. And so this is the period doubling cascade and it is like the period doubling route to chaos. This is the place where renormalization was introduced in dynamics. And we will get to more details about this but not at this moment. So what happens is, and it has something to do with the bifurcation pattern. And so if you have, and this is real world. So this is in real world systems with this friction. Then what you see is that you see all these period doubling bifurcations and they will convert to the boundary of chaos. And what you will see is that these bifurcation moments they convert to the boundary of chaos and they convert exponentially fast. And the rate is one over 4.669. Something like that. And this is in real world examples. So you can build electrical circuits and if you turn the voltage up then at some point it will create a period doubling cascade and it will turn chaotic. And then if you measure the parameters where you get the bifurcation you see 4.6. People measure this. And there are tons of mechanical systems with friction. And in electrical circuits they even found something like this in fluid dynamics. This is a little bit more tricky. So this is what they call parameter universality. For obvious reasons. Apparently these moments where you are bifurcating a period doubling bifurcation that the rate doesn't depend at all whether you are looking at an electrical circuit or a mechanical thing or even a fluid dynamics. There is something universal in this parameter behavior. And this is surprising. So apparently you have the renormalization connects the typical questions we know in dynamics but it gives itself new phenomena. There are intrinsic renormalization phenomena like parameter universality. This is by itself a very interesting phenomenon. These are things you can actually measure and people do measure this. Okay. We will come later to the history. More precise history of this fact. So let me show you the plan. So I apologize. So far we have been discussing like in a very informal and very sort of vague blah blah story. I have no illusion about it. You just have to let it come over you. This is just phenomena. But we are going to make this more precise in examples. And we are going to make the list of all the examples we understand well. Most of the examples we don't understand. But we will discuss all these phenomena in particular in this picture with examples. And that is what we are going to look at is the first thing is we look at different morphisms of the circle. And you are going to renormalize those things. You are going to look at unimodal maps. And this is our map. Something like that. Three. You are going to look at Lorentz maps. And you know that it's like the butterfly. You know how is that again? You know the butterfly. And you are going to look finally at hand-on maps. Okay. So this is clearly one dimensional. The dimension refers to the dimension of the underlying space. Unimodal is one dimensional. So Lorentz is... You know you can reduce this flow to a map on the interval. But it is a funny interval. So this is sort of one and a half dimensions. No, there is a serious reason. It's not a joke. Of course. There is something going on here. And then of course hand-on maps. It's really 2D and even 3D and higher. But we concentrated 2D. And this renormalization stuff, it is sort of difficult. And the difficulty is making this picture precise. This is the heart of the matter to prove hyperbolicity of renormalization. And it is something like... That is the easiest... Renormalization of circle-different morphisms is the easiest one. A little bit more difficult is Unimodal. And then Lorentz maps is still very difficult. And hand-on map is a little bit understood. A little bit. And so the complexity sort of goes down. It's becoming more complicated when you go in this way. Circles are simple. Unimodal more complex. Lorentz, you see. How are we doing this at the time? Ten minutes. So let's start to discuss circle-different morphisms. It's a simplest case, but we will need one or two days to discuss this. So it's not that simple. And also probably many of you have seen already the circle case. So it's like the first example, also to refresh a little bit your memory if you have seen it already. Okay, so let's do a circle-different. And so hopefully many of the blah-blah things will become more precise in this discussion. Of all the blah-blahs should become precise in this discussion. So we are looking at maps from the circle, the interval to the interval, and they look like this. This is 0, this is 1. So somewhere is what they call the critical point. And somewhere is the critical value. And it looks something like that. And so you're looking at maps which are something like that. There's a branch here, and there is some branch here. And what we want is that f is piece-wise smooth and monotone. So this piece is something smooth, and this piece is something smooth. So that's important. And then there is a little tricky thing which makes it into a circle-diffumorphism. But you want this. Oh, I made the wrong picture. Sorry. I made the wrong picture. Sorry. Let's do this again. And somewhere here we have the critical point. And somewhere here we have the critical value. And we want the two branches to be separated by the critical point. So then the second property we need, these branches should be a little bit related, and we want that the derivative of f on the plus side of the critical point times the derivative in zero, which is this derivative. It should be the same as the derivative on the minus side times the derivative in one. Yeah, it's really a circle. Yeah, because so let me say it like this. If you start a circle-diffumorphisms, and then of course you just a circle, and you could open the circle, and you get the interval, what you will see is that these brands, and these brands, they really sort of continue. It's a small thing. And this thing here, and this thing here is really the same. So they are circle-diffumorphisms. But now if you renormalize, and you will see the next step will be renormalization of such things. If you renormalize, it will not be like this anymore. It will not be anymore that this thing is just a continuation of the other side. There is sort of a knick. So here there is a knick happening. And also there is a knick happening here. And what we want is, we want to have a space of systems, and our renormalization acts on that space. So now you see if you will be really working with circle-diffumorphisms, if you renormalize, it's not a circle-diffumorphism anymore. So we have to enlarge the space. And if you do this, if you do this, if you use this definition, then that will introduce a space of systems which will be invariant under renormalization. That's a little tricky thing. Not tricky, but that's what you have to do. And you will see why this happens. Okay, so, yeah, this is like zero, and this is one. This is one. Oh, now you take the derivative here, and you multiply it times the derivative in one. I think it's correct. It's like this is that, and that is that. I think it's right. Let me introduce renormalization, and you will see why you need to do this. Okay. So renormalization. Exactly, yeah. No, no, no. And so we want to understand circle-diffumorphisms. So we are going to renormalize them, but the way we are going to renormalize, it will be such that if you take a circle-diffumorphism, it's not a circle-diffumorphism anymore. And then we are lost, because then we cannot repeat the process. So we have to enlarge a little bit the space where we are living, and then this enlarged space of maps will become invariant in renormalization, and then we can start to talk about hyperbolicity. And the enlargement of the space is this. And clearly, if you have a diffumorphism, it matches. So the diffumorphisms are included here. Okay. So let me define the renormalization. There are two cases, yeah, unfortunately. So it can be that C is smaller than... You know, let me give some name. The space of the systems is called D. And if the C is smaller than V, we call it D minus. And then how do you renormalize that? And so here we have C. And here we have V. And so here we have our map, something like that. And remember, when you renormalize, what we have to do is, you have your space, and here we have F, and we have to choose a U. And then here we will look at the first return map to define the renormalization. So how do we do this here? Excuse me? Yeah, yeah, oh, sorry, yeah. We have to define smalls, monotone, and orientation. Thanks, yeah. And like this picture, this picture. So the U we use in this case, our domain that's called the domain of renormalization, is U, and it is just this interval. So we are going to make a picture inside here. So U is the interval zero, V. And now we need the first return map to that interval. Now you see that's easy. This part is already returning. So the first return map on the right is just the original map. It puts it back immediately. And on the right, on the left, you see, you have to do this, then you land here, but then you're back. So you will get something like this, and that is the second iterate. And now you rescale the bottom to unit size, and you get renormalization. So that's how you renormalize. And that is in the case when C is smaller than V, and in the other case, I guess that means to stop. Do you allow me to finish the definition, and then tomorrow we take it? No, you're gentle. I understand. So the other case when C is larger than V, and we call it the V plus case, and we have a situation like that, and it is, this is the picture. So we need something like that, and we have something like that. And you can imagine what to do in this case. The U is the interval V1. So now we have to look at the first return map in this little square. And now you see the scope to be exactly the opposite. On the left side, you can just keep the branch. It's over the return, and on the right side, you see you have to go outside, and then you're back. And so we will see something like the second iterate. It's very similar to this case, just flipped around. And so in that case, in that sense, we have our renormalization operator. We should stop today. Let's continue tomorrow.