 Goedemorgen. En we continuen. Bevor we continuen met renormalisatie. Er zijn notes, handwritten notes, op de webpage. Dus als je naar de schedule gaat en je klikt op de titel, elke dag zijn er notes van de vorige dag. Oké, dus let's recall what we were doing. We had this very simple idea of renormalisatie. En wat het is, het is like a microscope. En wat was going on again, you have your dynamical system. And then you choose an area where you want to turn on your zoom in your microscope. And then you look at the first return map to you and then your rescale. And you get a rescale version of you and that will be in general, that will be X again. You have to set it up and set it away. And then you will have the renormalization of F. And so you should really think about this as a microscope. En wat we are going to do is, we are going to repeat this renormalization process. So we will do a little piece here and we will zoom out here. So that means that you start repeating your renormalizations. That means that you start to zoom in to a very tiny part of your original dynamical system. You should really think about renormalization as a microscope. A very simple ID. We then have turned out that it turns out to be, this very simple ID turns out to be, in the cases we understand, I always have been disclaimer, in the cases we understand. Most cases we don't understand. But in the cases we understand, renormalization is an extremely powerful tool in the sense that it connects many parts of the dynamics. So you can describe the topological aspects of the dynamics in terms of renormalization. There will be major consequences for the geometry in terms of renormalization. It's set up to do that of course because it is a microscope. It's perfect to study small-skilled geometry. Then it is also connected to the method theory. The final big branch is the bifurcation theory. Oké, so yesterday we did a lot of blah-blah about this. And it might be vague, so I apologize for that. So the first part of the course is going to be on-circle diffios. And that is, I think you can say, the simplest case of renormalization en in that circle you see this interaction between the different aspects of the dynamics in terms of renormalization, very clearly. So I would like to spend one or two or maybe three days on-circle diffios. It's the simplest situation to describe but you will see this interaction in a very clear way. So let's go. So let's first describe the space of systems we look at. So D will be met from the interval to itself. And you should think about them as something like this. And of all of them, so there is somewhere a point C and there is somewhere a point V and you will have a branch here and you will have a branch here. And what we want is that the pieces are piecewise smooth. Like this branch here should be a smooth map and this branch here should also be a smooth map. And sometimes we want to say how smooth they are so we can write a CK. It might be once differentiable, it might be twice differentiable and then we attach a K here. And then there is another condition and we discussed it yesterday. It's a stupid condition, it's an equation. And how was it? It's like the derivative in C plus. Like the derivative here times the derivative in zero and it should be the same as the derivative in C minus times the derivative in one. And it's a condition which reflects the fact that these maps are circlediffomorphisms. And without this condition funny things happen. So it looks technical but it's important. En dan let's see how we can define renormalisation. Let's go to D plus. We'll be defined on two subsets and you will get a new map of that type. And what is D plus? That is where the C, say D minus is where the C is smaller than V. Let me define you how the renormalisation works. So what you do is you have your map C and there is V and here we have our branches. And now we have to define renormalisation and the first step is to say where you are going to zoom in and so we have to say what is U. And in this case the U is this interval. En so U is 0V. En now you have to construct your first return map to this interval. So we need to draw the graph of the first return map in this box. So you see that the right part is already part of the first return map and the last part you see you have to map this interval outside but then it is mapped back immediately and you get something like this. So this is the second iterate. So this is the first return map which is like this little thing here and then we have to rescale back to the unit interval. So what we do is we take some affine map and this is like an F. So I is from 01 to 0V. It's affine orientation preserving. Of course there is only one way to do. So that's the rescaling. And then the definition of the renormalisation of F will be just, now it is going to live here so what you do is you take, you go to where you have to be you do this return map and you come back. So you do the scaling. You do, let's call this head, the return map here and then you have to scale back. So this is the definition of renormalisation in this D minus case. This is a very explicit construction en if you like you can write down four formulas for this. It's extremely explicit. So that is in the case of when the C is smaller than V. So let's go to the case when we are in D plus and that is where C is larger than V and then you see a very similar situation. So here we have C, here we have V how is it? So we see something, there is something here and there is something here. In this case we say that U is the interval V to 1. We have to construct the first return map. I'm doing something wrong. Oh, sorry, of course I do something wrong. We have to look at the first return map to the interval V1 so we have to look at this box. Sorry, we have to look at this box. So you see that this part, the right part is already returning. So here we just have F and on the right part you see I'm doing something really wrong. What am I doing wrong? Sorry, I'm scoring up here. Let's do this again. I'm sorry. So let's take our map and we have our C here and we have our V here. So we have V. Sorry, I understand. Ah, ok, thanks. This is also going on wrong. Ok, now you see. So this is V and the map is something like something is... Ops, now no. Something is sitting here and something is sitting here. That's how it is, ja. We want to have the first return map to the interval V1 so this is this one and now you see that on the left part it is just F en on the right part you see you go out and then you are back immediately so you get something like that. You see F2 here and then you reskill this again like this is your affine map and then the renormalization is exactly the same thing. So you use your affine map to go here you apply this first return map and you get the same formula. So this is the formula on the minus side and this is the formula, the same formula on the other side. Ok, so now we have defined our operator on the space of circle diffeomorphisms so there it is and now the goal is to explore this operator in all these branches. So let's start with the topology. Now you see you see the renormalization is only defined on the parts where C is smaller than V and where C is larger than V so there are also maps where C and V are the same so it's a little observation a note that if C is equal to V and so you see something like this and now this is the same V so you see something like that and you see something like that and so what you see is that the right interval goes to the left interval and then it is going back and then you see just diffeomorphism on the little interval and then you know that everything converts to a fixed part and so you see en maybe just an exercise for later this afternoon at the limit set so every orbit converts period we will be just jumping around and so this is really in this case and where we didn't define the renormalization the dynamics is very boring and you just converts to a periodic point so what we are really going to look at is say the set EI these are the infinitely renormalizable guys and that means for all N RNF is defined so where you can repeat this process forever and you never fall in the case where the C and the V are the same so these are the interesting cases if you stop at some point then the renormalization will be of this simple type and you will see that almost every point for every point converges to a periodic orbit ok so that is nice and now we get already a topological invariant and so what you can do is so let's take F which is infinitely renormalizable then you can define a number sigma N of F which is just a minus or a plus and the way it is defined is in the edit so RN of F is part of the sigma N of F so if you start to renormalize then sometimes you are in a minus case sometimes in a plus case so what you can write down if you have your F then you can start to record a minus situation and then how many times you are in a plus and then how many times in a minus and then in a plus and how many times in a minus so so this is the sequence of F you just keep track whether you are on the minus or on the plus side and now of course if you renormalize you forget where you were in the beginning so if you renormalize you just cut off a symbol and so what you get is something dynamic is like so we have our infinitely renormalizable maps we can repeat this process forever and we renormalize them and we see a new infinitely renormalizable map and now here we have this map sigma this goes into plus and minus sequences of plus and minus and here you code this is like the coding theory of renormalization and now here is just the shift and so if you take a map you see its sequence and if you renormalize you just forget where you started and you just shift the sequence and this commutes ok for people who know about entropy this is entropy this is like the full shift over two symbols this is a chaotic process so renormalization is not a completely trivial from a topological point of view is not a completely trivial exercise and remember that yesterday we had an exercising class and that we know that renormalization from d plus and minus to d is on to and that means from this you get that these codings are on to renormalization really has is a chaotic process by itself now yesterday we proved in the class in the afternoon that renormalization from each part is on to ja kijk if you don't do the exercise then but I didn't do the exercise but let's suppose that is true and that you can get every map that means you can follow that of all maps so d plus is where the c is larger dan v en d minus is where the c is smaller than v so at the moment there is no almost no we have only this formula so at this moment so the construction has nothing to do with smoothness this is all working for homeomorphisms there is no problem whatsoever no problem this is all for homeomorphisms you have a very interesting question so this this whole picture in topology is not as strong so smoothness and you will see later like maybe even today smoothness plays a crucial role everything is only going to work if our maps are twice differentiable so but let's go slow so we get some topology out of this renormalization operator um so now you can you can do a little bit more we can put it as an exercise this afternoon you have to think a little bit but this is like constructing by hand you can do this let's put it as an exercise okay so let's construct out of the sequence even another sequence en dat is going to be important so you can you can count how many minuses there are of how many how long the first block is then you can count how many is the second block and you can count how many is the third block and the fourth block etc so and then we get a new number we get the sequence of numbers so if f is infinitely renormalizable so if f is a map you can repeat this forever then you get a sequence a of f which is the sequence a n n larger than 1 and now you can do something even you can define a number row of f which is the continued fraction of this thing so it is like 1 over a1 1 over a2 plus 1 over a3 etc so and this is called a rotation number okay and so for people who looked already at at circle maps this is what you call the rotation number generally but now we introduced the renormalization of the rotation number by using a renormalization process so the typical topological invariant is already visible in the renormalization process I wanted to do but I wanted to do that as an exercise this afternoon no no it's too you know it's not so difficult but it's also not completely easy so if you take as a circle this is a rigid rotation over a certain angle over an angle like a row if you take that rotation then you can you can figure this out you can do this it's not completely easy okay so now we are going to be a little bit so now something interesting happens so we have our space and renormalization zooms in to a little piece en we zoomen in even a smaller piece smaller smaller pieces so initially we think okay if that is true then from our renormalizations we will only get information around this one point where we are zooming in but the little things you are looking at are first return maps so there are pieces which go around the circle so if you follow those pieces you are actually getting a picture of all the relevant dynamics so out of these renormalizations you can construct a picture of the whole dynamics and this is what we call dynamical partitions and so the goal is have recover the whole dynamics out of the renormalizations that's the goal so let me show you how that goes so let's see how this goes so let's take a map see be a bit careful so let's suppose we have we have something like that so this is our original map and here is our little v okay so the first renormalization it will be the first return map to this interval let's say this one this is the domain where we have to take the first return map and then that will give us after scaling that will give us the first renormalization but now so then we are in this picture and you see the v in that picture is this one so the next time we will have to renormalize to this picture so the next domain where the renormalization lives will be you too and now you have to scale this again and you get the second renormalization but now you see the next critical value is on the other side so the next time we will have to renormalize to this little interval so we get something like that that is the third interval and you rescale and you will get the third renormalization so if you have a map which is infinite renormalizable you will get the sequence of little intervals which are shrinking down and renormalizations correspond to the first return maps to those intervals and so what you see is so is f is an infinite renormalizable map and then there is a sequence of intervals and the end renormalization corresponds to the first return map you see it up to scaling it's always up to scaling that's also something these people who do renormalization in public they say you have to rescale but in their mind they never do that and you are just you are zooming in that's what you really need to do ok so now you also see the original point c is sort of in the middle so what you see is we have we have our interval un and sitting somewhere here is our c and you see the c cuts it the renormalization in two two parts and then what you see is like on the first interval we had the renormalization was like the second iterate here and then on the the yeah like on the next interval you would have like the fourth iterate here et cetera so what you see is that is renormalization so what you see is so these renormalizations they are first return maps and so what you know is that if you take this interval you iterate it a couple of times and then you are back and you iterate this one a couple of times it's the first return map and you will be back and what will happen is that there will be a certain iterate in interval n which puts it somewhere here and there will be another iterate on the right side frn which puts it somewhere here so it has the same picture as a typical circle map so this is up to rescaling a picture of the nth renormalization so what is important is the two branches of the renormalization are iterates of the original map and let's say on the left we have ln iterates and on the right we have ren iterates I don't know what these numbers are but they are there it's our first return maps okay so let's do the it goes back to Stefano's question the map from the space on to the sequence space is on to so you have to think a little bit about it but if you have on two maps you can take pre-images and they intersect and you use compactness and there is a limit so there are in particular if you take a richer rotation which is rotation number row then that row has a unique continued fraction expansion as you call it and that sequence of a's will be exactly the a's of the renormalization so the answer is absolutely yes okay yeah yeah yeah so what happens is I can write down the formula so it's a little less we have time enough so let's do a little so if you start with the a's and then you get if you have an infinite sequence then this becomes an irrational number but you can also cut off the continued fraction so you behave like what they call the the ends cut off of this continued fraction and it will be just 1 over a1 plus 1 over a2 plus an so you cut off the whole tail up to the ends position of course these are integer numbers so this is a rational number so this is something like p usually they write pn over qn okay and now what happens is and I think that's the answer to your question because like like this point is the image of this boundary point and that boundary point is the image so what I want to say is that these boundary points are in the orbit of the critical point so and maybe I should not get like this so what you get this ln is like qn and rn is like qn plus 1 that is how it works so at this moment how this return times are working we have an explicit way to do it but there is actually a very simple way to find those numbers and you can just read them off from the continued fraction and later a little later we will see that the number it's sort of interesting the number theory because you have heard about diomventine numbers and that has something to do with how fast these things grow and we will see that the number theory of the rotation number is going to have geometrical consequences it's beautiful it's really beautiful so a renormalization is going to connect geometry of circle difiumorphisms with number theory it's cool so you know like generally speaking if you have something like this in mathematics that you are working on a problem and to solve it you have to bring two very different pieces together to do it like to understand the geometry of circle difiumorphisms you are doing dynamics geometry and dynamics you have to get help from number theory so that's nice it's different and it's sort of this is a personal thing I like dynamics especially because I think you can say all branches of mathematics are needed to do dynamics of course topology plays a role of course geometry plays a role number theory plays a role analysis today we might see this analysis is going to play a role to call another branch of mathematics probability theory probability theory is going to be a play a major role so and this might be surprising even algebra algebra is going to play a role like symmetry and representation theory it is unbelievable I think this is exciting about mathematics it's beautiful sorry this was a personal thing you might have your own reason another reason would be that dynamics it started with Newton if it started with Copernicus trying to predict the motion of the planets and then Newton wrote down differential equations and then we had to understand these differential equations and then all this mess began and we are still not understanding it so another interesting thing about dynamics is that it has a strong relation with nature and what we are doing here and the numbers we are going to see they appear actually in nature so there is a strong relation in this physics and this is the real world so it's sort of a cool thing a very cool thing thank you for the question that was very no no we will come back to that a little later we will come back to how this number theory is connected to the geometry of the circle differential morphisms we were interested in our dynamical partitions we want to recover the whole dynamics we have this very tiny intervals with the two little pieces they go around the circle and eventually come back and this one also goes around the circle goes around the circle and eventually come back and you should think about it this is sitting really in a big circle our original map is a differential so these intervals this one maybe goes here in one step and then it goes around and maybe comes back somewhere here and it goes somewhere here and then it goes around another time and it goes around p and times and then it returns and this is the right part is something similar and so the right interval it will come somewhere here and it will come somewhere here and it will come somewhere here and then it will come somewhere here and then eventually it will return and so you see that the orbits of those pieces van die stukken starten ze in de cirkel te liggen, maar ze gaan eigenlijk om de cirkel te voelen. En omdat het de eerste returnmap is, zal het bezoek zijn. En dus deze renormalisaties gaan jullie een dynamische partij van de cirkel geven. Dit kleine returnmap kut de hele cirkel in veel kleine stukken. Dus don't try to imagine how they are really lying there. I'm not able to do that, it is sort of complicated. And it has something to do with the numbers. So don't try to imagine. So don't let your imagination go further than this picture. So just the interval flying around in the circle. How they are, it's really not so important. I will make another picture, which will show you sort of more interestingly what is going on. But let's first give a formal definition. So we have un is ln union rn. Let me be sloppy about whether it's open or closed intervals. So then what we say ln, that's going to be you take the left interval and you follow its orbit. So this is the collection of all those intervals. It is just fi of ln. Hever l, the iterate is 0, up to your back. Except for the last one, so ln minus 1. You see? It's just the orbit of this little thing. So for the same thing you have the r collection. These are all the iterates of the right one. Up to the moment you are back. And then let's take all of them together. And that is Pn. And that is the dynamical partition, which is just the left orbit union is the right orbit. And this is the dynamical partition. Can you read it? This is a bit high. So this is the dynamical partition. And then you see a picture here. The picture is complicated, but there is a schematic picture. And that actually has a little schematic picture of this picture. So that is the tower. So Pn is represented schematically by the tower. So what you do is, you take your interval Un, you cut it in the ln part, and you cut it in the rn part. And now you just make copies. So ln goes to one copy above it. It corresponds to the first iterate. But it is just a copy. You put it above, the second one, you put it above. And you make a tower up to, you make a tower, a building of height ln. Just copies. You should imagine the dynamics is something like this. And then it returns back. And the same for the right part. So you make a copy of the right part, a copy of the right part, another one, and you iterate, and it goes, and it goes. And this one might go a bit further. Even something like that. And then you get to the top. So it starts in the bottom, you go up, you go up, you go up. You go up, you go up, you go up. And then you return. En this has height rn. So this is the tower. You can really schematically represent this by something like this. Like a block. And a block. This is ln. And this is rn. So this is what you call the tower representation of this partition. This is a little historical thing. There is a field called ergodic theory. It's like the probabilistic theory of dynamics. And these people, they were using towers a long time before renormalization was done. So somehow, if you want to be precise about the history of renormalization, you should maybe look in ergodic theory quite early. It's not okay. Let's move on. Okay. Let me make some lemmas. The following lemma is going to tell you that this picture is actually a realistic picture. There are these joint intervals and they form stacks. So I'm erasing the real picture. It's too complicated. The tower is sort of easy to imagine. What you are doing is, you have the circle where all these intervals are flying around. And you cut it in pieces. And then you stack the pieces and you get something like that. It's just a cut-up version of the circle. So let me write down a little lemma. The first thing is that these pn are pairwise disjoint intervals. To be precise, the interiors of the intervals are disjoint. They might touch next to each other. So forget about that. So these guys are touched together and they are touched together here. Let's forget about that. So the interiors are pairwise disjoint. And the second thing, what is important is, if you take the union of all those intervals and you add where they touch, you get the whole circle. En so these dynamical partitions, ja, actually, of course you can. Let's not go there. Because now if you start to go to flows, you can do that. You can do that. Absolutely. You can do that. But let's stick to maps. Okay? Okay, so then, how are we doing with the time? You know, so, you know, actually, now so, the rotation number, ja, ja, you have to be able to renormalize infinitely many times. And that means your ace is an infinite sequence. And that corresponds to an irrational rotation number. So in the infinite renormalizable case, and this is always an irrational number. Like PQR. No, no, no. No, not at all. This is just topology for the moment. We welcome to that. There is a nice discussion. And that will lead to the notion of apiore bounds and nonlinearity and distortion. That will be the next subject of... And so far we... This lemma holds for homomorphisms. Ja, this is for homomorphisms. You know, let me... We are still at five minutes. So I think in five minutes, I can tell you why this happens. The cause of why the lemma is true. And that is... This is all combinatorics. So let me show you how this works. So what we are going to do is, subscribe renormalizations in terms of towers. And then the proof of the lemma will be by induction. So let's see how that goes. Okay, so let's see how this renormalization acts on towers. So let's see. So let's look and make a picture of the end renormalization. So we see something like this. Here we see u n. And here we see l n. And we see r n. And we see v. So we see something like that. See something like that. And now you see... So this iterate here is like the l nth iterate of f. And this one is the right iterate. En nu moeten we renormaliseren. We moeten dit renormaliseren naar u n plus 1. En we moeten kijken naar de eerste return map. En wat je ziet is... De eerste return map op l n. Je moet dit doen. En dan moet je dat doen. En dan moet je dat doen. En dan moet je dat doen. En dus dit iterate hier... Is dus f l n compost. Is f r n. En je gaat met f en l n. Je gaat hier. En dan moet je dit iterate doen. En dan moet je terug. En zo dit is gewoon f r n plus l n. En zo we zien van dit. Dat l n plus 1 is l n plus r n. En r n plus 1... Dat juist staat. Is dus r n. Dus dit is hoe dit werkt. En nu laten we zien hoe de touwer gaat. Dus laten we de touwerpictuur maken. Dus dit is u n. Dit is l n en dit is r n. En nu doen we het heel pictuurlijk. Dus je moet je... Je kritieel je v-value. Dus dat is een punt somber in r n. Dus dat is een punt somber hier in r n. En dan... Je ziet dat... De brand van de renamelisatie aan de reis... Is gewoon dit touwer. Want de volgende reis brand is gewoon de originele. Dat is een klein interfoul. En dit is een klein interfoul. Maar nu zie je... Dat de iterate op de linker... Dus je moet de linker interfoul nemen. En je moet het precies hier doen. Dus wat er gebeurt is... Dat deze interfoul... Eigenlijk gaat hier. En dan ga je op en kom je terug. Dus wat de renamelisatie doet is... Je hebt de originele en je blijft deze hier. Dus deze is deze blok. En laten we het zo draaien. Dus wat je moet doen is... Je moet deze touwer nemen. En je moet het gewoon op de top van deze. Omdat je het rate hebt. En dan ga je terug in deze touwer. Dus dat betekent dat je een touwer op de top moet nemen. Dus renamelisatie in termen van deze partijtie... Is deze kutting. Je kut deze deel af en je maakt die deel op de top. En nu zie je... Als dit de interfoul die hier blijft... Vormt deze deel op de cover van de hele cirkel... Dus wat je doet is... Je neemt al die stukken. Je kutt ze in twee deel. En je maakt die deel hier. Dus de nieuwe partijtie is weer een partijtie. Dus dit is een exerciceis. Dus in de avond... De exerciceis zal... Proven deze lemma. En dat maakt je gebruikt... Om hoe deze renamelisatie werkt. In deze interfoul. Oké, dus... Let's have lunch. En de noten gaan op de web. Dus als je wilt rekenen hoe de definitie was... Je zou er kunnen vinden. See you later.