 Goedemorgen. Het is oké? Ik oog veel van jullie een referentie. Het werkt niet, hè? Het werkt niet, maar ik denk dat dit de positie is. Hallo. Het is beter? Misschien is er een gevolg van dit. Nee, dat is het. Alles loopt, oké? Er is een red button hier. Hallo? Dat klopt, hè? Hallo. Gaat het goed? Ik zou zeggen iets, hè? Goedemorgen. Maar kan je me begrijpen? Oké. Laten we gaan. Veel van jullie vragen een referentie over de proof van hyperboliciteit van renabilisatie voor circlediffiummorphismen. Laten we het opnemen. Het is ook op de noten van vandaag. Laten we het opnemen. Referentie voor hyperboliciteit van circlediffiumrenormalisatie. En het is door G. Stark. Het is een lange titel. Maar wat is belangrijk? Ja, je kunt het uitkijken. Het is heel lang. Het is in non-lineariteit. Non-lineariteit. Nr. 1. En dat was in 1988. En de pages zijn van 541 tot 575. Als dit niet genoeg is, dan kan je gewoon naar de noten van vandaag, dag 8 en het hele titel zal in de top zijn. Dus je moet het kunnen vinden. Oké. Dus we hebben de circlediffiummorphismen afgemaakt. En de laatste tijd gisteren begon we te renormaliseren. Unimodal maps. En wat was de situatie? We bekijken de spas van unimodal maps. Het ziet er iets als dat. Waar dit iets is. C3. En waar de tweede derivative in de kritische punt is. Dit is non-zero. Dus er is echt wat kritische shape daar. Dus de spas van deze is de spas van unimodal maps. Maar dan zijn er renormalisable ones. En de renormalisable ones hebben de property dat er twee intervallen zijn, die worden geïnterveld. En er is een intervall u en een intervall v. En dit is geïnterveld met f. Dus als je een systeem hebt zoals dit, dan call het renormalisable. Dus dan zie je, als je renormalisaties doet, je moet de domain waar je zoekt. En we gaan deze intervallen gebruiken. En je ziet, de return map is gewoon de tweede intervallen. Je neemt de intervallen, het gaat hier en dat komt terug. Dus als je de graf van de tweede intervallen ziet je iets zoals hier, we hebben u en hier hebben we v. En het ziet er iets als iets zoals pad. Het ziet er iets als dat. En dan doen we, hier zie je weer een kleine unimodal map. Upsite-down. Als je dit in een f-niveau kan rescalen. En dan zie je weer een unimodal map. En we callen het renormalisatie van f. Oké. En dan hebben we gisteren de meeste theorem. Laten we gewoon het brengen. Dus de foto van dit renormalisatie-operator is precies wat je begint te zijn. Dus er existe een unieke fixpoint die een instable manifold heeft. Het is gewoon een curve, een 1-dimensionale object. En er is ook een instable manifold die een 1-dimension heeft. Dus en wat is belangrijk is dat de eigenvalue hier, de ongeving eigenvalue is de nummer dat we nog vroeg hebben. Dat is deze nummer 4.669. Ik remember gisteren we were looking at parameter universality en er was deze nummer 4.669 showing up. Dus dit is briefly de statement maar wat is belangrijk voor ons is dat als je deze stabele manifold deze manifold dat is exact de klas van infinitie renormalisable maps. En dit is een topologische klas. Dus als we terug naar onze diagram we hebben deze generelle foto van renormalisatie die de bifurcatiepattern die de topologie die de measuretheorie en die de geometrie uitleggen. En nu zie je van deze laatste deur van het hoofdtheorem dat de set van infinitie renormalisable maps die een topologische klas is gewoon een manifold. Het is een manifold een stabele manifold van deze operator. Dus dit explicent al wat over de bifurcatiepattern. De topologische klas van infinitie renormalisable maps is een manifold. En oké dus laten we zeggen een beetje meer over de dynamiek van deze dingen. Nou, misschien kan ik... Laten we deze eigen waal 4.669 gebruiken en explain waar de parameterdependant uit komt. Dus laten we een foto maken van alle unimodal maps. Deze zijn 3 maps met een non-degenerate kritische punt. En binnen daar er is een strip van de maps die een renormalisable is. Dus deze zijn de een renormalisable maps. En nu de foto van renormalisatie die hier is op een infinitie de foto. Dus wat is renormalisatie? Dus het zet de set van renormalisable maps en het maps naar iets zoals dat. Dus je begint te zien een hyperbolic foto. En daarnaast hier hebben we onze fixe punt en daarnaast hebben we onze onstable manifolds en daarnaast hebben we onze stable manifolds. En we weten dat de renormalisatie in de fixe punt een eigen directie en de eigen value is deze nummer 4,6 6,9. En laten we zien hoe dat nummer in deze parameter een universale foto is. Dus let's recall wat er gebeurt. En remember of you take your family if you take a family of unimodal maps then you can create kales, which we discussed yesterday and it goes according to a period doubling cascade. En we do something like this. First your dynamics is controlled by an attracting fix point. Then at some moment you get a period 2 point and then a little later you get a period 4 point. So it continues like this and then there will be an infinity value where the chaos begins. And then yesterday we discussed that these bifurcation moments where the period doubling happens converges to this boundary of chaos and it converts exponentially fast and the rate is 1 over 4.669. En so now let's use this picture to explain this parameter universality and you probably you see already how it goes. Let's do it with a little color orange. So this is our family here and this family, I don't know where it is but this may be sitting somewhere here. Some curve. En now we see that this strip are the once renormalizable maps but then there will be a thinner strip where you have maps which are twice renormalizable. And then there will be and then where the family enters this strip you have to check that is exactly this moment. So this corresponds to the moment t1 but then there will be a thinner strip where the maps are three times renormalizable and where the family enters this third strip it will be t2 and now you see the picture happening each time you go, if you have more renormalizations you get exactly these intersections converging to the infinity which is the intersection of this stable manifold. Ja, that is a big part of the theorem. And this picture is easy to draw it's just hyperbolic picture but to prove this is a big story. That is part of this theory ok But now you see if you have a saddle point in hyperbolic saddle point then you know that these pre-images they accumulate everywhere at the stable manifold is the item value of the fixed point you know from saddle point. So here you see explained why these bifurcation moments convert this is the item value of 4.6 Ok, so now this really explains the bifurcation picture and this is sort of where where the history began this 4.6 was the first thing they observed the tracer in Coulet and Weikenbaum and this is historically the beginning point. Ok, so but now let's go on to the rest of the dynamics 0 0 So you might think that this curve is the logistic family and here you have the infinite renormalizable map in the accumulation of period doubling and now we saw part of the theorem that the stable manifold is exactly the topological clans and so indeed these maps are conjugated and yeah and so this is like the third part Ok So you were saying that at some point you would describe fixed point the sum information we have about a fixed point and we'll give you some this fixed point is a complicated object and you cannot write down a formula it is some analytic function and it is something don't even try to write down a formula and hopefully today I can show you why it is something complicated and you cannot make this by hand Ok, so remember when we did the circle renormalization we had towers and we built a whole dynamical picture out of these towers of what are the towers for our period doubling renormalization how was it again so we start with this interval so u1 which goes here and comes back under two iterates within this map is again renormalizable so inside here you will again find two intervals which are exchanged by the second iterate and let's say this is interval u2 and that will be returning by f2.2 times f2.2 is f2.4 and then inside here there will be another interval this will be exchanged and it will be interval u3 and that will be returning into itself by f2.8 and so there are these intervals they are shrinking down that's a little theorem and they return by powers of the time return times of 2 to the n so you see that the tower from this you get the tower and the tower is extremely simple so here you have so here we have a picture of the nth renormalization like what we did in the circles and you see there is a bottom we call un and then there are all the images going up in the tower and the height of the tower is 2 to the n and when you come to the top you just return back so you see the nth renormalization is just a rescaling of 2 to the nth iterate ok, so remember when we did the circles the big tool was how does renormalization act on these towers so let's see how this works we do the same program as for the circle case and so how do how does renormalization act on towers and so let's see so here we have our little map and we have to take a u and we have to take a v and then we have to take the return to u now let's say that this is actually the nth renormalization and we have un here and we have vn so let's see how the corresponding tower looks like and it is something like this here we have un and we have 2 to the n layers and now inside here oh, I shoot this is un and this will be un plus 1 and vn plus 1 for the next renormalization no, it's a unimodal map and because it's a unimodal map so you start here which is this interval and there is a critical point and now from here to there you get something fold and then you just go up morphically and then you return the first step is the fold ok but now let's see where we have to cut so we have a picture of the nth renormalization and we want to make a picture of the next renormalization so apparently we have to take un plus 1 here en we have to take vn plus 1 en we have to cut along this the orbit of un and we have to cut along the orbit of vn and then we have to paste them together so we have here so here we have the part see the first part and here we have the part on top so you just take 2 substrips of the tower and you stack them in a different way and so now you get the tower of length 2 to the n and the bottom is un plus 1 and so renormalization is again if you look on the level of towers you cut off pieces and you reorganize them it's really the same as in the circle case in the long sense it is simpler because the tower is simple it's just one building so i think now we have all the tools to oh no, one more thing so let's let's start to discuss the dynamics of these maps and as in the circle case we get the layers of the tower which form our pn which is the dynamical partition is exactly the same as in the circle case we know there is a little problem that if you look at the intervals at the nth tower the intervals are pairwise designed but unfortunately the union of all those pieces is not the whole interval and you can see why that is true because when we cut off the pieces to construct an extrinormalization we forget the middle part so you cut off, you lose something so but at the same moment it is just an interval which runs around and then it is back so instead of calling it the dynamical partition we call this cn which is just the same as this partition is beside the nth renormalization cycle cycle just two to the n intervals which are running around through the interval in the case of the circle difgeomorphism it was a little bit complicated to draw the picture in the circle where all these intervals in the circle case it is a little bit complicated but in this unimodal case it is easy it is really easy and it is just constructing a counter set so let me make a picture for that let's see, let's first start with our interval en let's make the first the first cycle and now you know what it is, it is this interval and it is this interval and this one goes here and that one goes back so these two intervals form the first cycle but now inside here you are renormalizable so inside here there are two intervals and there is one here en they are exchanged by f to the fourth f to the two so that means if you take f they will get copies here taking this picture and you iterate by f and you get it here and now you see four intervals and that is the second cycle and now you know so inside here you will see again because it is renormalizable things which are exchanging under f to the eighth and this picture will be transported through through the cycle and it comes back here and it will be back and that is your third cycle and of course this process continues and in the end you will have some set here which is called B en which is just the intersection of all those cycles and of course it is going to be a counter set let's make a precise statement let me make a precise statement about the dynamics so a theorem so f is infinitely renormalizable and then let's see it will be indeed an intersection of all those cycles and then the C is a counter set and this is not completely obvious because you see how this is exactly the same as the construction of the middle third counter set you cut and you cut and you get two intervals it's the same thing but you are not sure that each time you cut off a definite piece so it's not like in the middle third counter set you always cut off a third and you cut off a third it could be that that you start off to cut out less and these intervals they actually stay large so you have to prove something that you always definitely cut off something and that is behind this part of the theorem and to do that you have to do again like you remember when we did circles with your morphisms we had to prove a key point was that the sizes of the intervals in the tower goes down to zero and to do that we needed nonlinearity and we needed that the map is C2 and so in this case we have a critical point and that complicates things and that's why we need C3 so there is some analysis behind here not difficult analysis in C2 probably a dust fill I think so Sarma knows I think it fills in C2 of course it fills in C1 but I think also in C2 another part of the statement about the dynamics is that this counter set is actually in a tractor so let me prepare that for every n there exists a periodic orbit and let me call it pn which is contained in the end cycle of period 2 to the n it is just some point which follows the tower and comes back so this is our periodic orbit so there are and the third point I cannot find that place so the next line should be under there but there is not so much space so the third point is that if you take an x which is not in all those periodic orbits if you take a point which is not periodic x is not periodic then the limit set of this point is exactly this counter set so every point which is not periodic will eventually get sucked in in deeper and deeper cycles and eventually it will be running around around this counter set ja ja ja ja ja, that is the same difficulty as proving that this is a counter set the existence is not so difficult and because you have an interval which runs around in 2 to the n steps and returns so you know the return map has a fixed point and these are disjoint intervals so the actual period is 2 to the n it is sort of simple and what will happen is I can make a proof you will see how it goes so you have your point and let's say after a couple of iterates you get into un so you get into un en nu let's look at the n plus first renormalization so what you have to do is you have to take the second iterate on this interval let me be brief let me not get very crazy about it so you get something like that so now your point lends in somewhere here but now this periodic point is expanding and now you see that once if you get in the gap you are not in the next cycle but now this guy is expanding and eventually you will get in the next cycle so you enter a certain renormalization level maybe in the gap but then inside this gap is this periodic orbit which is expanding this is like classical dynamics it's sort of this is not so difficult you need some tools you need to know what Swartzian derivatives are you need to know non-linearity things it's not difficult but you have to prove that ok so now we have a clear picture of of how the dynamics looks like and we see we see in our map this counter set and we know that this counter set is actually the attractor of the system so the dynamics is sort of clear so let's go and fill in the other parts of our diagram let's start with with the easy part the topological part topology and the theorem is it's the same as in the circle case so if f and g are infinitely renormalizable so they all have these towers they both have this counter set then f is topologically equivalent to g and that means that there exists a homeomorphism and this h of f is g of h so they are conjugated and remember in the case of the circle our towers filled the whole circle so the structure of the towers allowed us to build the conjugation as you know it is a little different our dynamical partition our cycles are not filling up the whole interval so that's a slight complication but the proof goes through and so for people who know needing theory and so it means that it says that there is a theorem by Milner and Thurston that says that if you know how the critical orbit how the critical point jumps around if you only know the combinatorics then f and g are the same and like from these cycles you can read off how the critical orbit jumps around en so this just follows from from an old theorem by Milner and Thurston in general you might have many pairs and the number of pieces might grow but the tower is always simple so we are done with this topology so let's go to the measure theory and so we see our orbits accumulating a discounter set en now let's see whether there is a measure which describes the distribution of our orbits so the measure theory and the theorem says that f has a unique invariant measure on our counter set and that means that indeed let's call it mu en dan if you take measurements and that will convert to our measure en you know the proof and by just looking at the towers and so remember that so the tower is a realistic representation of the dynamics we see intervals and intervals but we can do it more schematically by just replacing all the intervals and the maps by just a directed graph and so what you see is you get the end cycle and we just represent that by a directed graph of two to the end point and then there is the next cycle and of course that is included in the previous ones and you see you take the next one is included here and that will be represented by again a directed graph but now of to the end points you remember from when we did the circle it was more complicated we had two loops but now the situation is very simple so and the way this guy is included is just by winding twice around en so now you see remember we went this is the graph and then we went to the measure spaces which were just Euclidean spaces and we see R and we see 2 and we see R and in the inverse limit of these measure spaces we just see R and that means that the object here is uniquely ergodic so maybe somebody was not wrong last week so it might be a bit fast for that but we see and you use the combinatorial structure which you get from the renormalization schemes to produce the measure like in the circle case it's not just this that you also need to do if you take point outside the average is converged nah, that's okay yeah, but you still need to use the specific structure no, you know if you have something if you have a point which converges to something which is uniquely ergodic and then what you can do is you take your point and you put point masses on the orbit and then that measure on the piece of orbit will converge to some inverse measure on the limit of one, there's only one okay and this is the simplest situation you can imagine from a combinatorial measure here at the point of view nah, you know this is minimal because so it's minimal, you will get everywhere every point will accumulate we don't have the distance we don't have stable sense I think you do I think you do I think you do no, you don't it's complicated you don't you do it's the same thing yes and no I think you do I think you do and so what happens is I'm not 100% sure so if you take your point somewhere it's sitting here getting away from the fixed point and it gets in the cycle and then you start to jump around in this cycle so you could take any point and it would stay with a fixed distance to that point but then later you get to the deeper cycle this argument works I think what happens is I don't know I cannot really imagine probably if a point will get closer exercise, that's a good exercise it's not so difficult but I cannot come up with a definite answer you had a question say it again this is called an odometer it's like the limit of cyclic groups and if you go to other renormalisation schemes the limits there will always be a counter set and this counter set will always be the inverse limit of cyclic groups odometers ok, so ok, that's good so we are done with the measure stuff just a thing this measure is simple like this has a quarter this has a quarter this has a quarter so it's really a very simple measure and on the next scale this has an eighth so it's really the simple measure but the proof is the same as in the circle case so now let's go to I think it will be good let's go to the geometry where this excitement comes let's call it excitement now you will see where the complications are there is something going on here so let's there is something called scaling ratios if you take the middle third counter set then you take an interval and you cut off a third and this third is the scaling ratio but in our thing we don't know how this scaling goes so let's see what is going on so let's take the nth cycle and let's take one of our intervals there are many of them let's take like this one so then there will be exactly one sitting above so let's call this one I let's say this is this is our interval I and it is contained in one of the higher scale so let's call it T of I and then the scaling ratio of I I make it a big number it's important it's going to be nice it's simple, it's just the length of this I divided by something just how big these things sit inside here if you would be building the middle third counter set this would be just a third and now you look at at the set of a map F which suggests all these scaling ratios when you look in the nth cycle as you look at how these numbers tell you how the nth cycle is constructed out of the previous cycle so these numbers if you know all of them you know exactly the counter set and now you say sigma of F is the limit of this so how do you write that so you take the union of all those guys you take the closure and and you take the you know so this is the limit so all these numbers this will be something 0 en 1 en so this set and it is some subset of the unit interval and so all the limiting scaling ratios you can observe and this set describes like the deep the deep structure in this counter set and so how you this set tells you how asymptotically the construction is made I think I repeated myself ok and now there is a theorem let's first do an example if you do the middle third counter set you can define the same numbers and then you see that sigma n is just a third and you see that the limiting one is also just a third and so this set of the middle third counter set you see that this is a very simple scaling structure and this means very simple scaling structure you could call this a fractal and wherever you look you will see exactly the same picture now let's give you a thing about let's tell you something about how this set really looks like and one thing is that the set of scaling ratios is exactly the same as the set of scaling ratios for the renormalization fixed point r of f star is f star and so the way our counter set is constructed in an asymptotic scale is exactly the same as the counter set of the renormalization so f is infinite renormalizable and so our f is infinite renormalizable let's make a picture here we have our fixed point and here we have the stable manifold and our f is somewhere here and you know that these are the infinite renormalizable maps and so our f is infinite renormalizable we get an asymptotic scaling numbers and that is exactly the same as the scaling structure of the renormalization fixed point en of course that has something to do that renormalization converts there every renormalization remember is zooming in of course I only have one place but at least there should be something here which is inside there but actually they are the same so the asymptotic scaling structure of the attractor of an infinitely renormalizable map is exactly the same as the scaling structure of the fixed point so en nu the sort of the the mystery comes no, no, no no, no, no we are only doing pair doubling renormalization en remember we never talked about any other renormalization is everything is pair doubling dan is it not true no, no, it's not true ok so the fun part sort of the fascinating part is that this thing itself is a counter set so it is completely opposite to how you make the middle third counter set so in the middle third counter set you do everywhere the same and this counter set there is actually no two places where you do exactly the same so the scaling structure the way you build this counter set is highly extremely rich wherever you do something you have to do it according to that spot if you do it, if you zoom in somewhere here in a very different way and you can distinguish two places by looking how you are zooming in this is a counter set so it is an extremely rich scaling structure and this is what you observe what people measure in nature so you can imagine that in the middle third counter set you can make with your hands it's always a third, a third, a third in scale you have different numbers you will never be able to do that by hand but nature does it like that a consequence of this and you will not be surprised maybe that could be an exercise so if f and g are infinitely renormalizable and then you know that f is topologically conjugated to g and now if you look at the restriction of the conjugation to the counter set of f to the counter set of g then this thing is differentiable so we see rigidity again and you can imagine if these two counter sets have the same way of they are made the same way on an asymptotic scale and then you will see a fine relations between f and g so if you know this you can prove this ok, so let's continue tomorrow and then we start to discuss hand-on dynamics so the last two days will be hand-on dynamics and we will be mainly discussing the phenomena of sort of like today it will be difficult to squeeze out exercises things will get quickly complicated so let's keep this as an exercise for this afternoon so bye