 Een goede avond. Laten we recallen wat we doen waren we discussieën renormalisatie van circlediffiummorphismen. We hebben renormalisatie en we zien alweer dat renormalisatie je geeft om de typologie van onze circlediffiummorphismen te geven. Het is ook gevoeligd om de measuretheorie te constructeren, te constructeren in verschillende measures. En het is ook gevoeligd om iets te zeggen over topologische klassen en de bifurcatie pattern te gebruiken. Dus wat is er aan de linker, en we doen dat vandaag, is geometrie. Dus vandaag gaan we de circlediffiummorphismen finissen en dan de rest van de week gaan we aan de geometrie van handonmapsen. Maar laten we kijken wat we betekenen met geometrie. Laten we eens denken dat we een soort circlediffiummorphism hebben. En wat betekenen we met de geometrie van de circlediffiummorphismen? Er zijn veel geometrieze vragen die je kunt vragen. En bijvoorbeeld, een geometrieze vragen zou het zijn, laten we een punt nemen en laten we het itereren. En op een gegeven moment, de return, laten we dit 0 nemen, op een gegeven moment gaan we het return. De dichterste kun je krijgen en dan een beetje later returnen je even een beetje dichter. En dan later returnen je even dichter, etc. En je kunt vragen zelf, wat is de verhaal van deze verhaal? Hoe snel, wat is de rate, is er een rate van verhaal van deze returnpunt? En dat zou een geometrieze vragen zijn. En er zijn veel anderen die je kunt vragen, maar bijvoorbeeld om een idee te krijgen. Dus er is een theorem die een geometrieze vragen zorgt dat je kunt denken. En dat is een theorem met Herman en laten we kijken wat we krijgen. Dus we hebben, laten we zeggen, een C3 circledifium. En laten we zeggen dat de rotationnummer van F is gegeven door de continuïntracties van F. Laten we kijken wat er is. En laten we zeggen, zodat we weten dat we een homeoconjugatie hebben, F, met de rigide rotation. Dat is gewoon een conjugatie. En dus onze homeomorphisme, dit is een homeo. En dus deze homeomorphisme vertelt ons alles over de topologie we willen weten. Maar nu, Michel Herman proeft dat, eigenlijk, als de a n dus niet te snel naar infinity gaat, dan h is c1 plus alpha difium. En dat betekent dat h is verschillende en eigenlijk h, en het is invers. Het is verschillende en de derivatie van deze homeomorphisme is alpha hulder. En dat betekent dat je er wat controle hebt over hoe de derivatie veranderd is. Laten we recallen wat dat betekent. Het is een soort lip-sitz-condition. En dus, en nu, suppose dat je een C3 difiummorphisme hebt en de a n's niet te snel naar infinity gaat, dan is onze a n's een C1 difiummorphisme. En dat betekent dat op een heel klein gebouw dit conjugatie map is fijn. Dus dat betekent dat op een klein gebouw, een klein gebouw dieometrie van f is dezelfde als een klein gebouw dieometrie van de rigide rotation. En dus, als je hier wat je hier ziet, op een heel klein gebouw, dan kijk je dan op de correspondingspictuur voor de rigide rotation. En dus, wat we zien is dat de klein gebouw dieometrie van een difiummorphisme is dezelfde als de klein gebouw dieometrie van de correspondingspictuur van de rigide rotation. Onder een conditie. De a niet te snel naar infinity gaat. Dus dit gaat onder de naam van rigide. Een klein gebouw dieometrie kan niet bevormd zijn. En je kunt zeggen, oké, laten we onze difiummorphisme nemen en we zien deze orbit gaan rond. En misschien, door de difiummorphisme te schipen, kan ik vertrouwen hoe deze orbit is accumulatie. Nou, apparaat, dat is onmogelijk. Dus een klein gebouw dieometrie van een klein gebouw is rigide onder deze conditie. Oké, dus laten we het proeven. Laten we het proeven in het simpelste geval. Oké, dus laten we zeggen de proeven van rigide. Hermann Theer, rigide. In het simpelste. En dit is het simpelste geval. En dat is als alle a's zijn 1. Nu plus, minus, plus, minus, plus, minus. Oké, dus laten we kijken hoe we dit doen. Nou, remember, in this case, renormalisation has hyperbolic behavior associated with... Miss this row is 1 over 1 plus 1 over 1 plus et cetera like this. En remember what was happening again? We had 1 rigid rotation and that was renormalised to another rigid rotation. And here we had the family of rigid rotations. It was an expanding curve and the same curve will pass by to the other guy. And moreover, this is the expanding part, and there is also a stable manifold in both pictures. And now our map we are going to look at is sitting somewhere here. And remember that from renormalisation we learned that the topological classes are stable manifolds. And so we know that our map is lying in the stable manifold. So what we get from that are two things. The renormalisation of F converges to these two points. It will convert that and then come back. And this convergence will be exponentially fast. It will be exponentially fast. So let's make a picture. Let's make a picture. We know. Let's make a picture of the end renormalisation. And that is some difumorphism. Let's make a picture. So something like this. And somewhere is the critical point. This of course depends on where you are in the renormalisation. And somewhere is a vn. And somewhere here is the diagonal. So it converts to the rigid rotations. And for the rigid rotations this is a straight line. Let me first make the end renormalisation of this one. And it will also look something like that. And now there will be a c and there will be a v. And that will be exactly a straight line here and exactly a straight line here. En de c-nummer en de v-nummer are fixed. En zo en nu de end renormalisation looks very much like one of this one. En het betekent dat deze brand hier is almost a straight line. Er is een beetje twigel in het, maar niet veel. En de andere kant ook. Het is almeas een straight line. Dus je ziet dat de deel tussen deze foto's is, zeg maar, dat op de n. Dat is iets exponentieel klein. Dat kan een beetje meer precies zijn over dat. Dus wat we betekenen is dat... Hier hebben we onze vn en hier hebben we onze v-star. Dus de foto's zijn exponentieel dicht, maar dat betekent dat in particular c-n en deze perfecte nummer zijn exponentieel klein. Dus dit is 0, 1. Dit is 0, 1. En deze punten zijn bijna op hetzelfde plek. En hetzelfde voor de vn. De vn zijn exponentieel dicht naar wat het is supposede te zijn om de ritzende rotatie te geven. Oké. Er is nog een andere plek van informatie. Lemmen we dat we aan het renormaliseren maar er was een andere tool en dat waren de toren. Dus laten we de toren maken. Laten we de toren maken van f. Dus we doen hetzelfde. Hier hebben we l-n, hier hebben we r-n. En dan hebben we onze toren. En hier hebben we de n-trenormalisering van f-rho. En dat heeft ook een toren. En er is een l-n hier. En er is een r-n hier. En er is... een toren hier. En er is een toren hier. En natuurlijk, in deze toren weten we dat als je de bodem hebt en je naar een bepaalde niveau gaat, dan is deze map f-vijn. En dat is de ritzende rotatie. Dus alle iteraties zijn iets f-vijn. Dus als je naar de bodem gaat, naar een bepaalde niveau, dan krijg je een f-vijn map. Dus van deze exponentiele convergente van bekijken in de proef van dit weten we dat in dit geval deze map exponentieel dicht is. Dus dat betekent dat de toren ook dezelfde lijkt. Niet alleen de positie van deze twee punten en de twee branders, de pixel van de renormalisatie, maar de touwen, zijn ook exponentieel dicht. Oké. Dus nu gaan we renormaliseren. En wat was renormalisering? Je moet het nemen. Dit zal belangrijk zijn. Dus laten we red gebruiken. Dus wat je moet doen is renormaliseren. Je moet een van de gebouwen kutten. En de kuttingpunt is van vn of vn-star. Oké, dus laten we hier v-star en dan zie je dat de kutting precies in dezelfde positie gebeurt. Als deze had de deel 0,4 van het hele ding, dan zou dit ook 0,4 van het hele ding zijn. Want deze map is ook oké. Dus nu gaan we onze v1, vn en we moeten ook hier kutten. Dus we moeten hier vn en we moeten hier kutten. Dus nu gaan we kijken wat we krijgen van deze en deze informatie. En de fact dat deze iterate van de bodem op een niveau is al oké. Dus let's take this interval en let's zoom it out to the unit interval. So somewhere here we will find v-star up to scaling we get to the same size. And the proportion of this to that is the same as the proportion of this to that. And let's do the same thing here. Let's take this interval and let's zoom it also to unit size. And then the cutting point will be sitting somewhere here, vn and this distance will be exponentially close. So the place where you are going to cut in the tower of F to construct the new renormalization is exponentially close to the cutting place of the construction of the next renormalization of the written rotation. Oké, so this is about the towers but there is a circle. So let's use the towers to go back to the circle and look how the conjugation goes. And remember how we built the conjugation. Let me make a picture and you will remember. So all these intervals here and those they are actually lying in the circle and they form what we called the dynamical partition. And here we also have the corresponding dynamical partition of the rigid rotation. En the conjugation has to be something like that. Whatever the conjugation does it has to take corresponding pieces to corresponding pieces. In particular this level has to go to that level. With that this picture as if you take this level this has to go to the corresponding level there. So all layers are preserved. En we use that to approximate the conjugation. So let's see how that goes. Here is going to be a graph this is going to be the graph of our conjugation the approximations. So let's take this little interval and I don't know where this is in the circle but let's say it's somewhere here. Let's say it's somewhere here. Somewhere in the circle. En let's take its friend the corresponding interval of the rigid rotation. En we know that the conjugation has to glue them together. So maybe here is the interval of the rigid rotation. En now we know that our conjugation is something inside here. There is also stuff here but we don't worry about it for the moment. But remember that we constructed the sequence hn converging to our conjugation. En hn by construction was made in such a way that it is just a straight line on this level. Ok, so now let's make a picture of hn plus 1 now that you have to do by taking the cutting point here this is exactly this cutting point and you have to take the corresponding cutting point of sorry you have to take the corresponding cutting point in the interval of the rigid rotation en those have to be matched together so the next these have to be matched together and the rest is fine so this is hn plus 1 and this is hn and now we are almost done and because these points they are in the middle 0.4 or something 0.6 and now you know that relatively they are exponentially close so that means that that the distance here is exponentially close but because they are sort of sitting in the middle these slopes these two slopes are also exponentially close to the slope you had before you see that so here you use that these points are exponentially close we are using this piece of information and we use that this guy is nicely sitting somewhere in the middle it's not crazily in the border it's sort of nicely in the middle and so then you really see you see this picture the next approximation has a slope which is exponentially close to the slope of hn and so let me write down what we got at first we see that hn plus 1 and hn in just in in distance is exponentially close but we also see that the derivatives is exponentially close of course you have to be a bit careful because in the corners they are not differentiable and so this means where the affine pieces they have exponentially close slopes ok and now from this exercised for tomorrow it is really not difficult but to do it right you have to deal with techniques here exercise for tomorrow that hn that the dhn converges to dh which is continuous that's the exercise if you take the limit of course you can see that this is not going to be a big deal and you see almost uniform convergence of the derivatives so you will have a candidate function which is going to be the derivative and then you have to show that dh is dh in the sense that you get a limiting function and the second part of the exercise to show that this limiting function is in fact the derivative it's all little things but don't forget we are doing dynamics and we are dealing with the analysis of dynamics and here you see it's not going to be difficult but there is some analysis involved and the third part is showing that this derivative is holder and that is also not very difficult because what you know is these intervals they swing down exponentially fast you have to believe me you can see this from the picture at the mesh of pn goes to 0 exponentially fast maybe not with the same lambda but with some rn which has something to do with the cutting place but then you see that the fluctuations on the scale of dh on the scale ii is lambda to the n so you see an exponential a fluctuation of order lambda to the n on a scale of r to the n so what we need is that lambda to the n is proportional to some rn to the alpha and like the fluctuations should be bounded by the distance to a certain exponent and then you erase the n and you get an alpha so like the little technical part which you will have fun doing it is sort of nice you can finish the rigidity theorem ok so let's go to the bad news I remember so the theorem of Hermann it says that if a n is not too fast dan rigidity h is c1 plus alpha en nu er is een counter-example misschien niet een counter-example maar er is een example en het is gemaakt bij Arnold en dat zegt de volgende er exist een analytiek analytiek analytiek circle-diffio het is infinitief renormalisable en de invariant-measure remember we were able to use our renormalisation and the loops you remember to construct an invariant-measure and moreover we proved in the circle case there is a unique invariant-measure so this is this invariant-measure of our circle-diffiumorphism and that one is singular with respect to the back-measure singular with respect to the back-measure so it's a measure concentrated on a set of measure 0 ja if there is something to do with being not necessarily now what can happen is that you see you have to make a little correction here and if we have so this was a plus renormalisation you see we are cutting on the plus side so suppose that we have to do plus renormalisation for a long long long long time so that means that the next time there will be a cut here and there will be a cut here and these proportions will be sort of the same but then you have to start to make this thing here and then there is each time a little error but if the number of times you have to repeat this procedure is much larger than the errors you have the thing can blow up ja, so an an is exactly the number of cuts of right cuts and so if you have many cuts to do here then there are exponential small errors but you have to do tons of cuts and that can screw up the thing what will happen is if you have to do many cuts here then eventually this guy will be sitting almost in the corner your suggestion but then on this side there could have been many little errors and it could be that that point is still way behind and then you see that that the conjugation has to take this little piece and has to put it in something really big and that means that the derivative is going to be enormous and so that's something to do with accumulation of errors now in our case we had an is 1 and so we know that the cutting point is like the Fibonacci number we know exactly where it is if an gets large then this guy gets close in the corner here and then it jumps uniformly through here but then on the other side it starts to jump around but it starts to accumulate errors and then the thing is screwed up and in the example I'm going to show you is it's really going to happen but you have to take the A and it's very large it's going to be ridiculous to infinite it's like you cannot imagine how fast it goes yeah yeah yeah you know what is going to happen is so I made this picture here and your orbit is just coming through and these are the consecutive plus cuts these are the plus cuts and this is sort of and this is not sort of this is a uniform cut so each little piece has the same length that is for the rigid rotation so in this example what is going to happen is that the cuts you are going to see here are like this so it starts out nice and here there are like really a billion guys and then slowly it gets then suddenly it gets out again and so and of course if you take your conjugation these clues these pieces to that pieces it will be very tiny pieces compared to those which have an enormous derivative and the thing blows up and this can happen if you get by accumulation of errors so there is a very precise condition there is a very sharp it's like the Bruno thing it's a very precise condition so this comes from this formula where lambda is the the rate of conversions a rate of renormalisation conversions and this r is the geometrical decay and this is how the renormalisation conversions and this is how fast those intervals are shrinking down and the alpha comes in exactly in this equation so the the holder exponent has a very precise meaning so do we still have do we still have time or what time do we 5 o'clock ok so let's discuss this example the idea for the example is this picture somehow you have to do something like this first remember that let's see what happens so take your analytic and look at the family and it is the one we had in our first exercise you just rotate a little more and then the construction is by induction of course renormalisation level to renormalisation level so each renormalisation level you are going to adjust how much you shift a little bit so let's say we are on the end renormalisation level and we see a picture like that something nice here and maybe something nice something nice like here this is the picture of the end renormalisation and now comes the first adjustment and so let's say this is F T N so now comes the first adjustment in this parameter and the first adjustment is just to push it up until you create a tendency so the first adjustment is going to be something like this something like this here we still have our thing and now it is something like and we created a fixed point and of course we had an analytic map so there is only a finite number of those periodic points yeah you just rotated no it is yeah but the key to the exercise was monotonicity so if you increase T then this graph will start to move up this at least T this is sort of a monotone now yeah it's not obvious but it is topology you can do this by purely topology no, because here we use that the guy is analytic so a guy with an analytic map cannot have an interval fixed point so this is the only moment where we use that the guy is analytic so that is the first adjustment so now look this is a periodic point and let's draw the whole orbit in the circle and so this is the whole circle and now we just have a different morphism this is a periodic orbit and you see that on the left side it will be expanding it will drift away but on the right side you are coming in and every point will be coming in eventually and like of course if you take a point on the right side very close on the left side it will take a long time to enter from the other side but eventually that will happen so you can take an enormous iterate and take tn enormous such that almost every point lies in this very tiny let's say this is of the size 1 over like very tiny and the whole thing here is very tiny a total length is smaller than like 1 over n but after this number of iterates almost every point landed up somewhere on the right side and so after that number of times you see that 99% of the points after this number of iterates ended up in this very tiny little neighborhood and so that means that if you think in terms of measures the measure is supposed to describe how the orbit is distributed so you see at this moment this is essentially the discrete measure concentrated on the periodic points which describes the distribution so it's extremely singular it's sitting essentially in a finite set of points and more singular than that you cannot get it so and then the second adjustment you know what to do now remember this is this is the picture of the entry normalization and you remember that this graph is like a certain iterate of the original map so if you go back to the picture of the original circle you will see rn points ok so there is really a periodic orbit and almost every point after 1 million steps is unbelievably close to this periodic orbit it looks singular and now you know this was the first adjustment now let's make the second adjustment and you know what to do so second adjustment second adjustment let's go back here we have our we have our map something like this and something like that and now let's make an adjustment this adjustment is really unbelievably fine and what you do is you do this you know what you do and now you do this it's the same picture except that you erased the periodic point so what is really happening here if you zoom out a little bit so if you zoom out this picture what you really see there as the diagonal comes through and your graph is something something like that and then you see so the periodic point is gone and now what happens is that your points are coming in but this is so small it's going to take forever essentially a and steps to come through this little tunnel and so you will see in that picture that you still have almost every orbit concentrate along this periodic orbit if you look at this time and now you repeat and on the next scale you make this even smaller if you go on then you will see smaller and smaller sets in measure with carry 99,99,99,99,9% the next time even more and in the limit you will get a singular measure it's a dense set because it contains one point has a dense orbit so this is a funny thing it's a measure which is sitting I cannot imagine how this looks like beyond this picture that it is essentially sitting on pseudo periodic orbit something like that let's go home