 – Me našli postavilo retromalizacijo na rotacijo. Sreč je nekaj malo je posleda, da je to, da se je vse vse vserobilo, in zelo vsebo, da se predelimo potrebo, da je bilo na nekaj nekaj zelo, zelo vsebo nekaj zelo, nekaj nekaj nekaj nekaj zelo, nekaj nekaj nekaj zelo, ki ustavimo način začeliti nekaj, da je to, da je težko način, da je to, da je tudi nekaj nekaj nekaj nekaj, plugged or get about rotation for a second and just say present this as an algorithm. So algorithm for let's say some geometric conversion of the continuity fraction, geometric continuity fraction. And of course it's related to what we were doing yesterday, but I just ali jerelax veliko začajel oto ostavil razložijo, da sem yeni ricjel po crashed Jozen, veliko so prihalj lässtanje. Zarednja je ne n barba čavljenja, meni spotrečaj z error ot , dar se statistični un undravijo čadroga, i zero is the union. Delta zero, delta one. Picture here. Zero, alpha minus one minus alpha, delta zero. I already did it wrong. Delta minus one. I want this to be minus one. Thank you. Delta minus one. So we take alpha, we build these two intervals, we want to define. Now, inductively, intervals i n, which are made by delta n, union delta n minus one. This chalk keeps breaking. Okay, so I have two intervals, and let me always remark that as n grows, the intervals become smaller. So there's always a big, a large and a small. So in this picture, always delta n is the small interval. And delta n minus one is the large interval. And the position of small and large flips with n. So if n, let's do by case. Let's say if n is even. What did we have? This is the picture we had there. If n is even, n was zero. If n is even, then you have, the small is on the right. So small on right, delta n here, and delta n minus one here. And what I want to do, I take the small, every time I take the small, and I cut the small from the opposite endpoint. So here the opposite endpoint is to my left. And I start cutting as many full copies as I can, until I'm left with the reminder. So if you want to cut, cut, cut, cut. So let me write it, start and cut delta n from, in this case, the right endpoint, from the left endpoint, as much as possible, call delta n plus one, the reminder. So this little reminder is delta. So you take the small and chop it from the opposite side. And if n is odd, the same, but the picture is flipped. If n is odd, delta n is to the left. Delta n is left, n is odd, delta n is to the left. So I need to cut it from the right. So cut from right. And the same. Maybe this is already the reminder. So here there is no rotation, no nothing, it's an algorithm. You have two intervals, you cut the small from the left, you are left with the reminder, now the large becomes small and the small is the other and you do the opposite. You cut. Now the next step, you have the reminder and now you will cut the reminder from the opposite side. Then you have a new reminder and you cut the small reminder from the other side. And once you cut from the right, once you cut from the left, once you cut from the right, once you cut from the left. What does this have to do with rotations? And now let's connect it to yesterday. So if I look at the first, so let tn be the interval exchange, exchange of delta n and delta n minus 1. So this is just a map which swaps the two intervals rigidly. If it's a map t to the n, it's a map from in, back to in. And one is mapped to the opposite hand and red is mapped to the opposite hand. And remark, if I look at t0, let's go back to t0. t0 has length, i0 has length 1 and this t0 is nothing else than a circle cut open at alpha. And this map we already saw is just a rotation. So t0 is actually on from i0 to i0. If I identify this one with a circle, this is the rotation by alpha. Our usual rotation by alpha. So the claim is that these maps that I'm building are actually induced maps of this original rotation on this sequence of shrinking intervals. And this is what we kind of did in class, but also I asked you also in part to verify. So let me say proposition. This is again, I'm recalling essentially from yesterday. Tn induced map so first return map is the induced map of t0 t0, which is equal to alpha which is a rotation opened up on in. If you want to have my original rotation and then I have a sequence of nested intervals which are becoming smaller and smaller and containing zero. If I induce my original map on this small interval what I see is this tn. And now this is this idea of zooming. So I'm zooming on smaller and smaller scales. But now if I want to renormalize I need to open up this small interval. And maybe you can do it. First we can zoom it and then we can glue it back to a circle. So maybe you can do it like this. This is you have your small interval you have t to the n and you can kind of zoom it and rescale it to length one. So if you rescale so rescale in to length one to length one. So say you just multiply by one over the length. So you make it again length one and this endpoint will be actually this endpoint will be gauss to the n of alpha. And this will be minus one plus gauss to the n of alpha. This is what we showed that the ratio between the two intervals is gauss to the n of alpha. Rescale and then if you want you can glue glue back to a circle. And what you see is a rotation by gauss to the n of alpha. Ok. Is it clear what I'm doing? I'm inducing then making the map unit length and gluing it back. If you want yesterday I said the opposite. I said here you can glue this back to small circle and then rescale the circle of length one. It's the same in each order. So let me say again. So this is the algorithm and hopefully today it's more clear. I mean I'm not proving that it's the induced map but you can just sit down and again you did it in some sense yesterday yourself if you did the exercise. And let me recall you the idea which is this idea of renormalization so maybe you will not care about this specific renormalization for rotation in your life but in dynamics there are many instances the picture is kind of like this. You are interested in studying the rotation and you consider the space of all rotations or you consider the space of all dynamical systems that you are of the type that you want to study and in many examples we did mostly entropy zero dynamical systems in many examples there exists an operator so let's say x is the space of rotations and here you can identify space of rotations with zero one because you can think that a rotation is given by the rotation number so the space of rotations will be just the interval zero one thinking that each alpha is a rotation by alpha and this renormalization is indeed this operator here that zooms and rescales renormalization is induced induced plus rescale in this case induced plus rescale so for us here the renormalization of a rotation is really sending me to the rotation by gauss to the n of alpha so in this parameter in the rotation number parameter I am doing the gauss map so some people apparently yesterday asked why do we need to why do we need to do from right once and from left another time couldn't I just when I rescale I could also choose to flip the interval and then I always cut from the right so you can do that it's perfectly legitimate algorithm and it has some advantages to actually flip at every stage but I like not to flip at least this week I like not to flip because let me show you again the rotation picture so because remember in this rotation picture that we saw yesterday what I am doing I am inducing so I am first inducing on this yellow interval and looking at the Poincare map when I am doing this cutting from the right cutting the red from the right what I am looking at first return map on the yellow interval of the rotation so I need to look at iterates of these pieces to see when they come back so this is how do I work does it work is there a pointer no here it is so this picture this is what we were doing in the algorithm you see I have I have the red and the red to the left and the green to the right cut the red from the left of the green three times and I am left with the reminder and this is showing you what is happening on the rest of the circle this is what is happening on the rest of the circle when I am cutting from the left actually I am inducing so I am rotating until I come back here and then I am rotating until I am coming back there and then this is my new smaller interval with previously small and reminder and then I do it again this is kind of cutting to the left and then I have a smaller interval to induce and so on so I am inducing a rotation of smaller and smaller intervals and when you do this inducing you will notice that actually you are always rotating in the original picture clockwise but the induced map once it is a rotation backward and once it is a rotation forward so the induced map it is kind of related to the fact that convergence of the continuous fractions are one to the right and one to the left so sometimes you overshoot sometimes you undershoot and when you undershoot you have the effect of rotating back so that is why I like to keep the flip because it kind of records this backward rotation every once in a while ok but I want to move one step further so I want to explain that not only we have an induced map but we do have indeed some partitions of my original no I have some partitions of my whole space that I began with so this is a general phenomenon when I do inducing so let me tell you in general I want to tell you something about Kakutani or Rocklin Towers skyscrapers ok so say that I have a function a dynamical system from x to x assume that what did I want to say I want to assume maybe it's convenient to assume that f is invertible which will be the case for us assume that it's invertible and then you remember yesterday I look I can look at a subset y and assume that f of y from y to y this is the first return map or the induced map is well defined that only means that every point comes back so this means for every y there exist n such that fn of y is back in y for every y in y ry of n is the first return time I can look at the level sets of the first return time function so look at all points which take the same time to return so let me define yn the set of y in y such that the return time to y of small y is equal to n so these are the points that take time n to come exactly times n to come back ok, so then let me say as a fact if you want you can try to prove it as an exercise it's a little bit like similar to the Poincarere current exercise that Stefano left yesterday but it's a little bit annoying to check and you really want to use invertible the fact is that I can write my space x my whole space as the following union I will write it and then I will give you a picture of this union so I can write it as a union of n of the union from n sorry union from n union from k from 0 to n-1 of f to the i of yn I'll draw you a picture of this it makes it immediately clear what this means yes sorry it's a union and actually the claim is that this is everything and also that the union is this joint so everything that you see in this union all these intervals all these sets are per what is joint it's a disjoint union so every point falls exactly in one of those sets so let me show you the picture so the picture is that maybe I'll draw you first flat sorry so let me chose first you have my space x and then maybe my space x is very big and then you have y and then I look at some portion of y which comes back in time n so this thing means that it will go out this will be f of yn this will be f-square of yn this thing will be out for exactly n-1 iterates and then will be back so these iterates are here so for each n I take yn and I translate and I shift it exactly n times and this will be ah I'm sorry f to the k thank you so as k goes from 0 to n-1 I look so this part is so this part is what I'm drawing I'm drawing these are the iterates which are out and then come back and maybe let me draw it and this part is actually called a tower, a dynamical tower and there is a more visual representation which is this one so let me say your space I'm representing it as an interval and here I'm representing e1, e2, e3 dot dot dot en so e1 takes one iterate and then it's back in y y2 goes out of my space for one iterate so it goes out and then it comes back to y yn y2 goes out for twice and then it comes back yn goes out for n times and then it comes back I can kind of draw symbolically these sets as towers over yn so this is just this tower just represents the images then back for larger and larger so these things are called towers so the union for k from 0 to n-1 of fk of yn is a tower the union of towers so this whole union is many towers form sometimes this is called the Rockling tower or Rockling tower and many towers form a Kakutani skyscraper so basically I'm inducing and I'm representing my space as towers up to the first return time so the claim is that the Kakutani skyscraper gives you all points in the space every point has to be either in a base or in one of the floors in one of these towers so I'm trying to do a little bit of general dynamical system theory when I'm doing my specific example so let's go back to the rotation and maybe it's more clear what everything is so for example when f is the rotation and y is our interval in we have only ry has only two values we saw that it also only has two values qn and qn-1 the return time of maybe I should say that x is 0 and yqn is actually delta n-1 ny qn-1 is equal to delta n so these are only two intervals and one returns in time delta qn and the other returns in time qn-1 so there are only two towers in my skyscraper sometimes I see that people are confused if you are confused do you want me to ask and here we really have intervals so we can really draw towers so I can really draw this picture so this is delta n this is delta n-1 really I can draw qn-1 copies of delta n on top these are intervals and then I can draw qn copies of the interval delta n-1 yes sorry if so I'm actually assuming that the map is well defined that all points come back to y but then you will only have a unique tower for the return time some of these towers are going to be empty in that case you have only one tower which is the empty tower the return time will be sorry sorry sorry so you are saying I might forget some assumption if x is identity and y is some subset it's not invertible did I say invertible? oh yes who can find maybe I need some kind of minimality or transitivity or some kind of in the case of the rotation everything is fine because orbits are dense so this is irrational rotation sorry if the rotation is rational by the way this picture the tower will still work but at some point you will have some identity tower ok let me think I don't know he is worried about if I take identity and y to be some part of space but I'm not spawning the old x I'm not spawning x so there are I think you are right there are points which will never return and you have to put somewhere in that picture you have to add some part of space which is not swept in assumption like the map is sweeping or something that if you take the union of the images of y they contain x I think you are right I will double check but I think he is right so when it's an assumption yes no I really want to find x and that's why he is right that there is some issue so I don't only want to find y I mean certainly I can but I want to I think you need something that y is a swiping set for x which means that when I take y and I look at the images of y all the forward images of y contain x sorry but this example is fine so ok maybe ok so here I'm saying I have a rotation and I induce it on this set in this is my IN it's the base of this picture is IN and then I say look at the what is the return time of my rotation on IN the return time we proved it yesterday or it was part of the proof that we did yesterday that when I induce the induced map is an exchange of these two small intervals and there are exactly two returns one interval return in time qn in time qn-1 and I'm just claiming that if you look at qn iterates of one interval and qn-1 iterates of the other interval these iterates are all these joint up to their first return time and the union is all the circle but I will show you this so this picture is basically I can show you the same picture on the circle so these two towers the picture that we have been looking at at all time so let me go forward maybe so this is indeed the picture so the red is one tower and the green is another tower so just here I'm plotting the the intervals stacked upon each other in their dynamical sequence well here they are somewhere in space so this is exactly the picture is this union so maybe we can even write it with some notation so actually let me put some notation because I saw so let's call, I don't know let's call delta n i delta n1 delta n and delta n i is alpha to the i of delta n and the claim is that the circle the whole circle is union the whole circle is the union from 0 to qn minus 1 of delta n union this is one tower this is say the red and then there is the union from 0 to q sorry this is qn minus 1 minus 1 and this is qn minus 1 of delta n minus 1 i so this is another way to say this is the green sorry? no no no sorry maybe that's what is confusing enough here so this n and that n is not related this n is the return time and this n I'm looking at level sets of the return time function and here there are only two level sets which are non trivial the level set which corresponds to qn and qn minus 1 the n is like the height of the tower and here there are only two towers with two heights one has height qn and one has height qn minus 1 so this n is not the same n so this qn is one of the ns and the qn minus 1 is another ns all the other ns contain nothing because nothing returns in other time does it make sense so I just want to say this picture that you see has this green are all intervals which are images of the original green and the red are all images of the original and similarly here in the next level in the next level again I have my small green and my small red and their images up to the return time together span the whole circle and yet sorry, yet another one I have only a small red and a small green in my induced map but if I look at the images up to the time they come back here the images of the red and the images of the green all together span give you a partition of the circle does it make sense I said that I am confusing ok so this picture is the dynamical picture so if I go from the previous one there are also so this green maps here, here, here and back the red maps here it jumps then here, then here then it jumps to the next then it has a complicated dynamical structure but in this simplified pictures of towers I'm ignoring in which order they appear and I'm kind of stacking them in their natural order so in this picture the rotation actually moves up so this floor is the image by the rotation of the previous the next floor is the image I'm pictorially draw the images by the rotation on top of each other so r alpha moves up up to the top and when I'm in the top from this floor I need to go back to the base according to the induced map then back to base the induced map tn so this algorithm this renormalization produces for you three things it gives you an algorithm one of us short and long intervals and it gives you these dynamical rockling towers it's actually nice to also see what is the algorithm on the towers so now I will do it's a little bit boring to draw it's a little bit annoying to draw the floors so let me draw a rectangle for a tower so you should imagine that this rectangle has floors but I will stop drawing the floors for the rectangles so we have two towers delta n is here delta n minus one is here and actually sorry let me make the second tower larger let me not draw all the floors and again this tower represents the dynamics the base at the base I have the induced map the original so r alpha moving everything up and then coming back according to this map so let me make a connection has anybody ever seen finite rank transformation and cutting and stacking just one so this is a construction which is very much used to study many kind of dynamical systems dynamical so in general you can have many towers countably many but dynamical systems which admit representation with finitely many towers are sometimes called finite rank so the rotation is an example of a rank 2 dynamical system because you can represent it as a sequence of towers so this is just the algorithm that we are going to see now on towers this is actually an example of cutting and stacking so many people in dynamics in symbolic actually in spectral theory many areas use cutting and stacking so this would be a special example of cutting and stacking so I want to say what are we doing on the base do you remember the algorithm of the base on the base what I'm doing I'm taking this interval small interval and I cut it from the right as many times as I can so this operation I was doing it only on the base but actually you can do it on the whole tower like this you can take the small tower and cut so cut the small tower from the opposite side so what does it mean this small tower I cut it here I did too many copies because I'm not going to be able to stack it ok my example is too big so let me make it a little bit smaller so I want it big but not too big so let me say that my tower is like this so I cut actually let me cut for the reminder I'm lying a little bit on the size so I can cut three chunks of my tower of equal width and then I'm left with the reminder this is the reminder tower from opposite cut small towers from opposite side so this is what I mean the base was fitting a whole number of times before having a reminder and I'm just cut do you have is it clear what I'm doing I'm cutting my tower at the points where I was cutting the base and now the algorithm is called cut and stack cutting a stack so what am I going to do next I'm going to stack I'm going to cut so stack them all stack them on top of each other so what am I going to do I still have a small tower I still have a reminder the small tower was blue the reminder was red and what am I doing I'm kind of stacking the first time that I cut I wish I cut the movie actually I don't have a good movie I stack it on top of sorry I cut the piece I stack it on top of the small one I cut the second and I stack it on top I draw you the picture so let's call this is one this is two, this is three and this is reminder and I'm going to stack one, two and three and now what do they have and I stack them in the order in which I cut I cut and stack I cut and stack so let's actually check so this height was qn-1 this height was qn how many towers do I cut how many times can I cut in terms of continuous fractional entries a an so I cut an times because we did it yesterday we checked maybe an plus one sorry an plus one ok, so what is the new base and what is the new height the new base is qn plus one and qn this height is the same that I had is still qn what is the height of the big tower the big tower has qn minus one which was the small tower plus an plus one the big tower which was qn so what is qn minus one plus an plus one times qn everybody knows new terminion this is qn plus one so you see this cutting and stacking gave me the picture at the next stage so essentially this so this is stage n and this is stage n plus one and what you see so this is a representation of the rotation the whole rotation on the whole space as an induced map on this interval in on the base so this is how I see this union is the circle and this dynamical towers show me how points move under the rotation they move up until the first return map this is also a representation of the rotation so the union of these towers is still the whole circle well I'm just cutting and stacking so I'm just moving my intervals around but I still have the whole circle and here the induced map is the induced map on a smaller interval and then giving a dynamic representation of how the rotation act as induced map on this interval so to induce on this interval I go out up to the height of these towers and then come back and why does it work, why this cutting and stacking I'm doing nothing else than following the dynamics because if I start from the small blue if I start from the small blue my rotation moves me up and then where do I come back well I have to go back according to the exchange and then I come back here then my rotation moves me up when I'm at the end of the first copy my induced map moves me exactly here so maybe I will take a color like a yellow or a red so my rotation my dynamics is like this up up up back here up up up up and then back here up up up up back here and that's it so that's why I'm stacking because stacking is following the dynamical movement okay so it's this so wait, when did I start and when I'm supposed to finish I think I'm supposed to finish now so this is a very concrete example in the case of rotation that you can build with your own hands but it's really an example in dynamics that is used very often in this inducing are two fundamental tools in many in the study of many dynamical systems so we spent quite some time to do this cutting and stacking and so let me just tell you tomorrow I want to give you some applications so I will give you a hint of several results that you can use you can prove using this renormalization and cutting and stacking I won't give you details so so that's the plan for tomorrow so thanks