 Fano, nekaj ne bo, ali je vseč v pošku. Znamo, da se počeš, da je tudi ozvodnjen organizacijo skučnega. Prepoč, da se počeš, da se počeš, da se počeš, da se počeš, da se počeš, da se počeš, da se počeš, da se počeš, da se počeš, da se počeš, da se počeš, da se počeš. Ami Wilkinson, kaj je z njega ziletiracijskih, ne bi tudi tudi tudi. Tukaj, Hanna Rodriguez-Hartsev, včešljamo drugi lektur in tudi sem tukaj. Zdaj smo tukaj, da imamo površenje tukaj. Tukaj, da imamo površenje tukaj, da imamo površenje tukaj. Tukaj, da imamo David, Irene, Oliver in Lucia. Tukaj, da imamo Kadim. So, they will be doing a lot of things during these two weeks in particular. They will be co-teaching one of the three courses of this week. So, the course Introduction to Ergodic Theory, they will co-teach. And they will also be key figures with all of us in the tutorial sessions. So, let me say a little bit about the structure of the day as you might have seen from the program. imajo 3 hrvosti lektučnih dnev, hrvosti Hannah, hrvosti mi, in hrvosti tutorskih, in hrvosti Amy na svoj svoj. A potem, prišlično, vse lektučnih. Na dnev, na 2.00 p.m. na svoj dnev, rekovamo tudi, na toj svoj svoj, na zašličnih. Tako, svoj svoj dnev svoj svoj svoj dnev na problemi. Vsem, da stresimo, as general say, in the evenings and in the weekends you have plenty of time to do tourism or to explore the city. We hope to see you here in the morning but even more important we really hope to see you. We expect you to be here every afternoon and try to work on the problems because really you should know this already. You don't learn mathematics if you don't do things, so you can listen, you can hear but you really need to do problems vsi nekaj izvajsno problemov, ki so na svoj pristini do competitorsi, skaj počan, ki so pribyt s tebe koncept vse, da ne jeGO, živ jo jekega možda, kot idem s neku nič nekaj. Jaz imam tebe vse podiv, nekaj nekaj, da sozajeli se vse, zato bodo vse zelo začeli na prave in na nas zelo zelo jel in na toga in, zelo ima izgleda, kako se tako puni, da bo in što, zelo izgleda, nekaj, da se tako čaturno skupi. Zvonča, da se prišli, to je vse vse, ne vse konferens, zvonča se postoči na mende. Vsi ne znamo, da sem tudi tako, da sem tudi, da so svoj, da sem tudi postoči na svoj več. In na terse, da se počuči, da se nači in izgleda, da je svoj, da so vse očil, da je to vse vse. In ne znamo, da se postoči na svoj več. ki se minuso za dve vrši, je in osev optimje. Zbili smo, da so nekaj skupel, da so njega nekaj vzivno. Zbili smo, da so njega nas nekaj način od雜i, da smo z njega všetneh izgledaj, da ste nekaj vzivne, da so nekaj vzivne, in da so tudi nekaj našli izgledaj. Zbili smo, da so nekaj vzivne, se prdešne upelje, trave, forbidite začučne prdrečenje, vsega skupanje, v početnih osvetih. Kaj je nočno, imamo še socializujte in všečnih ovdje. Koakat, nekaj potem sem post敛ala še socializujte z pomembnoj o podpravmi vrtečnih. ...If we have groups, we will explain everyday. And then there will be a presentation of ICTP... ...We will see the program... Another important part will be panel discussions. There will be panel discussion this week. And other panel discussion next week around topics like... ...being a PhD student, a successful PhD, applying for jobs as a PhD or a postdoc, becoming an independent researcher. So there will be different activities geared... Kaj so vzpevno na mjelj particov, se da se na najbolj delovim participacij, ali tukaj mi všeč vse pozdravili. Zgledaj, ki je to za všeč, da sem bilo površen, če pa je včasna, tako ki sem začala, da bodo tudi najbolj površen in maybe have seen more, maybe people, there are some experts that maybe should not be here. But this week is really for people who are new to dynamical systems. So we want to start from the very basics and give you hopefully really a good idea of some of the basic key ideas. So first of all let me write dynamical systems. So what are dynamical systems? Dynamical just means systems which are changing, are changing with time. So anything in evolution, which would be anything from the weather, which is changing, molecules of gas in a box, or the financial market, or a population in biology, all of these are dynamical systems. And so first of all, am I writing big enough for the last row? Can you still, should I like bigger? Bigger? OK, good. So dynamical systems can be pure and applied. It goes really the whole way of the spectrum. They can be very pure, they can be very applied. And change in time, and often they are chaotic. And as you said, apply dynamical systems. Maybe I can say they can be applied systems come from physics, from economics, from biology, from what did I forget? Astronomy, historically, planetary motion, and so on, and so on, and so on. And they can be extremely applied. But they can also be very pure, and they are kind of an active research area, which sits between, in some sense, analysis. We will see some measure theory will come into play this week. But they also have geometry, they also have interaction with number theory, and also combinatorics and probability. So as you see, and I can still put dots here. Actually, one of the things which fascinated myself when I was an undergraduate student and I learned about dynamical system is really how interdisciplinary the theory of dynamical system is, and how many tools from different areas of pure mathematics are used. So this is really what I like about dynamical system. And I really use, for example, geometry, probability, analysis. So there are connections with number theory, so it's really rich. So if you are interested in this part, I'm going to disappoint you, because we will really not go into any applied dynamical system in these two weeks. We will really stay kind of on this side of the blackboard. But if you are interested in that side, you will learn a good foundation. So you will learn the basic theory and some basic concepts that hopefully one day it could help you in study very concrete models. We will stay on more the pure side. And again, dynamical system have two other aspects. They can be discrete, or they can be continuous. Okay, so what does it mean discrete and continuous? It refers to time evolution. So here time moves in discrete units. First, this is already too small, probably. Can you still read something like this? Yes. So here time, and actually my n is the Italian n. So time in the time evolution is measured into integer. Time intervals could be minutes, seconds, years. But you look at discrete pictures of your system. And in the continuous picture instead time moves into continuous. So the time is a real number. So here the motion is encoded by points. Here the motion by being encoded by trajectories moving continuously. And again, mathematically this means iterations of a map. And this means orbits or trajectories of a flow, or solutions of differential equations. Trajectories of a flow, or solutions of an ODE, for example, a PDE, ODE. And again, about these two words. So we said we will all be on the pure side, and we will really mostly focus on discrete form. I think all two weeks. Maybe we will some point mention continuous, but then reduce it to discrete. So again, we will not say much about those. And what is a discrete dynamical system for us? A discrete dynamical system, as I said, is just from a space x into itself. And you should think of this map as describing, this is your system. So x is the system. So a point x in your space is the initial condition. Describes the time evolution. I will give you a basic example in a second. F is the time evolution. So if you look at f of x, this is the system after one unit of time. And maybe I will start erasing this side. So if I apply my map f, it tells me how my quantity of interest is changing in a discrete time unit. So for example, you could think that x is the interval 0, 1. And f is, I don't know, let me say this map, that I will not use again, but instead I will not do anything applied, but I will start with an example, which is motivated by applied word, maybe. No, not only. OK, so for example, it could be 0, 1. And this map, if you plot it, is parábola, down-facing parábola with the top at one-half. And this map is actually called, it's in the logistic family. It's a critical logistic map. And x, you can think of this x as the percentage. For example, this is a model used in biology. You could think that you have a population of some species which changes with time, and there is, I don't know, n is the maximum population capacity if you have an environment. And I don't know, x times n is the time, the individuals in your populations. And f of x, I don't know, could be x of a million butterflies that your greenhouse can contain. And then f of x is just a population after, I don't know, one day. In this sense it describes the evolution of your system. And you could encode with a variable, whichever quantity you want to study, and with some function, whichever time evolution describes your future variable. And now some very basic definition that, again, I'm sure I'm boring some people, but there are some people who have not seen dynamical systems. So we need to start from somewhere. So what are you interested in? You're interested in orbits. OK, so we said we have x, f of x. After two units of time, we have f squared of x, which from now on we will denote f of f of x. And in general there's a notation when I write f n of x. So in this course, in dynamical systems always, power is not a derivative and derivative f n is the nth composition. So f n always means f composed with f composed with f n times. And if f 1 minus x, absolutely. Thank you. People are paying attention. Good. So please, any comment that you have, any question at any time, I'm happy for you to interrupt. If it's an easy question, it's very good to answer immediately. If it's a longer question, maybe we'll discuss later, but do fit to interrupt. And pick up typos, which I can always. OK, and if f is invertible, so if you can invert it, you can also write f minus n, which is the composition of the inverse n times. And then with this notation, so the forward orbit of an initial condition of a point, I will denote it as o f plus, so orbit of the map f in the future, the orbit of the point x. This is the collection of f of x, so the dot f n of x, dot, dot, dot, dot, s n, bariz among natural numbers. OK. So this is the recording of future states of your system as time goes by in discrete units. The key question, so, OK, maybe stupid. OK. If you look at my map and you take, I don't know, three-fourths, you apply 4x times 1 minus x, which is the right one, and you get 4-3 times 1 minus 3-4 times 1-4. So this is 4-3 times 1 minus 3-4, so you get 1-3. Did I do it right? Ah, yes, OK. 1-3 and then so on, so on, so on. So you just apply your map. I had written one third on my notes, so I had different names. What do we want to know about our time evolution? So one basic question is to try to understand what is the nature of these orbits. For example, you could ask, does there exist any fixed point? So any fixed point, this is a point, an initial condition, which does not change under time. Is there a state, which is invariant, time invariant? Or you could ask, if there exists, sorry, this is, does there exist a symbol? There exists a question mark, a periodic orbit. So the next, you might not have a point, which is fixed, but you may have a point, which i.e., that means a point x, such that there exists a time n, for which n of x is equal to x. There exists a point, that after finitely many many steps comes back. And in this case, the orbit is finite, in the sense that, in this case, you see the same finitely many values repeating it. In this, you can write an infinite list, but the infinite list consists by finitely many things repeated over and over again. I tend to speak fast, yes, there is a question. This is supposed, this definitions work for any function on any space. You can apply to the same example, but in general it's the questions that you ask for any, I said, did I do it wrong? It's very possible. Forex, it's actually fixed point. Let me correct example, which shows, I think the fourth is the periodic point, fixed point. Yes, I think I did it silly myself. I cannot compute this. Okay, sorry. So I think the example, let me write. So I wanted to do one third. One third goes into four thirds, minus one minus one third, which is eight, nine. And this one you can keep going. I don't know. The next one, it's 3281, and so on. And this, I think it's an infinite one. And if you indeed, if you compute in the example, in the example, I think f of three fourth was indeed three fourths. So this is indeed the fixed point of my map. Thank you. Yeah, I do complain if I have. Okay. So you can ask these questions, but you can also ask the opposite. And these are somehow the simplest orbit, fixed points or periodic points are as simple as one can hope for. At the farthest end, you can actually ask if there exists a dense orbit. So what does it mean? Maybe let's put it as a definition, but it's a little bit vague definition at this stage. O f plus x is dense. It's not, okay, I will tell you. If it gets arbitrarily close to any points in your space. So to say this arbitrarily close, you actually need your space x to be a metric space, to have a notion to measure distances, which I'm kind of skipping now. If you think of your example of zero one, it's clear what close means. So it's dense if for every other point in the space and for every epsilon positive number. That's how we measure closeness. There exists an n such that fn of x is epsilon close. And to say this, you need a distance. So you need a notional distance. So for example, in zero one, this simply means that fn of x minus y is less than epsilon. Okay. So these are points which you cannot hope that you go everywhere but you can hope that you go arbitrarily close to everywhere. And there's lots in between. There can be lots in between or cannot depends. There could be situations where you're not periodic but you're not dense. So maybe you are confined in some part of your space. Maybe. And even if you are dense, it's not the end of the game. You can ask much more. You can ask if I plot an orbit and it goes close to everywhere, how will it spread? Will it actually be equidistributed? So I will just write it without defining it because we will see this later this week more formally. You can ask if it's equidistributed which intuitively you expect means it spends equal amount of times in equally large parts of space. Is it equidistributed? And let me keep it vague. Maybe we will define it tomorrow for the rotation and certainly maybe Davide will do it in equidist, I don't know. We will see it later. You will see it later. And more formal way to make this precise. Ok. So very basic questions. So I just wrote the logistic map because I wanted to tell you something about. You can think this is a population in a biology model and if you haven't seen the dynamical system it gives you the idea of time evolution. But again, we will not do application so the logistic map will not be our main example. But let me tell you what will be two further examples this week. So example 2 which is also not going to be my example but it's going to be Hannah starting example or main example. No. Is let me stay on 0, 1 for another little bit. This is just the following map. f of x which is 2x if x is less than 1,5 and 2x minus 1 if x is greater than 1,5. So if you plot it this map goes from 0 to 1 in the first half and again from 0 to 1 in the second half. It's a piecewise defined map and this is called a doubling map. Let me write two maps now. I will also write the rotation in the 0, 1 setting is example 2 and this is example 3. Again, let's stay in 0, 1 for not much longer and let me fix a parameter alpha between 0 and 1 and then the map I will call it r of x maybe r alpha of x this is just x plus alpha if x is less than 1 minus alpha and x plus alpha minus 1 if x is greater or equal than 1 minus alpha. If you plot it it's also in this picture it's a piecewise map so let's say alpha is large or small so this is alpha it's identity shifted from alpha so it's identity this is x plus alpha and here from 1 minus alpha to 1 you go back to 0 and just the time to get to a height alpha and this is going to be a rotation in 0, 1 so these two maps first of all it will become our main examples at least at the beginning of this week and they might look maybe similar to you but they are actually very different so can anybody tell me what according to you is some difference between these two maps some feature which is different what is different what do you think is the main difference between these two maps? Yes, the other one well, depends on alpha could have fixed points alpha is not 0 OK, good point alpha is this one one has fixed point the other doesn't good one more differences yes very good difference exactly one is invertible one is not and I can even add more this is actually 2 to 1 and it's not invertible because it's 2 to 1 so two points there are two preimages and this is actually 1 to 1 and this is indeed invertible 1 to 1 more differences pick up a little bit in isometry this is a good really good one so this is and it's actually let's write it here piecewise isometry but it is in isometry but you have to be careful of which distance piecewise isometry so actually the derivative is 1 it's identity with translations OK and what about this one it's not an isometry which feature does it have that's really the key difference all the others are very true but this one is piecewise expanding expanding which really means that the derivative is strictly greater than 1 at every point OK and thank you so all these two I think of them are really at the heart of why these two maps have very different dynamical features and the doubling map is really kind of the model of first of all piecewise expanding maps and this is what Hannah will be telling us all about this week she will go to also the nonlinear case and next week we will see hyperbolic dynamical systems in Hannah and Amy's course and somehow the doubling map is at the heart also of the hyperbolic dynamical world so if you have in one dimension you just have space for expansion in higher dimension you have space for expansion and construction but somehow this is the feature of the hyperbolic world if you know what that means you know if you don't know you will learn but so and this map is in some sense very chaotic map that you can have it has lots of periodic orbits it has lots of dance orbits and it has lots of intermediate features between dance and chaotic and it also has this sensitive dependence on initial condition which you might have heard about the butterfly effect if you change your ax a little bit your future evolution can be completely different so this is like the prototype of a very chaotic map and again Hannah will kind of push towards this probably she will tell you about the piece so as expanding she will tell you some key ideas of dynamics starting from this map like symbolic coding and structural stability am I right? Hannah yeah you will say more so from now on I will ignore this map and leave it to Hannah and my favorite map is really the rotation and this is a key example if you want to study entropy zero so I'm using this word but I don't assume that any some people know what entropy is and then you know this is entropy zero if you don't know what entropy means ignore it we will not do entropy in this course but what I want to say this is low complexity so in some sense it's much more predictable than the doubling map but nevertheless it does have some chaotic features it does have for example, ergodicity we will see and next week we in the really piecewise isometry it's a key feature and next week we will generalize this into interval exchange maps and the word of piecewise isometry of the interval so this will be our model this week and the key idea that I want to do this week is really to explain some basic features of renormalization so the way to study these low complexity systems involves a key idea in dynamics which is the one looking at your system on a very small scale and like having some other dynamics which acts as a zooming lens to study the system on different scales so what my goal for this week will be is to explain renormalization for rigid rotations for these maps and we will go tomorrow the day after tomorrow to the Gauss map and continued fractions and see them as a mechanism to renormalize these maps ok, so these two words Hanna and Amy myself we will split these two weeks like this and I think I have time to say a little bit more so I started with 0,1 but in reality these two maps can be both seen as maps of the circle and let me say that we have circle in the title of this week so a more compact way to write this map is actually just f of x can you see orange or can you see these different colors? you can write it as 2x mod 1 so let me remind you mod 1 means simply ignore the integer part so mod 1 just means remove integer part so for example if you take 3.14 mod 1 the integer part is 3 and this is 0.14 mod 1 and similarly here you can write this map more compactly as alpha of x is x plus alpha mod 1 but another question this corner is bad I assume is it bad can you see this corner? you can see the corner ok, no problem then so you see every time your output is greater than 1 you want to remove 1 in this case so that you are back into 0,1 your map maps to 0,1 it's always have to be less than 1 and when you overshoot you remove 1 and indeed both these maps are maps of the circle and if you read the title of our conference this first week it's called a circle of concepts in dynamics this was a pun that Amy suggested because we work on the circle and we'll tell you a circle of concepts of basic ideas so so our space it's nicer instead of thinking of x being 0,1 it's a little better to think that x is the unit circle so let me introduce just the unit circle so this notation s1 if you haven't seen it is just the circle of radius 1 in the plane so you can think of it as the space of x,y such that x squared plus y squared is equal to 1 as a subset of R2 and you can also think of it as cosine of 2 pi theta comma sine 2 pi theta that's how you parameterize points on the circle where theta is a parameter angle parameter between 0 and 1 so I can parameterize it with the angle parameter theta and write these points on the circle as cosine sine so this is the circle and sometimes it's also useful to think of the circle as a complex plane so you can think of it as complex numbers so you can write it as z in c complex number of modulus equal to 1 and this is the same then saying the same then saying that z you can write it is e to the 2 pi i theta so this is cosine of 2 pi i theta plus i sine of 2 pi 2 pi theta plus i sine of 2 pi theta and this again as 0 less than equal to I can put less than equal here less than equal to theta less than 1 ok, complex coordinates are in real plane and this circle is not so much transparent from an interval so if you take your circle and cut it open and straighten it you get an interval so what you should think that this s1 is actually the unit interval 0,1 in this angle coordinates I'm just wrapping a unit interval around my circle but I need to glue 1 with 0 so it's just a unit interval and identification which we write like this 0,1 module tilde where tilde just means that 0 is glued to 1 so 0 and 1 are the same point with glued endpoints really take your circle cut it open and you have an interval but you have to remember that these two points were the same on the circle and alternatively like equivalence classes this is r mod z so it's real number, module of the integers so this is the space of equivalence classes of the form x plus z where x is a real number and this equivalence classes means what, that you take everything up to integers so x plus z is equal to y plus z exactly when x minus y is an integer ok, so you can think instead first of all, one more time can you all see the bottom everybody? you can use everything in the board and you are all happy good, otherwise complain or otherwise sit in front because it's nice to have people in front so don't sit too far away if you sleep I don't care but hopefully I shout enough to keep people awake but maybe I also go too fast and people fall asleep too slowly too boringly but I don't get offended ok sorry, we have to do I want to make sure that everybody is on our same step so maybe you've seen this many times maybe not so it's good too so can I just have a reality check how many people have seen the circle or how many people have seen ok, the circle good, essentially everybody ok, how many people have seen doubling maps before in a lot ok, not everybody how many people have seen rotations before almost everybody ok, how many people have seen continued fractions before quite a lot do the opposite how many people have not seen continued fractions is it true? ok, it's very few and how many people have seen the Gauss map I need the opposite how many people have not seen the Gauss map ok, I think maybe we can speed up a little bit tomorrow class at least ok, so how am I doing with time I still have 10, 15 minutes probably I'm going until 9 no, I'm going until 10 there's no Stefano anymore Hannah, I'm going until 10, right? perfect, ok so let me then say something let me write a theorem and let me write our first theorem about ok, we have the circle let me write our first theorem about rotations ah, first of all maybe one more comment so our alpha the map that I wrote in 01 on the circle is maybe very simple is just counter clock counter clock clockwise rotation by 2 pi alpha angle so this x plus alpha mod 1 is doing nothing else than taking a point z and rotating it to this so x plus alpha so maybe if you write it in complex notation ah, our alpha 2 pi i theta if I had an angle theta what is happening is just I'm adding 2 pi theta plus alpha I'm adding alpha mod 1 ok, so this is just multiplying 2 pi here I miss an i 2 pi i theta this is exponential of 2 pi i theta and this is e to the 2 pi i alpha times e to the 2 pi i theta so this I multiply by a complex number of modules 1 and phase alpha ok, it's useful to know also the complex version and on the circle really the rotation is an isometry and if you want to think of it as an isometry of 0, 1 you need to pick the right distance so you need to remember that 0 and 1 were glued together on the circle so you need to remember that on 0, 1 you want the distance that tells you that 0 and 1 are close to each other so you need to take a distance mod 1 ok, so now I can write this first theorem is this dichotomy the first thing you want to know about rotation is this dichotomy so there are only two possible behaviors for a rigid rotation and it really depends on the rotation number it really depends on this angle alpha that you are adding so there are two situations I'm looking at our alpha from s1 to s1 either alpha is rational either the rotation number is rational and then what happens for every z in s1 the orbit of z is finite and actually not only that and our alpha to the q is the identity map so all points have the same periods, all points are periodic with the same period ok that's the first case very easy for example if I rotate by pi fourth after four rotations my circle is back to itself pointwise the second case is much more interesting alpha is irrational and then for every point in the circle the orbit of every point is dense so the other case whichever orbit I plot it will fill the circle ok, densely and there's nothing spurious there's nothing else in between there's nothing which is not neither dense nor periodic ok so this is really the starting point from a rotation so now I want to know also how many people have seen this theorem and how many people can prove it ok so I think I will not prove I think we will see the proof but let's see ten minutes I think it's enough to start at least if we want ok so first of all the first part I will not prove or maybe we can write one line ok proof first part is just if alpha is p over q if I write it both 1 r alpha of x r alpha to the q of x I just to add q times p over q and take the result modulo 1 but q and q simplifies and x plus p modulo 1 is identity for every x so if I add p over q q times I go back to where I started ok so the real thing to prove is that if I take a point ok part 2 I want to prove part 2 so I take alpha irrational and here there is a first step one which is just that this orbit has infinitely many points infinitely many points so this orbit will never close up so this is the first step this is my claim and we can check it by contradiction so if not imagine that there exist a k and l that x plus k alpha modulo 1 is equal to x plus l alpha modulo 1 imagine that there are 2 points when I rotate so this is r alpha to the k of x and this is r alpha to the l of x so imagine that there are 2 points which are the same that means that there exists an integer such that x plus k alpha plus n is equal to x plus l alpha so we can simplify x and we can solve for alpha and I can solve for alpha unless that alpha is ok I have a minus doesn't matter n times minus l and here I am using that k is different than l do distinct points which become the same but what's the problem here exactly but this means that alpha is rational this is a rational number contradiction so for 2 points to become the same and it can only happen if my rotation number is is rational if my rotation number is irrational all points will be distinct and now I don't know how fast I don't want to go too fast so the step 2 is the key one so who can tell me the pigeonhole principle or the shoebox principle anybody has heard of that anybody knows what the pigeonhole says because I there is well I don't know if you for the rotation there is Dirichlet theorem which is based I think of a really good this is a very basic principle that if you have n boxes and n plus 1 shoes you need to put 2 shoes in the same box or if you have n pigeon n cages n plus 1 pigeon you need to put 2 pigeons in the same cage so n boxes plus n plus 1 object implies that you need to put you need at least one box that exists a box with 2 objects so I think I will do step 2 and then leave you to think about the principle I don't want to do too much solving so if you haven't seen and many of you haven't seen the proof I will do step 2 and then leave you to finish and give you the solution tomorrow so let's do step 2 step 2 is that I claim by step 1 so let me say ok let me say that I fix an epsilon and this epsilon is what I will use later to prove density and then I can find an n such that 1 over n integer number is less than epsilon ok and now I look at z r alpha sorry I am doing x or z it doesn't matter let's do sometimes I write z of it as a complex number sometimes I write x if I think of it as 0,1 ok look at z r alpha of z up to r alpha to the n of z and then look at the unit circle so I have how many points do I have this piece of orbit it's an orbit of length n plus 1 distinct distinct points and this is crucial because for a rational rotation I could run back to the same point but for an irrational this by step 1 will be all different points so I have n plus 1 distinct points and then I am going to divide my circle into n equal parts and the claim is that by pigeonhole if I divide my circle into capital n boxes so here I have n equal arcs and n plus 1 points well there must be an arc which contains 2 of them pigeonhole so by pigeonhole there exist k and l with 0 from 1 less than k different than l less than capital n maybe I should do I'm sorry nobody complains I start this is 1 I need to do n plus 1 I need n plus 1 because this is 1 no no no no that was correct this is 1 2 yes that's fine that's okay this is between 0 because r alpha to the 0 is like identity by convention r alpha to the 0 is identity so this is r alpha to the 0 of z which is identity okay there exist k and l such that r alpha k of z and r alpha l of z are epsilon close where epsilon close is in the arc lengths so the arc between them in arc lengths the angle angle so basically if you want this angle the angle is less than 2 pi epsilon okay and I will stop here because I think it's time for me to stop but also this is a good point for me to stop the theorem is not proven yet I just found 2 points which are very close epsilon close and I want to use this to show that the orbit of z is dense so it will get very close to any other point on the circle and I think it's very good if I don't do it today and I will actually ask you to think about it so if you have seen it great but if you haven't seen it I think it's a good thing to try to do and then we will solve it tomorrow it will be conclude that the orbit under r alpha of z is dense okay there is still one more step to do using one of the properties of the rotation that we mentioned was mentioned today so try in the afternoon you will try great so I'm happy to stop thanks