 First of all, so welcome to everybody and let me thank Fanny and Emanuel for the invitation. It's a great pleasure to be here and it's a honor to give this course. And so we'll have, okay, eight hours together. So I want to give a glimpse of the research which is going on, on area preserving flows on surfaces. So maybe let me just say why. So first of all, I will, I will try to convince you today. Flows on surfaces are one of the most basic dynamical systems. So this is kind of, will be a course in dynamical systems and will, there are some people that I understand are not in dynamics and you will get a glimpse of some dynamical problems in this specific contest. So it's quite fundamental in dynamical systems. And then it's the topic of research in which, which is very much linked to my own history and my own research because I started giving a little contribution in this area since my PhD. Then I kind of went back to do other things for several years, but there have been significant advances, okay, in the past decades and especially right now in the next, in the last few years, there has been a revival and lots and lots of new research going on on deeper and deeper chaotic properties which I will try to outline today. So it's a fundamental topic in dynamics and it's very active, especially right now as I will try to explain. And let me tell you, we have, I have eight hours. So my goal for today is to give a very gentle introduction to somehow I want today will be very general structure of the course. We will give some basic definitions and I will kind of present stating maybe a little bit informally, but many of the results which give a view on what, what is happening in this area. So we will have some basic notions and also some informal statements of several results. So you should think of today as an overview of what will we do in more detail later on. So this is today. And then we'll have this week, somehow it's a more gentle week. So the second, the second class which will be Thursday. I will also introduce some background and basic tools. So especially it will be kind of fundamental tools in dynamics and related and to the topic. And then there will be one week break and in two weeks from now there will be two more lessons. And if anybody really wants to go deep into or get a sense of really techniques and proofs, I would like to say you have to wait techniques. So there are not so many experts. So there are many people that I think will enjoy this first week, but then I hope you will keep coming to the second part. But I was going to say if there was any expert in the audience, I would tell them you should really go into the second week to see really proofs in depth. Okay. So what, what is the topic of the course? This is structure, but so as I said, we are talking of dynamical systems. And dynamical systems are also, you can think of them as group actions. But the two basic cases are iterations of a map. So if you have a map in the space, this gives you a dynamical system. And this is what is called sometimes a discrete time dynamical system. You can think of this map as describing the evolution of the system in units of time, which are discrete. And if t is invertible, you can think of this as a z-action on the space x. To n, you associate t to the n. And this acts on the space x. But the other depends on some people like discrete. Some people like continuous dynamical systems. So the second example, when the time is continuous. So continuous dynamical system. Continuous time dynamical system. It's given by a flow. So I will write this notation with parentheses to indicate that it's a one parameter family of transformations of your space. And the index t is parameterized by r. And another way to say a flow, it's essentially an r-action. So to each t, we associate a transformation of my space. And we have the property that for the phi 0 is the identity. And we have the group property. So if I flow for time t plus s, this is the same than flowing from time s and then flowing for time t for every t and s in r. And so this course will be mostly on flows. We have flows in the title. But the two study flows, not today, but next week we will also study some one-dimensional maps, which are interval exchange transformations. So we'll have an excursion into maps as Poincaré maps of flows. But mostly, we are concerned with flows. And what should you think of the main example of flows? Flows arise as a solution to differential equations. One example. And let me give you the one example which will be relevant for what we will do later today. So say that I give you, let me give you the example of Hamiltonian. I want to give you an example of Hamiltonian flows. So let's start from the plane. We will go to surfaces soon. But say that I give you a function from r2 to r, which is smooth. And this is what sometimes is called Hamiltonian. And I give you just the following in two-dimension here. Just the following differential equation. x dot is dh dy. And y dot is minus dh dx. So let me call it h for Hamiltonian. This is an example of a differential equation. And I can look at the solution. And the solution will define a flow. So I can define 50 of a point x0, y0 in r2 as xt, yt, where this is the point reached in time t by the solution x0, y0 by solution of h such that x0, y0 is x0, y0. So flows are solutions of differential equations. And this is just one example, Hamiltonian. I want to mention it because we'll see soon flows on surfaces, which are, in some sense, Hamiltonian. We'll get to that. And Hamiltonian system, of course, are appearing in many problems in mathematical physics, in celestial mechanics. But this is not the topic of my course. And more in general, you could also have a manifold. And let me not write a manifold, the vector field. And then we can look at the integral curves of the vector field. And that will also define a flow. So we want to focus on this course. I want to focus on smooth. I'll put smooth. I'll put it in parentheses because sometimes we leave the smooth word, but smooth, low-dimensional dynamical systems. OK, so smooth, low-dimensional. So which dimensions do we have? So if you go to dimension 1, well, I don't even want to do flow. Let's just do maps. If my space is the circle, it's a compact one-dimensional manifold. Then, and you look at the transformation from S1 to S1, which is the smooth, smooth differential. This is one of the most basic dynamical systems you can try to study. But you enter a very well-known theory, so the theory of circle diffeomorphisms. So many people might have encountered the circle diffeomorphisms. Sometimes it's an introductory course in dynamical systems. Arthur Avila, who was now in Zurich, he just became colleagues. And he just taught a graduate course on the theory of circle diffeomorphisms. Of course, if you wanted to know about this, you should have gone to. Someone was there. He was a student of Arthur, who was there. But today, you're not going to do dimension 1. We are going to do dimension 2. So all the course, so this course is all about dimension 2. And it's all about flows on compact surfaces. So let me say throughout the course, basically, I just want, from now on, every time I write S on the board, maybe we write notation. So our space X will be S. And S throughout the course will be a closed surface. So I want something which is a smooth surface. So smooth, connected, orientable, orientable, and compact, and no boundary. So let me just plot. Let me just draw this. Two-dimensional, smooth manifold, compact, no boundary. So a surface. And just also notation. So G is always going to be the genus of my surface. And we will not work on spheres. So we will start from the torus. So the genus will be greater than 1. And mostly, in some sense, I will talk a lot on passant about the torus. But somehow, what I feel more affectionate to and what I mostly worked on is higher genus. So genus from two above. So the focus will be on surfaces of genus two and above. And OK, so we have a space. It will be a surface. And we will look at flows. Let me just define. And as I said, mostly they will be smooth flows on surfaces. And I will get into that in a second. But what I want to say is that I want to do ergodic theory. So within dynamical systems, people in ergodic theory want to study dynamical systems which preserve a measure. So I will assume, basically, we'll only talk about conservative or dynamical systems. Sometimes they are cold. And the 50 will be area preserving. Let me write it. I will be more precise in a second. And let me just say that this is the ergodic theory set up. So as I said, there are some people who are not in dynamics. So maybe I will also give basic definitions, even though maybe I don't spend too much time on that. So what is the ergodic theory set up? So the ergodic theory, you want to look at a space which is a measure space. Probability, actually. So you have on your space, you also have a sigma algebra of measurable sets and the measure. And actually, I will restrict to the case of finite measure. And you can assume probability measure. And when you look at a flow, if your flow preserves a measure, so what you want to ask is that, first of all, these transformations are measurable. So for example, pre-images of measurable sets are measurable. And the measure of a set doesn't change when you flow. So measure of a is equal to measure of 50 of a is equal to measure of a for every a in my sigma algebra. Or if you have a transformation, you want something similar. But in general, if the transformation is not invertible, it's important to look at pre-images of sets. So pre-images have the same measure than the original set. And that's the type of, these sometimes are called conservative dynamical systems. And I need to erase. So I guess it's here. I'll leave the title. We are building momentum for, and because we are going to work mostly with smooth dynamical systems, I will basically, maybe I will put it here. For us, the sigma algebra, it will be Borel sigma algebra. So we will work on many faults or spaces which have a topology. And the measurable sets will be generated by open sets. OK, time to give a very basic example. And we'll build also in the examples. We'll build, OK. And sorry, maybe one more word. So in this ergodic theory setup, we will investigate chaotic properties. So maybe I will leave a general philosophy how chaotic can a smooth, conservative, low-dimensional dynamical system. Maybe this could be a little bit the leading subtitle. And I borrowed this. Actually, there is an instant paper by Basam Fayyad and Giovanni Forni and Adam Kanegoschi who starts with this sentence. How chaotic can a smooth dynamical system be? And they show you that for some genus ones, smooth flows, you can have the back spectrum, which is a property of very much probabilistically chaotic dynamical system. So our objective will be looking at some kind of typical smooth flows and see which chaotic features they can display. And chaotic properties, I will define soon. Mostly, we will talk about mixing. We will talk about many other things in between. And I will define them as we need them. So OK, so general overview. And now let me give you a first example of an area-preserving flow on a surface. And this is the most basic example, perhaps. First of all, am I really happy I'm not going too fast? Or am I talking loud? I mean, complain if I'm too fast, too slow. If you cannot read, if you cannot hear me. Usually I shout, so OK, so linear flows on the torus. Again, apologies if some of you works in dynamical system. You probably have seen this example, but today is the day for putting everybody on the same footing. So I'm mostly going gently also for people in other areas. OK, and then it's also good that we'll use some things later on, so OK. So basic example, so space is a surface. And I want to look at genus 1. So I'm going to look at the torus, R2 mod Z2. And I'm just going to draw a square. And you should think it's just, in your mind, it's just R2 mod Z2, the fundamental domain. You can take a unit square. And you need to think that opposite sides of the square are the same, mod Z2. So if you glue them by translations, topologically, this is a torus. So this is actually a flat torus. So it's a torus which we show as Euclidean manifold. And so I'm going to fix an angle in S1. So this is an angle. And then you can just look very basic flow. I'm going to write it as differential equation. x dot and y dot are equal to cosine and sine of your angle. So here, I'm just constant linear differential equation. Maybe it's overkill to find it like this. So the solutions. So phi t theta, phi t theta, I'll denote this theta to remember the direction. Our solutions is the flow given by solutions to star. And it's called the linear flow because clearly, basically, what I'm doing. If I fix an angle theta, I'm just moving on lines in direction theta. So cosine and sine are constant. The velocity is constant. And because I'm on the torus, I'm just have to use identifications. So every time my line leaves the boundary, it comes back by using the identifications, right? And this picture, really, if you plot it on the torus, then I'm traveling over lines which wind on the torus. Straight lines in the plane. And clearly, I'll write it here, phi t preserves Lebesgue measure. So mu is equal to Lebesgue area on the torus. That makes sense on t2. And this is one of the most basic dynamical systems. And we know everything about it. And if you want to be interesting and do something in genus one, you can start perturbing it. So I can start adding some small perturbation to this constant linear flow. And then you enter the real of KAM theory. And you want to put the yofantan condition on theta. You can ask if, when I perturb, I get something genuinely new. If I can have new different properties, or I can get a dynamical system which conjugated to the original one. This is not the topic of my course. It's a very well-studied area. And people maybe could do Hamiltonian or KAM. Some of you told me they are no. Everything about this. I don't want to do this. I want to move to higher genus. I told you my focus is genus two and above. So second example will be genus two. And let me, maybe before I do the example, maybe let me put a remark. So if the genus is greater than two, so this was an example of a flow which is non-singular. So the derivative is non-zero at every point. It's constant. So if I go to higher genus, let me make a mark that I need to allow for fixed points of my flow. So for singularities of the flow. So let me just say, so any smooth 50 has fixed points. So I will call them singularities. I will give you a picture in a second. And let me say that we always will assume that these are isolated. They are isolated finite order, order zero. And in particular, we are on a compact surface. So I want to have finitely many singular points. This will be another standing assumptions. So what are these fixed points? So basically, you can think of this differential equation. You can think of points where the derivative is zero. And they have to be isolated. And then you can just classify how they can look. And this is, again, some basic, if you. I don't like to teach dynamical systems on flows and some linearization. But if you ever see any type of some basic course, you can kind of compute how a phase portrait of a represerving flow looks near a singularity. So let me just draw the picture. So this is an example of a fixed point, which is a center. So this is a point which doesn't move. And around, I have some closed trajectory of my flow. And this is also an island of closed trajectory, closed flow trajectory. Or they could be, for example, simple saddles. So this is the plot of a sea. Maybe I should orient also my trajectories. This is another fixed point, which is a saddle. And a simple saddle, because it's a single zero of the differential. And the Hessian, for example, is non-zero. So if you have a, say, you think of a differential equation with non-zero Hessian, the terminate of the Hessian in non-zero. And then you have a local phase portrait like this. So that's how your orbits will look in a neighborhood of your point. So they will have these red lines, which go into the singularity. I will call them separatrices. So we have center. We have simple saddle. And maybe I will do, there is another example, which has what I will generically called multisaddle or degenerate saddle. And I will draw one, but it could be, basically you want an even number, but greater than four of incoming, outgoing separatrices. So this will be, you can orient them one in, one out. Did I put an even number? Yes, out. No, in, out, in, out, in, out, in. OK. And then you can orient your flow. So this is the type of picture that you will see at, finally, many points of your flow. And again, we want a concrete example. So I'll give you two types of concrete examples. And maybe let me, why did I say that there are, if the genus is greater than two, there have to be fixed points. So let me recall you what is Poincare Hopf theorem. If you haven't seen this, it doesn't matter. It's just a fact. So to each of these fixed points, you can associate the index of the fixed point. So the index is kind of telling you how a little unit tangent vector, what is the degree, if you take a unit tangent vector and move around the fixed point, how many times you rotate around yourself. So if you don't know, it doesn't matter. But take as a definition that the index of a center is equal to 1, index 1. The index of a simple saddle is equal to minus 1. I hope I didn't mess up the signs. And the index of the multi-saddle, so they say that there are 2k prongs. Prongs are the separatrices. Then it's minus, I hope it's minus k. Let's check. So no. Index minus 1. No, I should have written it. Minus k-halves? Is it? No. Let me check if I wrote something on my notes. What's the index of a saddle of degree 6? Anybody? It's I wrote minus k if there are 2k plus 1 prongs. OK, that's how. Let's check now. If there are 2k plus 1 prongs. So here I had 4, 2k plus. Here there are 4 prongs. So k is equal to 1, and the index is minus 1. And hopefully if I get 6, I should get index 6 prongs. You should have index minus 2. Doesn't match? Yes, 6, then k is 2. OK, 2 plus 1, 3, 6. OK, good. OK. And what Poincare I hope theorem will tell you is that the characteristic, Euler characteristic of the surface, which is 2 minus 2g, has to be equal to the sum over the fixed points of the flow of the index of P. So if I add them up, the indices should match 2 minus 2g. OK, so you can basically have no fixed points only when the genus is equal to 1. OK, a regular point would have index 0. And OK, now let's do the second example. The second example I want to do, we did the unit square. Now I want to take the unit, or a regular octagon. So O will be a regular octagon. And I want to build a surface, which I will denote SO, by taking O and identifying opposite parallel sides, very much as I did in the case of the torus. So this tilde is glue parallel side by translations. Colors are useful here. Drawing all opposite sides of the same color. And you should imagine that they are identified by the unique translation, which maps one to the other. OK, the arrows here have no meaning. They just mean that they are glued. OK, no deeper meaning. And then I'm going to look again at 50 theta. So I'll fix an angle. First of all, if you glue an octagon by basic topology, you can check that you get a surface of genus 2. So SO has genus 2. And if I fix a direction, I will look at what is called also. I will also call it linear flow, which is again solution of the same equations I had before. So I just move on linear lines with constant speed in direction theta, and solution of star. And sometimes it's also called linear flow. OK, let me now say it later. So what should you do? Again, you move on a straight line. And then when you hit the boundary, like before, you just use the identifications too. So if I hit here, OK, you keep going, hit the red, and then. So trajectories are straight line. But I'm drawing trajectories which don't hit the vertex. So from my definition, it's not so clear what happens when you hit the vertex. And which color, red? Do we like red? OK, so you can make an exercise and check that when you glue by identifying opposite edges, I'm following what happens of a point. So this point is the same than that point, which is the same than this point, because red is glued with red, which is the same than that point, because green is glued to green. You can verify that in this picture, in here, so here, the vertices of all are all glued together, and they produce unique point P on the surface S. So there is a unique point which is glued by all the vertices of this octagon. And what happens of this point? Now I want to plot. So how many trajectories enter? So there are, I'm fixing the direction. And I assume it's not the direction of a side for simplicity, or? You see, there are three trajectories which end in this point. And there are three trajectories, which we should plot here, which go out of this point when I flow in my direction theta. So I claim that this point, what it looks like, is a multisaddle with three prongs. There are three in going and three out going linear trajectories from this point. So if you've seen it, you've seen it. If you haven't seen it, maybe you're not used to think of it. So is it clear what I'm saying? So I'm showing you a saddle. I'm showing you a flow on a surface, which has linear trajectories on every point. But every point is regular, has a unique straight line trajectory passing through it. But one point, the red vertex here, which is one point on the surface where I see actually a saddle. So the trajectory is on my flow. If I plot them on the surface, I see three of them going in and three of them going out. So the other trajectories will look honest, regular trajectories like in the torus. So they will have, I don't know what they will do, but once I fold my octagon into a surface, they will wind online. So if you have never seen this before, I give you an exercise. So convince yourself what happens if I do the decagon. So you take decagon, maybe I should call it d. And Sd is the associated surface by gluing, parallel sides, and verify that the genus is again, actually, do you know what the genus of the decadence is? OK, well, maybe I'll verify. I'll give you already the solution. The genus is also 2, so I will plot Sd. The genus is again 2, but this time verify that there are two points which are singular, p1 and p2. And each of them will be a simple saddle. Genus is equal to 2, and there are two simple saddles in the linear flow. So not all vertices of the decagon are glued to each other. There are glued in two equivalence classes, and each of these will give you a four-pronged saddle for the flow. So do this if you haven't seen this picture, and ask me at the end if you. So first of all, I hope that you see, we said at the beginning, why Poincare hop theorem, if the genus is greater than 1, we cannot have a flow without fixed points. And we can check that the characteristic, the index here will be what will be minus 2, and 2 minus 2g for genus 2 is minus 2, fine? So I can have one index minus 2 fixed point or two index minus 1, and they also add up to 2 minus 2g. So check the formula. So if you have never seen this before, let me tell you. So this by this, I mean 50 on SO, is an example of linear flow or directional flow, linear or directional, on a translation surface. And this is not what I want to do, but I want you to know for culture if you don't know what this is. So this is so-called flat surface. So it's a surface which was glued out of a piece of the plane. So this is an example of a surface which carries a Euclidean metric. So there is a Euclidean metric at all points which come from the interior or from the interior of the sides. But this Euclidean metric actually has singularities, flat surface with so-called conical singularities. And these are singularities of the flat metric where the negative curvature is hidden. So I will not enter into this, and there is actually beautiful surveys, for example, Zorich. Also, Yokoz taught many courses on flat surfaces. And I just wanted to mention that this example is in the world of flat surfaces. And this is really the geodesic flow for the flat metric. So I'm moving on a straight line in the plane. But there is a remark. This linear flow is discontinuous, is not continuous, is not even not smooth, is actually not continuous. Why this flow is not continuous? Because maybe let me just write this and plot something. Singularities are hit in finite time. So I will just write this. So because I'm traveling with unit constant speed, at some point, if I am on a separatrix, if I am on one of these red lines, in finite time I will hit this red dot. And when I hit the red dot, I claim that this saddle splits trajectories. So the flow is not continuous. So maybe I will just plot. So if I have two trajectories here, which start very close, but one is to the right, and one is to the left of the separatrix, one will go here. The other will go there. So you see this flow has a singular point that in finite time, split the future history. And of course, you can see it here too. So here blue, if I start here, blue will follow here. And if I start there, why? This will follow the green and go here. So the trajectories are split in finite time. OK? So if I want to study smooth flows, this is not an example of a smooth flow. It's not even a continuous flow. But let me tell you that the ergodic theory, actually the ergodic theory of these linear flows on translation surfaces have been very actively studied since the 80s. So we have someone working in Tecmuller Dynamics. And many people, I don't know how many people. So let me write somewhere. So the ergodic properties of these linear flows on translation surfaces are very well understood because there is a beautiful connection with Tecmuller Dynamics, which acts as a renormalization on this world of translation flows. And again, Jean-Christophe Yorkos, who unfortunately passed away, has taught several courses at Collège de France. And maybe some of you took. I don't know. Maybe not. But if you want to know more about this topic, there are beautiful lecture notes written by your calls. There is an upcoming book by Viana. There is a survey by Zoric, a measure. So it's a beautiful topic. But again, it's not going to be our focus. Even though there is a lot of connections between what I will teach and translation surfaces. And occasionally, I will tell you where are some similarities. And we will use, from next lecture from Thursday, we will use interval exchange transformations, which are very much related to translation flows on surfaces. And we will also use some inputs from Tecmuller Dynamics, which I will tell you out of the context of Tecmuller Dynamics. So I just keep in mind that this is a sister story of the story I'm telling you today. So now I need to do example three. So example three is going to be smooth flows on higher genus surfaces. Or they are somehow also symplective flows in dimension two, or they are also locally Hamiltonian flows. And so let me define. This will be our main object. So first I will give you the abstract definition. And then I will give you a more concrete definition. So here I want to define, let me put a title, smooth flows on higher genus surfaces, also called locally Hamiltonian flows. So here, as we will have, our surface was smooth. So it's a smooth two-dimensional manifold. And if you want, I want to take, I'll say the word symplectic, but there will be nothing symplectic in this course, if you think of symplectic geometry. I'll say it and not omega will be a symplectic form. And basically, just omega is a smooth, non-degenerate two-form, area form, smooth area form. So think, actually, more than think, so omega locally in charts will be something which looks like some function vxy dx wedge dy. So this is what a smooth two-form, where v is a smooth. And if you choose right coordinates, you can actually, you can assume it's really locally dx wedge dy. So standard area, Euclidean area form. And now the abstract way to get area preserving or flow which preserves omega is to take a closed one-form now. And the flows will be in correspondence fixed. When I fix omega, smooth flows will be in one-to-one correspondence with closed forms. So again, I will be more concrete and get to locally Hamiltonian in a second if you don't like this, but I want to give this definition first. Am I? OK, so given eta closed one-form, so d eta 0. And let me stress that I'm not writing exact. I'm just writing closed. Then I can define a vector field on the surface. There exists a unique x vector field such as that. And again, you just want to write that this notation, the contraction, so this is sometimes also called ix omega. So this is the contraction of the two-form by the vector field is equal to eta. So what is this contraction? So I have a two-form, so it acts on vector fields. So if I can contract it, I just basically plug in the two-form x comma dot. And this becomes a one-form. And I want this one-form to be my one-form. So this is the meaning of this contraction. Closed one-form such that the vector field such that when I contract the two-form, I get my one-form. So sometimes I could call it x eta. So it's uniquely determined by eta, fixed omega. And 50, or if you want 50 eta, the flow given by eta is the flow along, which integrates this vector field, along integral curves of x eta. Very abstract definition, but let me be. This is also a definition that I will not use explicitly, but we will soon go to a much more concrete description and especially Thursday. But this is the definition of, OK, maybe let me say something. So first of all, let me do one more thing in this context. So this flow, 50 eta, 50 eta preserves omega. So in this way, I basically get a flow which preserves the two-form. Maybe I should say, if mu is the area given by integrating omega, if I have a two-form, I can integrate it and I get a measure. I get an area. This is my area form. If mu is the area, then when I say, basically, I have that this flow preserves mu. So it's area preserving in the sense I defined at the beginning. So it's a way to get area preserving flow. And essentially, it's an if and only if. So every smooth area preserving flow can be written in this formalism of one-forms and two-forms. And let me just say what is the proof. So first of all, preserves omega. It's basically equivalent, this again, to say that the lead derivative, this is a little bit of differential geometry if you want, preserving this area form is the same than saying that the lead derivative is 0. But now, if we compute this lead derivative and lead derivatives, then there is a chain formula if you want, or product formula for e-derivatives. I don't know if you'll write it like this. And in this case, this is equal to 0 because the omega is 0. And by definition, the contraction of omega on x is actually eta. And this is by definition of x. x was such that the contraction is eta. And eta was closed. So eta being closed is equivalent to this flow preserving the other form. So this is the one-line proof. So let me now be more concrete. And we'll be even more concrete on Thursday. But one way to think of this is basically to think of these flows are locally given by Hamiltonian equations. So alternatively, so if I have omega, so in local coordinates, in local coordinates, so if I have a small open set of my surface, in suitable order, you can find local coordinates. If omega in suitable order coordinates can be written as really dx, wedge dy. So you can always choose coordinates where your function is identity equal to 1. This may be sometimes, I guess they are called the Darbouk coordinate, or symplectic coordinates if you like symplectic geometry. And eta, locally, by locally I mean on u, small open set, simply connected open set, is exact. So a closed form on a simply connected open set will be exact. So this one locally will look like dh for some smooth h from u to r. So locally, my closed form will be of the form dh. And I claim, and if you want, you can try to prove it as an exercise that if you want, it's an exercise. And this flow of 50 eta, in this coordinate, in this coordinate, the solution to h, our system of Hamiltonian equation. So let me write it again. So this u, in this coordinate, let's call them x, y in u. So if I call my point, it's a coordinate. So we are in R2. I call x, y my coordinate. And what I'm really doing, I'm moving along dy minus dh dx. These are the standard Hamiltonian equations. So hence, 50 eta is called locally Hamiltonian flow on the surface s. So again, preserving a smooth area form is equivalent to being locally Hamiltonian. So being locally solution of this Hamiltonian equation on neighborhoods and patches of my surfaces. But I'm not saying it's a Hamiltonian flow. I'm saying it's a locally Hamiltonian flow. And now, warding NB, a global h might not exist. So it's not a Hamiltonian flow, because I don't necessarily have a globally defined function h on my surface, such that I only have locally defined. So what is somehow well defined is dh. So dh is globally defined. And maybe let me give you an example. Again, I have a question. So has anybody been to other of these courses? So do we have a break normally or not? No, but I want to know. Maybe you want to break. I don't know. So I'm happy to do as I say, five minutes break, but I just don't have all this up here. And then I haven't told you the fun part. I'm still going slowly on definitions. And I want to give you in the second hour a sense of what is happening. So OK, let me finish. I will give you this example of non-global Hamiltonian. And then we have a five minutes break. And then I will tell you what has happened about what we know about chaotic properties of locally Hamiltonian flows and why should we study them. OK, so let me just give you an example. And the example is just what we had before. So if 50 theta is not Hamiltonian, because if there existed an H from T2 to 2, it's locally Hamiltonian, but not Hamiltonian. So if there was a globally defined H, H has minima or maxima. But in this flow, but the H here is never 0. So in this linear flow, you don't have 0s. So if it was Hamiltonian, I cannot have a function on a compact surface which doesn't have critical points. So this already is an example of a non-Hamiltonian. So and maybe let me tell you another remark. Then we can stop. So Novikov, which will be a starting point of the second hour for the motivation to study one motivation for locally Hamiltonian flows. Novikov calls this flow, calls them in the 90s. Maybe actually I should say locally Hamiltonian. I might be the one who started calling them locally Hamiltonian on a regular basis. I think the name that I think it locally Hamiltonian conveys well this notion of locally Hamiltonian. But Novikov would call them multivalued Hamiltonian flows because you can define an H which is multivalued. So you could define a Hamiltonian H, but once you do a loop, a non-trivial loop in homology, your function has to be a translate of what you had before. So it's well-defined up to constants. So the one form is well-defined. Yes? Exactly. You can think you can go to the universal cover. And on the universal cover, your closed form is exact. So on the universal cover, it's well-defined. So if you descend, it's multivalued. And the multivalued are related to the periods of your. Exactly. OK, so after we have five minutes break, so just to relax a little bit. And then I will tell you why Novikov wanted to study these flows and what we'll be proving about them during the rest of the course. OK? Thanks. Have a few minutes break to relax. OK, so we spent a lot of time motivating the study of dynamical systems as flows. And we want to care about smooth flows on surfaces. And now we got our third real example of how you can get the smooth area preserving flow on a surface. OK, I will always call them locally Hamiltonian flows, but it's equivalent to preserving a smooth area form. OK? And I didn't plot a picture, but if I were to plot a picture, I could have plotted actually the same picture we had before. We could have plotted the same picture that I got from the octagon. But there is a crucial difference if you now look at this locally Hamiltonian flow. So maybe let me remark this. So there will always be saddles. There will always be singular points. It's a matter of life. We will always have either centers or single saddles or higher saddles, but 50 locally Hamiltonian. Basically, if I'm traveling on a separatrices, it takes infinite, infinite time to reach the saddle, to reach a singularity. OK, let's write saddles. Center you cannot reach. So if you have a singularity, basically what is happening is that you never reach it in finite time. And if you have a trajectory which comes close to the center, to the saddle, it slows down. It kind of takes infinite amount of time. So it's not that there are no saddles, it's just that they are not visible in finite time. That's the difference, OK? Subtle, but that's the difference. And then, OK. So why should we care about locally Hamiltonian flows? And I'm still in the introductory move. So I did this course like a long seven. So you start with a long introduction, then we will do lots of preliminary, and then we will get the proofs. So today it's lots of motivation day. So let me write some motivation. So motivation 0, I've already said it. It's a basic, smooth, low dimensional dynamical system. So this I'm just rewriting it. But it should be obvious that if you are a mathematician, area preserving flows on surfaces are very basic object we want to understand. But there is a physical motivation. If you're like physics, there is a motivation from solid state physics. So this is recorded. So I cannot say when they ask you why do you care. People like to say, oh, solid state physics. I think I'm perfectly happy to say they are fundamental mathematical dynamical system. But also in solid state physics, you might really want to study this. And this is indeed the motivation of Novikov's school. Novikov and his school in the 90s. So Novikov was the one who suggested multi-valued Hamiltonian, I should say that, and multi-valued Hamiltonian is a model for, I'll just write this, electrons, transport of electrons in a metal under a magnetic field. So you have a metal, and you have a magnetic field, and you want to study electron transports in this metal. And he has a model for electrons in a metal under a magnetic field. And maybe let me write. Don't ask me what this means, because let's ask someone else. But let me write what is in semi-classical approximation. And that means that you want to use quantum mechanics to treat the electrons, but classically, you want to treat classically the magnetic field. And whatever that means, I claim that in this model, basically this model really reduces to study locally Hamiltonian flows on surfaces. So his motivation was this one from solid state physics. And let me tell you one key word on Fermi surfaces. So basically, the surfaces come from energy levels. Basically, you have some energy. And the level sets of these energy functions are give you the surface. And your electron is constrained on traveling on the Fermi surface. And the magnetic field gives you the restriction of also being in the magnetic field to give you this locally Hamiltonian flow. And to give you a sense of where it is coming from, this is very interconnected with two. So let me write two and try to explain. And let me try it through pseudo-periodic topology. And so I will tell you, this is connected to one. So what is the question in pseudo-periodic topology? There is one special case. So you take S, which is a periodic surface in R3. And by periodic, I mean Z3, periodic, say. Say it is a lattice of symmetry in your surface. And let me plot how a periodic surface might look in a unit cube, which is the period. It might look like this. I don't know. This is, I don't know. Imagine some kind of fundamental unit of surface. Here there is a boundary component. But I have to repeat this periodically on R3. So I cannot draw. This is where you, if you open, if you search on the internet, I don't have pictures, search Fermi surface, gold Fermi surface, or some metals Fermi surface. And you will see these pictures of a periodic surface in R3. So Fermi surfaces are actually Fermi energy level surfaces, are periodic surfaces in R3. And if I look on R3 mod Z3, I get a closed. If I identify these open components in the block, I get a closed surface. So this is the surface I was having before, which is a block of my periodic surface. And what is the pseudo-periodic? Then you take a plane. So intersect with a plane. Actually, more than on a plane, intersect with the family of parallel planes. So you give some family of planes in R3, and you slice your surface with these planes. And these planes, you should really, this is kind of an abstract version of one. So your surface could be the Fermi energy level surface. And traveling on a plane is related to the magnetic field. So you're kind of intersecting a constraint from energy surface with the constraint of having a magnetic field. And the intersections will be one dimensional. So I have a surface sliced by planes. And intersections are trajectories. How do they look like on the surface? I have this surface, which is the quotient of the periodic one. I intersect with the plane. And if I get a foliation, and these leaves of these one dimensional foliations are trajectories of a locally Hamiltonian flow. So that's another instance. And again, I'm not going to say more about this, but there is a nice survey by Anton Zorich. I think it's also in some more recent survey, but there is an old volume published, which is called pseudo-periodic topology. And there he tells you more in general. Pseudo-periodic topology has to do with your periodic surface could be in Rn. And you could intersect with hyperplanes. And there is higher dimensional version, but we'll stick to R3 and two dimensional planes. And OK, so he has a survey. How do the leaves wind around? How do the closed leaves wind around the surface? And I must say, this is my understanding of the situation is that Novikov was very much interested in how these trajectories look like on the universal cover, in R3. So if I travel along one of these one dimensional trajectories in R3, on my periodic surface, what do I see? Do is there some direction which I asymptotically follow? Are there deviations? And these questions I have to do with kind of also how this flow acts on homology of the surface. And Dinikov has kept working. There are also some recent results of Sasha Scribchenka and Uber and Arthur Avelin. And also, somehow, but historical Zoric also was, as a student, was interested in these questions. He wrote when he wrote this survey. And then at some point, if you just care about these questions like asymptotic direction, you don't care about how you move on the trajectory, with which speed you move. You just care about the topology kind of there. So in some sense, it turns out, as I will explain probably Thursday, that this locally Hamiltonian flows can be seen in many cases as time changes of translation flows. So sometimes they have the same trajectories of a linear flow, just that I move differently. For example, I slow down when I get to a saddle. So this speed matters a lot when you study chaotic properties, but doesn't matter for other questions. So I think this was one of the motivations that gave a push to take Muller dynamics into the study of translation surfaces and linear flows. And there was the Konsevich Zoric's conjecture that then was solved by Avelin Forney and then there was Forney's work on deviations of organic average. So a lot of this was motivated also by this Novikov and pseudo-periodic topology. But for chaotic properties that we will study now, you cannot cheat and go to the world of translation surfaces. You need to stick to locally Hamiltonian flows. And that's what we will do in this course. And maybe something which is indeed special of locally Hamiltonian flow. And this is my third motivation, third and last. I want to say that first let me start with a remark. So 50 area preserving on the surface have zero entropy. And I'm not going to define entropy. It's many, so do you know what entropy is? Curiosity, yeah. So if you know what the entropy is, these are an example of zero entropy dynamical systems. It's a measure of how chaotic a dynamical system is. And actually, this was proven by Lysan Yang probably in the 70s. I didn't check the smooth area preserving flows. Or in general area preserving flows have zero entropy. So they are, in some sense, let me add digression. So dynamical systems very roughly are classified. Boundaries sometimes are smooth. But sometimes there are three big categories. Hyperbolic dynamical systems, elliptic dynamical systems, and parabolic. So first of all, hyperbolic dynamical, I'm not going to give you a definition. But one feature of hyperbolic is that they have positive entropy. So hyperbolic dynamical systems have positive entropy. Parabolic and elliptic have entropy zero. And I claimed that this, sorry, I didn't say, area preserving flows. Maybe I will say it now. OK, maybe I'll say it. So area preserving flows are in the parabolic categories. So what's in here? Elliptic, for example, linear flows on tori and perturbations of linear flows on tori are elliptic dynamical systems. And hyperbolic, for example, the geodesic flow on a negatively curved hyperbolic surface is an example. Or if you transformation a nozzle of the thermorphisms, I don't know. If you don't know dynamical systems, don't worry. I'm just trying to, for those who know, I'm putting some boxes. So we are in between. We are in this parabolic world when we study higher genus area preserving flows. So a feature is that we have zero entropy. So we are not hyperbolic. But one rough way to distinguish these categories is look at the butterfly effect. So look at sensitive dependence on initial conditions. So if your system is chaotic in the popular image that you have, this butterfly effect tells you that if I start with initial conditions which are nearby and move them, these conditions will diverge. So elliptic system, so this is also called no butterfly effect, no sensitive dependence on initial conditions. On a linear flow on the torus, two nearby orbits stay together forever. In a hyperbolic and parabolic world, on the other hand, there is a butterfly effect, or sensitive dependence on initial conditions more formally. But the key difference is that in the hyperbolic world, this divergence of nearby trajectory is exponential. So it seems diverge very fast. Instead, in this parabolic realm, there is divergence, but somehow slower than exponential. So maybe I would say parabolic flows have entropy 0, 2 exponential, sub-exponential. And actually, this is usually polynomial or super-inomial, sub-exponential divergence of nearby trajectories. You can take this as a meta-definition of parabolic flows. And I claim that area-preserving flows, on smooth area-preserving flows, or surfaces are, to me, one of the fundamental classes of parabolic flows that you want to study. So I think my own research is not only about flows of surfaces, but I think it's only about entropy 0. So I like parabolic dynamics. I'm trying to understand what are features of parabolic dynamical systems. So I very much like this as one of the key examples. And just again, to put things in the context, I think there are three main examples. And if you don't know what they are, it's not the time to define. So first of all, this is for people who do know some dynamics. So on hyperbolic, on a surface with negative constant curvature minus 1, you have geodesic and horocyclic flow. So geodesic is the hyperbolic flow. The horocyclic flow is probably one of the main examples of parabolic flows. And in general, you have unipotent flows in homogeneous dynamics, unipotent flows on semi-simple groups. This is a key example. And in homogeneous dynamics, you also have nil flows, nil flows on nil manifolds. And this I'm telling you basic parabolic flows examples. And the last example I want to write here is a smooth area, locally Hamiltonian on surfaces. OK? And we were writing the introduction of a joint paper with Davide and Giovanni, who is here, who was my PhD student, and then Giovanni Forney. So we meet tomorrow to finish Friday's introduction. And we had a whole discussion of what should we call them. So horocyclic flow, in analogy to the hyperbolic world, we could call it uniformly parabolic flow. And nil flows are something called like partially parabolic flows, because they are not parabolic in every direction. And these Hamiltonian flows are non-uniformly parabolic flows. So to me, I want to say it's a key example, because it's a non-uniformly parabolic flow. OK? So to me, this is the most important of all motivations I gave you so far. And OK, that's enough. I hope you're motivated, are you? Yeah. But this is really, I'm kind of trying to think. Don't understand parabolic dynamics and what are features of parabolic dynamics. And these three classes are main models. And then you may want to look at perturbations of these three classes and time changes. And this is another story that I'm not telling this time. But OK, chaotic properties. So I need to get to state something, otherwise I cannot let you go without having stating some theorems we want to prove. So Arnold Conjecture. Arnold Conjecture. So shortly after Novikov was with his school was suggesting studying locally Hamiltonian flows, Arnold wrote a paper which started some key study of chaotic properties. So take 50 locally Hamiltonian on the torus, on a surface of genus 1, on T2, genus 1. OK? And we already said that linear flows, we don't want to study linear flows because they are elliptic. And I want to, even if I don't have to have singularities, I can look at flows with singular points on the torus. So with one simple saddle. And if I put a simple saddle by Poincaré Hopf, I'm bound also to put some other singularity and one center. So I have simple saddle and center. So I have area preserving flow on the torus with one center and one saddle. So Arnold in the early 90s. So here I marked that you have to have actually, if you have a center and you have area preserving flow, you also need to have a saddle loop. So in this picture remarked that there is what I call saddle loop. What is a saddle loop? It's a separatrix which comes back to the saddle. So a trajectory which goes from saddle to saddle making a loop. So this is a saddle loop. So it's a trajectory. So equal separatrix connecting saddle back to itself, OK? Saddle loop which bounds the island of periodic orbit. So the phase portrait of the trajectories of your flow. Around the center, we already said they are closed. And this is inside the saddle loop, OK? And the rest, I don't know yet. So this is sometimes called this center filling a saddle loop. This is called the trap. So here nothing interesting happens. My orbits are nothing interesting. If you like closed orbits, you are happy. There are many simple closed orbits with periodic trajectory. Here your flow is periodic. But Arnold conjecture is about what happens in the complement of this trap. So what's the dynamics in the complement of the trap? So let me write it like this. What's dynamics outside the trap? And let me say that topologically, actually it's the same than a linear flow in the torus. It's like a linear flow. So let me remember what the picture of a linear flow we had before. So I start from a linear flow. And let me say that I make a hole in my, at some point I put here a saddle loop like this. And I maybe deform my trajectories a little bit to fit this trap. So what am I plotting here? I'm first plotting a linear flow. And then I'm making a little hole. So it's like I had a torus with a hole. And on the complement, I have trajectories which look like a linear flow. Really think of what we had before. Just draw inside at some point. Transform one of this trajectory into a separatrix with a loop. And make the form a little bit before. And the main, so actually 50, restricted to the complement of the trap, is nowadays called also Arnold flow. Arnold didn't call it Arnold flow. But because of this paper by Arnold and conjecture by Arnold, sometimes when I would say Arnold flow, I mean the flow on the complement of this trivial island region. So the conjecture of Arnold is about the chaotic properties of this Arnold flow. And stated conjecture. And I write Arnold flow, the complement of restriction to the complement of the trap. I will write typically. And this typically I will comment on later. So this Arnold flow is related to a linear flow. And typically means for irrational directions which are maybe satisfy some diophantine conditions. So sufficiently rational rotation numbers, if you know what that means. But typically for the peak one, a random conjecture is that Arnold flow are typically are mixing. OK, so now I will define what mixing means. Definition, again this is not a dynamic course, but I'm trying to give basic dynamic definition. So by far curiosity, has anybody who knows what mixing means? High for more than half, more than half. But I will give the definition nevertheless. But because there are people who do not. But maybe not. OK, so say that I have set up a periodic theory. Measure preserving. So measure space mu of x is 1, so probability measure. And 50 from x to x preserves measure mu, reserving mu is mixing for every a, b, and b. For every a, b, and b, for every two sets, if I take a set A and I flow it, and I look at how much of A intersects some sets B. So I have a set A somewhere in my space, a set B somewhere else in my space, and I start moving A. And look at how much A intersects B after time t. So if it's mixing, what happens is that this measure of this intersection converges as t times infinity to the product of the measure of A and the measure of B. So another way to say this is that A and B, so 50 of A and B, let me say some keywords for those who haven't seen mixing, become independent asymptotically. This is kind of a probabilistic notion. Or if you like a geometric notion, 50 of A spreads uniformly with respect to the measure mu. So the set A kind of mixing really gives you this image. When I flow A, it kind of spreads in all space in proportionally to mu. OK? And again, equivalently for every f and g in L2, another way to say it, if you like functions more than sets, and you can check as an exercise of measure theory, for every f in L2, if I look at what is called the correlation, t correlation between f and g, which is equal to the integral of f composed with the flow times a g in d mu. So I take two functions, and one I compose with the flow, the other I don't, and I look at this correlation. This converges as t tends to infinity to the product, to this kind of the correlate. So this correlation integral converges to the product. This basically has a characteristic function will give you the previous definition. And sometimes people, if you look at special classes of functions, you can talk of decay of correlation. And this quantity is decaying. If the function's at mean 0, it goes to 0. And you can ask how fast they decay for certain subclass of smooth functions. OK. So reality check, this is a very strong property to ask for a flow on a surface. So maybe I'm running a little bit short of time. So let me just say it quickly. So linear flows on t2 are not mixing. So if I just go on with this straight line unit speed, you can convince yourself that if I take a ball and flow it, nothing interesting will happen to your ball. Your ball will not spread, will travel at constant stable. OK. So this is not true for linear flows. And for linear flows on translation surfaces, so we had our basic example of the octagon, where again I travel with the straight line with unit speed are never mixing. And this is one of the very first results on translation flows proven by Anatoly Katok, who was also died recently last summer. It's one of the great names in dynamics. We're going to miss a lot. And he has an early day paper on where he proves that this absence of mixing. I will say this a little bit of this probably in one of the next lectures. So I want to say example one and example two that we saw today are not mixing. So mixing for a flows on a surface area preserving is sometimes not something you expect, having seen these two examples. But nevertheless, are not the conjecture mixing for this locally Hamiltonian flow on the torus. And let me remark for non-dynamicists. So mixing actually implies ergodicity, which I will not define now, and actually implies because the measure is smooth. It implies that almost every trajectory of the flow is dense on the surface and uniformly distributed. So it implies that typical trajectory will fill the surface and spend in each set some time proportional to the measure of the set. So it's a very strong chaotic notion. And maybe we will go back to mixing as we move forward. I want to state something. So first of all, this specific conjecture in genus 1, so Arnold conjecture was proved shortly after he formulated it by Jakov Sinai, who was my PhD advisor in Princeton. So actually, he was the one who started me on all this story of locally Hamiltonian flows. And Hanin, also in the 90s. And I will also say that there are many several papers by Kocher Gin, who actually improved on the result of Sinai Hanin. So proved a better, almost every, a better typical, better class. But again, we will state to these theorems precisely, as we probably already tomorrow I can state it precisely. So OK, great. But Arnold conjecture is only for genus 1. But I told you since the beginning, I'm interested in surfaces of genus 2 and above. So if genuses 2 are typical locally Hamiltonian flows, mixing. So this is kind of the general question that stayed open for me. So as I said, the genus 1 case was solved, I think, just two years after Arnold asked the question. But the higher genus case, now we have a really full picture. And that's what I want to finish the lecture with. And what I will want to explain. But it stayed open for 20 years, essentially, after Arnold posed the question. So I want to finish this lecture with some results, with some summary of results that will be the main core of what we will go through. OK, so now we have, first of all, maybe I should say typical. So on locally Hamiltonian flows, one can put a topology. And all of this we will do later in the course. So I'm just saying heuristically what we will do. So there is a topology. Essentially, you can perturb them by adding an exact form. So it will be perturbation by adding a small exact form. And this will give you a neighborhood, a low-pensate. And there is a measure class. Measure class. So there is like a measure class on the space of the flows. We will do this carefully later on. And typically, when I say typical, I mean almost every or full measure. So it's in the measure theoretical sense. So I want to say what is the typical in this measure on this species. And now there is a full classification of the mixing properties of these flows. So I will draw, and maybe let me put another assume that the singularities are non-degenerate, i.e., all the centers and simple central. So if you have actually, or maybe I should say in the full classification, maybe let me say like this. If there is a multiple saddle, if your flow has just one multiple saddle, the flow is mixing. And this was discovered much earlier, even before Arnold. It's Kocergin in the 70s. And somehow this is non-generic. So the degenerate saddles are kind of, so there is an open set where you don't have degenerate saddles. You can easily perturb the degenerate saddle and only non-degenerate saddles are open. And this was indeed beyond before Arnold. Conjecture Arnold says, well, Kocergin proved that if there is a degenerate saddle, the flow is mixing. But our flow has only a simple saddle. The methods of Kocergin do not apply. Kocergin has also some results on non-mixing. So Arnold asked his conjecture. So say for a second that we are in the non-degenerate case. So the picture that, again, it's a preview, but because we formulated more precisely. But the picture is that the answer is, it depends on whether or not there are saddle loops. So the picture will be that there are two open sets, u1 and v2. So the picture which we know today is that there are u1 and u2. There are open sets, open set where there are only simple saddles. For example, on genus 2, two simple saddles. So this is an open set in the suitable topology. And u2 is instead the set where there are saddle loops. Homologous to 0, I should say. I will go back to this. Saddle loops. For example, Arnold's flow is in u2. So there is a picture like this. And homologous to 0 means that your loop separates the surface into connected components. So here I have a disk and the rest. And that's how the saddle loops are persistent in this suitable topology. So if I perturb, I still have a saddle loop. We will go back to this. So you have these two open sets. And so here you have all these simple saddles. And here you have these saddle loops. And for example, so if there exists a center, we saw by Arnold, well, we saw. I told you that this implies that there is also a saddle loop. So if you have a center, you are in u2. And I will wrap up. And the picture that we know today is a full understanding of what is going on. So basically, the summary is that in u1, almost every typical locally Hamiltonian flow is not mixing. Maybe I'll write it like this. It's not mixing. It's not mixing. But on the other hand, this works, we will go back and define. But minimal, ergodic, and weak mixing, which is something just less than mixing. And in u2, the typical for u2, so maybe I should write for u2 prime open set, open and dense set. There is an open and dense set of u2. And such the typical fit is not minimal. I didn't. I was a little bit too fast. So it's not minimal means that there are, for example, there are traps or there are areas which there are not all trajectories are dense. But I will actually, I'm too much towards the end to explain properly. This is just a picture that I will explain later on. But let me say mixing on each minimal component. So this is the case of Arnold flow, where I have an area with closed orbits. But in the complement, I'm mixing. And what this means, I will explain. And actually, much more is known. And you can even say mixing of all orders and mixing with quantitative estimates. And these results, where do they come from? Not mixing is actually something which I proved now almost 10 years, no, 11, what is it? Eight years. I think it was really proven 10 years ago. And it's a paper which appeared in Anna's or Mathematics, which is answering this global Arnold conjecture in any genus. And minimal and ergodic come from translation flows. So this is a result of Meisur and Simotanus Livic for translation flows, which I will not dwell into. And weak mixing is also something that I proved in 2009. What about mixing in U2? So for, and I should say that there were special results by Sheklov in genus 2 and by Kochergin in some not exactly smooth case, but related case. And mixing here, it's actually I had a special case. This was my PhD thesis actually in a special case, which I will describe Thursday. But the general case, it actually was proven by Davide, who is a Ravotti, who is sitting here. No pressure. What year? It appeared in 16, maybe no. 17, OK. And also with quantitative mixing estimates. And turns out that mixing happens, but the decay of correlation is very slow. It's something like sub-exponential or sub-pollinomial. Logarithmic. And very recently, there has been a lot of new progress. So actually, maybe I will write in U2. You also have mixing of all orders. And you have results on, OK. Then there are results, related results on the spectrum of these flows for some cases. And there is the results on, what did I want to say, spectrum. And OK, there's a lot of recent activity pushing beyond mixing in these flows. So this is somehow a vague, I will have to stop. But this is somehow what I want to go into is, in some sense, explaining this picture. So my goal, I will kind of slowly connect locally the Hamiltonian flows with their Poincaré map, interval exchange transformations. We will do this Thursday. And explain special flows representations, which are a basic tool in dynamical system. And I want to basically start next week to define some basic tools on interval exchanges and Birkhoff sums and study of Birkhoff sums for non-integral functions. So some basic dynamical tools will be set up on Thursday. And I will try on Thursday to give you a precise statement of this dichotomy. So these two states went on mixing or absence of mixing in a simpler language, so very concrete language. So Thursday I will state formally, I will do some preliminary work and state a little bit these theorems. And then in the week after, I will actually sketch a proof of those results. So I will sketch a proof of some mixing and absence of mixing in these two setup. And behind this, there is some nice geometric arguments on shearing. There is some analytic tools on estimates on Birkhoff sums for functions which are not in L1. And there is some arithmetic number theoretical tool. There are die-offend time conditions for interval exchange transformations. So all these tools, I want to give you a flavor and the foundation of what they are and how they come into play. So my goal is to explain the tools carefully. And sketch the outline of this picture of classification. So I just got the introduction. And hopefully I gave you an idea of the topic. And then on Thursday, we will start slowly with describing Poincaré maps and special flows representation and continuing from there. Sorry for running a little bit. We have like five extra minutes of the break. We took them now. Thanks. Yeah. Good. Thank you. Thank you.