 Okay. Good morning again, everyone. For those of you who were here last week, I hope you have a nice weekend and spent all the weekend preparing for this week. No, I'm just joking. Hope you also had some visit to the local area and enjoyed your time, your weekend in Italy. For those of you who just arrived, very welcome to the second week of the workshop. I'm just going to steal a few minutes from Professor Wilkinson's first talk because I realized that I never really introduced the faculty in the first week when I gave the introduction, so I just want to take this opportunity to also emphasize the extremely high level of the professors who are teaching these classes. I'm sure you're well aware, but it doesn't help just to remind you that the three organizers and directors of the schools and the main lecturers are Hannah Rodriguez-Helz, who got her PhD from the University of Montevideo and is a professor of Universitat de la Repubblica in Montevideo, Uruguay and has recently moved to the Southern University of Science and Technology in China. She's also the Vice President of OAS, which is the organization for women in science and developing countries. She gave a presentation about this a few days ago. Corina Ulcigray got her PhD from Princeton University under the direction of Yakov Sinai and she's a professor of University of Bristol and this year she will be on leave at the University of Zurich this coming year. And Professor Amy Wilkinson got her PhD from Berkeley and is a professor of the University of Chicago and was an invited speaker at the Last International Congress of Mathematicians, which as you know is a very prestigious honor to be able to give one of these talks. So there's would be lots more to say but you can easily find out. I just really want to emphasize that in relation for example to one of the panel discussions where we talked about what is good research and the difficulty of distinguishing good research and bad research, I think having the time here and having the possibility of talking to them and the research that they do, you can be sure that whatever they do is good research. So I encourage you very much to once again to take the opportunity of their availability to talk to you during this whole week here. We also have some fantastic tutors most of whom you know by now. Oliver Butlerly was my own PhD student in Imperial College and is now a post-doc here at ICTP. Irene Pasquinelli is finishing I think her PhD at Durham University. Davide Levotti was a PhD student of Corina Ulcigray and Lucius Simonelli was a PhD student of Giovanni Forney at the University of Maryland and is now a post-doc at ICTP. And Hadim Wall is just arrived. He's here, maybe you can stand up Karim because you know they don't know you yet. And he is also one of the tutors joining us for the second week. He was my PhD student. He was a diploma student so he was a he's from Senegal and he was a student in the pre-PhD diploma program that I talked to you about last week and he then stayed on to do a PhD with me and is now a post-doc at the University of Borkum in Germany. So and he will I think give some of the lectures also this week. But just as important the cast and I want to emphasize something that is really special which we have always have a lot of countries represented in ICTP conferences here we have 50 countries I think, I hope. And I think that's really amazing. I don't know how many places have so many countries I'm going to do even if it takes a couple of minutes more. I want to do a little fun exercise which is I'm going to call out the countries one by one and I want the person from those countries to either put their hand up or stand up or shout here I am or somehow make themselves available. Okay, let's see. Maybe not everyone is here right now but let's try. Algeria Yes, two. Armenia Yay, Belgium Ok, Benin We don't have a Bolivia with us Brazil, sorry Brazil, great, Canada Yes, Cameroon Yes, Chile, yes, China Alright, Colombia Ok, we're missing the Africa. Yay, Croatia I know we've got a Croatia but we don't have any Cyprus, yes, Egypt Alright, France Georgia, we've got Georgia somewhere, Ghana Hungary, yes, India Ok, Indonesia Alright, Iran always a big group Israel, it's nice to be able to see people from Iran and Israel sitting next to each other. Thank you Yes Italy Korea, Macedonia Yes, Malaysia Mexico, yes, Nigeria Yay, Pakistan Ok, Peru Poland, ok, Portugal Russia, Senegal Serbia, yes Spain, Sweden Thailand, Togo Africa, we're having a bit of a problem in African Trinidad and Tobago, yes Tunisia, ok, Uganda Alright, Ukraine Ok, UK, USA Alright, Uruguay Ok, Uzbekistan Yes, Vietnam That is just fantastic to see, really, really, as you see it's just one or two people from most countries but also some bigger groups and it's really great to have you all here and as I said in the first week this is really an important chance to interact between each other, take every opportunity you can to get to know each other both from a human point of view and from a scientific point of view this is very important and finally let me conclude to remind you that tomorrow is African night so at ICTP this is a social evening that is organized actually independently of our conference it's kind of in our social program but it's an ICTP event it's just a lucky coincidence that we have it on our week and it is usually a really great party up on the terrace here live African music and African performance and a chance to wear your African clothes if you have them and a chance to dance and to see your professors dancing ok, so thank you very much and welcome to everyone and so I will introduce Professor Amy Wilkinson who just arrived for the second week and she will give the first lecture now, thank you Amy so I'm going to waste even more time from my precious lecture minutes, how many people here if you could just raise your hands, just arrive this week, this is your first, ok keep them up, little exercise everyone in the room, turn to the nearest person you don't know or someone who just arrived this week and introduce yourself, say where you're from and a little hullop, so this is just beginning get to know you session in another 5 minutes new people raise your arms for a sec I hope you'll continue these conversations into the break I almost don't want to start because I see such wonderful lively conversations, so please continue these later, so I'm giving the first lecture in a 10 lecture course originally the lectures were going to be by me and Hannah Rodriguez Hertz we decided to mix it up a little bit and War is going to give a couple of the lectures as well the subject of this course is smooth ergodic theory, so the point of these lectures is to examine the property of ergodicity and related properties, statistical properties of dynamical systems in some examples and the first examples we're going to consider will be linear examples or what we might call rigid examples and you've seen some of these if you were here last week like rotations of the circle and expansion by 2 on the circle or on the interval if you like so today is going to be a rather gentle, at least the first lecture will be a rather gentle introduction to some of these examples especially for those of you who were here last week, I hope to challenge those of you who have seen a lot of this material before this is what your opportunity is at this conference, you're first you get to meet other people who are interested in dynamics, come from a variety of backgrounds, I guarantee you that two people who know a lot of dynamics and come from different places know a lot of different dynamics so for those of you who've already seen a lot of this material you still have an opportunity to meet other people and learn from them you have an opportunity to teach other people who have not maybe seen dynamics before at all and give them your perspective and you have an opportunity to be challenged by some of the exercises that we give you this week, the plan is to give some exercises that should be accessible to most people, some more challenging exercises and maybe some exercises that you're not going to finish in the problem session but you'll have some time to think about them for the rest of your life excuse me for coughing I made the stupid decision to try to run up those steps to get here today so I kind of winded myself anyway, okay, so one more thing again trying to waste as much time as I can so my t-shirt or dress or whatever depicts my relationship to mathematics I eat mathematics I am voracious, I bite but my teeth are made of hearts because I love mathematics so I'm in some ways very gentle to it but this is also a kind of reminder to you although it's a kind of twisted reminder that I don't bite and I welcome questions and I welcome people interrupting and saying I don't understand or asking for clarifications and coming and talking to me after lectures I'm staying at the Adriatico so you'll see me a lot breakfast, dinner, etc I'm going to start with linear or rigid systems and then we're going to move on to more realistic kind of real world systems which are dynamical systems that are differentiable but can't be, they're smooth, they're nice in a lot of ways but there's no nice easy formula to write them down or if there is, the formula involves like trying to figure out what's going on the formula involves like transcendental functions it's not nice and linear like the systems we're going to consider at first. Okay so the setting for these lectures is the world of measure preserving transformations and to introduce notation what are we looking at we're looking at a triple of objects called a probability space which together form a probability space x is a set, e is a sigma algebra probability measure meaning that the measure of the whole space is one. So it's a measure defined on b is the set of objects that you can measure they consist of subsets of x with various properties a measure preserving transformation or MPT for short so that's measure preserving transformation is a map just a set map from x to x it's measurable which means that the pre-image of a measurable set is measurable, the probability measure mu is preserved and we write that f star mu equals mu f star mu is just the push forward, measures push forward so if I take a b measurable f star mu of b is just defined to be mu of f inverse of b. Okay so this property is the same is the same as saying that the measure of f inverse of b is the measure of b for all b measurable okay now for the key examples that we will consider typically x will be at the very least a compact metric space b is the sigma algebra generated by open sets a Borel sigma algebra and then mu is some measure defined on the Borel sigma algebra and so we write to denote a measure preserving transformation we package all this notation into a single picture and we write this to indicate that f is a measure preserving transformation it preserves all of this structure the one weird thing is we take f inverse of b it doesn't have to equal b it needs to be contained in b okay that's important I shouldn't have said equals and you might think that's weird why not f of b contained in b well that's because f inverse taking inverse images preserves the operations of intersection and union and sigma algebras have to be closed under intersections and comparable unions so this is the sensible way to measure our measureability just as continuity is defined by saying f inverse of the set of all open sets is contained in the set of open sets so an example when x is a compact metric space b is a Borel sigma algebra generated by open sets of a measurable map would be a continuous map so typically our maps will be continuous so first I'm going to say what is ergodicity I think first I'm going to talk about some examples alright so one family of examples that you've seen last week are rotations of the circle think of the circle as the quotient of the real numbers by the integers so the integers the integers are a subgroup of the real numbers I add two integers I get an integer I invert an sorry an additive subgroup so minus I don't know why that was 3 that should be minus 1 and this quotient is also a group if I want to picture the circle again this should be reviewed from last week if I want to picture the circle as this quotient I can just take all the numbers between 0 and 1 inclusively and none of these numbers differ by an integer so they're all distinct in the quotient but 0 and 1 are the same and so in the quotient and everything else in the real time can be brought into by translating by some integer can be brought into this interval so this interval is what we would call a fundamental domain and we can think of the circle as just the interval with the two endpoints identified so what's a rotation of S1 well I just take an element of the group so I take alpha on the right of R mod Z well if I like I could just take alpha in R because what I'm going to say will be well defined and I define a transformation F sub alpha from R mod Z to R mod Z that sends a point X to X plus alpha where we have to think of if we think of this as being a map of R1 that takes a point X to X plus alpha and then you have to take it mod 1 or mod the integers and why is it called a rotation well it precisely is a rotation if I take a point here and I add let's say alpha is a small number I add alpha I've moved it now if I add alpha again I'm over here but now that's identified this is identified with the point here and so if I think of R mod Z as a circle of circumference 1 then rotation by alpha then F alpha takes any point and it just rotates it through 2 pi alpha the angle 2 pi alpha now I've given you a transformation I haven't given you a measure I haven't even given you a sigma algebra but let B be the Burrell's sigma algebra so just the sigma algebra generated by intervals and what's mu well mu I think you can't see that I'm going to go over here mu is the Lebesgue measure how is it defined well mu of any interval AB is defined to be so here AB AB are chosen are chosen in 0 1 for now with A less than B so it's chosen so that the measure of the interval AB is B minus A now of course if I have an interval that is not contained but it has a length less than 1 I can always translate it over and get an interval in 0 1 and that's the measure and I take the measure at the length of that interval once I have defined again I apologize if this is some review but once you've defined a measure a sigma algebra or let's say on a subset A is the set of intervals in general if I define a measure on the set of open sets and B is a Burrell sigma algebra then something called the Karatea-Dory extension theorem that in fact this extends so I have defined a measure on the set of intervals so it's not really a measure and it has some nice properties so it's additive etc so it has properties as close as you can get to being a measure but once you've defined it here with the right properties then the Karatea-Dory extension theorem implies that this measure this extends to a unique measure and that's the Lebesgue measure and in fact this measure is a very special measure is in fact the unique measure that is invariant under all translations mu this particular mu I've just defined is the unique measure I should say I want A less than B and I want B-A to be less than or equal to 1 here mu is the unique probability measure r mod z such that for any alpha in r mod z F sub A star of mu is mu so in other words this is a measure that's actually invariant under any rotation and how you prove this very quickly is you do the following and this is I think a good exercise for you know if you haven't done this before proof first of all go to Google and look up the actual Karatea-Dory extension theorem okay step two show that F star or F alpha star of mu and mu are the same measures this subset the set of all intervals okay well that should be clear right if I take the measure of an interval take the pre-image of that interval under any rotation the length doesn't change so it assigns the same measure to intervals the push forward in this show that they are the same on A and then the uniqueness of the Karatea-Dory extension theorem tells you that the two measures that they generate have to be the same okay so cool because I didn't really I just gave kind of like hand wavy kind of basically incorrect definition or statement alright so therefore so this is a very interesting fact and it has consequences but since we're still talking examples let's just go back to what we were saying if I take any alpha in R mod Z the rotation F sub alpha acting on R mod Z B mu where this is Lebesgue this is a measure preserving transformation okay so that's our first most basic example and I wanted to talk you through it slowly before we move on to example two which is another example from last week so this is the doubling map but more generally we can do expanding maps so the doubling map is also I'm going to call it F again but I won't put a subscript that's also a map on the circle and it's defined by F of X is equal to 2 times X now you might say well how does this make sense well I need to take this mod 1 or mod the integers why does what I wrote make sense well it makes sense because if I have R up here and I have the map that sends X to 2X and I have this projection where I take the quotient by the integers so this is the projection okay where X a point in R is just sent to the coset if you like X plus Z then this diagram commutes meaning like if I take an X here and I add some integer N and I double it I get X plus 2N and that's an integer so I can project that down here okay so just from like that kind of point of view this makes sense if you want to look at this map however as a map on the interval the fundamental domain 0 1 think of this as a map I'm going to draw the graph of F I take the map F of X equals 2X as depicted this is not a map from the interval to itself it's a map from the interval to the interval 0 2 but now I try to make this picture look better so there's one in these numbers I have to make them inside the fundamental domain 0 1 and this is what the graph looks like and since these points are identified I'll just draw it like that okay exercise proof so this is level 1 exercise part 2 so let mu again be Lebesgue measure measure let B be the Borel sigma algebra for this doubling map F star of mu equals mu prove that mu is invariant now you might think that doesn't make any sense because if you double you double length so how on earth could I be preserving Lebesgue measure but you remember that F star mu of that is mu of F inverse of the set for those who were here last week I assume this has been done as an exercise in class okay so it's a good review but I just want to note if I take F inverse of let's say an interval I actually get two intervals of half the length of the original and so the measures of those two intervals add up to the measure of the original these are this again is review I hope but I'm also trying to introduce some new notation example 3 is now in higher dimensions so you guys have like seen the circle to depth so now let's talk about higher dimensions so let's let x equal the torus and we're big boys and girls so I'm going to do this on an n dimensional torus but then I'll give you some examples I'll give you I'm going to talk you through an example where n is 2 so x is the n dimensional torus it's the quotient of R n by the subgroup z n well mu is going to be Lebesgue measure on this quotient how am I going to define this well I'll define it this is a product of R mod z with itself so exercise n times so this is the torus it's a product of n circles mu is going to be Lebesgue harm measure we can define it as follows mu of a rectangle of area less than or equal to 1 is going to be that's a rectangle it's a little product set if you like so product instead of saying a rectangle of area 1 let's just say a product of intervals j j equals 1 to n or even a product of sets why are we being so why am I being so gentle here so b j is a copy is a measurable set inside of b the measure of the products of the b j's is going to equal the product so this is a product of sets so maybe instead of writing it this way I should write this as that's a product an enfold product is the product of the measures this is on the circle of the individual sets okay so this is a construction of a product measure exercise if you've never seen this before is to show that this actually defines a measure so once you defined these are kind of like a product set so sort of rectangles rectangular sets these generate the sigma algebra of all Borel measurable sets so I should say b equals and now we're going to do something that generalizes that generalizes multiplication by 2 so we're going to take a set a sorry a matrix n by n matrix integer entries and let's assume that the determinant of a is not 0 very important then takes well if I look at multiplication by such a matrix and I take a vector whose entries are integers then I get a vector whose entries are integers so I get that this quotient that sorry that a of z n is z n but now that means we get a commutative diagram as before here we send a point x we have the quotient but now this is the quotient by the integers well if I take an x and I add some m m in z n and I apply it by I apply an integer matrix to it I'm translating again by an integer and that means that there's a well-defined quotient map to this quotient and I'll call it f sub a or maybe I'll just call it a sometimes because it's very similar to a and this is what's known as homomorphism of the torus so and it's non-trivial so let's give let's name this quotient call it t n f sub a mini exercise this is a group and this is a homomorphism a little bit harder is to show that f sub a of Lebesgue is Lebesgue it actually preserves Lebesgue measure so that's also an exercise but do it in the case where the determinant of a is 1 that's the easiest case because determinant of a equals 1 this implies that f sub a is actually invertible in fact it's almost equivalent because determinant of a could be minus 1 this implies that f sub a is invertible and so to prove that the measure is preserved you just need the key observation from calculus actually from linear algebra is the following fact if b is a subset of r n measurable so Borel set the Lebesgue measure of f inverse so let's draw a picture is invertible we have the so-called change in variables formula we're going to see this again but in the non-linear setting where we replace a by a determinant here's a ah sorry here's b here's a inverse of b there's the preimage of b I'll draw it over here the Lebesgue measure of b sorry I can't see that the Lebesgue measure of the set b is the same as the determinant of a the Lebesgue measure times the Lebesgue measure of a inverse of b this is actually the change of variables formula or the key and I should say this should be the absolute value of the determinant because maybe the determinant of a is negative that's a key property of linear maps and that's in non-linear dynamics that's what you do a non-linear version to prove that various systems preserve measure or have certain properties with respect to measure okay how measure is transformed Lebesgue measure okay now so just to summarize this n-dimensional torus with again the Borel sigma algebra I keep using B kind of for different things but this is different spaces this is the Borel sigma algebra of tn and this is Lebesgue measure on tn this is preserved by the map f sub a as long as the determinant of a is non-zero so determinant one it's pretty easy higher determinant you have to know something about what's called the degree of the map so the fact like with multiplication by two the number of pre-images was two and higher dimensions the number of pre-images under a map linear map a on the torus number of pre-images is the absolute value of the determinant of a so it's a very similar calculation but don't worry I'm not really going to talk much about higher dimensions except for dimension two for now I need to show my slides and there they are brilliant okay so now in the last I just want to slowly walk through I want my clicker to work but I want to slowly walk through an example and this is the example we're just going to beat the crap out of this week you're going to be so sick of this example by the end of the week but we're going to use this example to generate other examples okay so here we are dimension two just want to make sure we're all on the same page so the torus the two dimensional torus is r2 mod z2 okay there's r2 there's a distinguished point in r2 that's the origin linear map right linear maps linear transformations preserve the origin inside, I can't move basically inside of r2 just like we had the integer points on the real line we have z2 those are the points with integer coordinates and just to give you a sense of why that quotient is a torus and how we find the analog of zero one with its endpoints identified let's look at what happens when we take a line segment in particular the line segment here between zero zero and one zero and we add something in z2 so for example if we were to add the vector one one that line segment would move over there and in the quotient those two line segments are the same really that's the same that's the same and so on okay so in particular those two line segments are the same because I can move by one zero and I get the same segment well similarly if I take this segment segment zero one I'm sorry one zero I can translate it around by elements of z2 notice how the endpoints always end up in points in z2 because in the quotient all those black points become the same point and in the quotient all of those blue vertical lines segments become the same segment and similarly for the red now inside of that square no two points inside of that square are identified with each other right on the other hand any other point in r2 like if I went over there can be translated back into that square right so if I had a point over there somewhere in that square I could translate it back so that square is what we call fundamental domain for the action of z2 on r2 that is to say the torus the two-dimensional torus is just a square probably many of you or if not all of you have seen this before it's just a square with the opposite edges identified by translating either vertically or horizontally now by the way that's not the only fundamental domain for the action of z2 on the square in fact I could take well I could take for example a parallelogram of area one and translate it around and that could be with the opposite edges identified that's also a fundamental domain in fact if I take any matrix with integer entries and determinant one and I apply it to that square and I glue opposite sides I get the same old torus it's also a fundamental domain so that's just a little point it's not so crucial to what I'm going to say I'm going to go about five more maybe this is in the way or something so now I'm going to consider a very special linear automorphism of a two-dimensional torus the quotient and that is where I take a matrix a equals to this integer matrix okay notice it has determinant one so it's going to produce an invertible map on the torus an invertible what you might call an automorphism because it's a homomorphism that's invertible so we call this a linear automorphism of the torus we also call it the cat map and I will explain why at the at the end of this lecture so let's just see very very concretely what happens when we apply 2111 to the torus here's our fundamental domain now let's just apply a to that square we get a parallelogram and in fact well the two columns 21 and 11 describe the images of those two corner points right and then that's 32 that's the sum of these two so that guy there is also a fundamental domain for the torus and I'm going to simplify opposite edges but now let's reassemble it to see what the image actually looks like from the point of view of the original torus this is like when we graphed x goes to 2x and then we cut this piece down and showed the graph on the interval so there I'm going to overlay four isomorphic fundamental domains of my original map the image into pieces and then reassemble them now that picture that I just gave gives a good indication of what this invertible map look does for example and we'll see more of this later the eigen directions for 2111 well it's a linear map right they're actually perpendicular in this case well when I apply the map one eigen direction which is the eigenvalue greater than one so there's two values here one greater than one the other the inverse less than one one is going to get expanded and one's going to get contracted and you can sort of see that just by one iteration of this map that we get this sort of squishing in this direction expanding in this direction now why does this map preserve area well there's some set sitting inside let's just look at an open set and let's just see what is the pre-image of a set like this really concretely well if I apply f to that set that's inside I don't know why I did it this way ah yes I'm sorry so if I look at that original set but I express it in terms of this new set I'm sorry f of this set it looks like this and now if I apply f inverse to this object I'm going to get a set that well because I did f inverse it's going to be kind of stretched along the contracting eigen direction and expanded along the expanded along the contracting eigen direction for a because it's the inverse and stretched I'm sorry and contracted along the span expanding it okay so in a case you looked at a picture of this inside of the plane this would be what the pre-image of that set there looks like under 2111 and it has the same area for precisely the reason that the determinant of the inverse of this map is the same as the determinant is one of the determinant of this map which is one you can see that those if I drew them reasonably they have the same area so we're going to examine a lot of properties of this map later perturbations of this map here's why it's called the cat map so there's a book by Arnold and Aves it's a classic book it's a bunch of lecture notes of Arnold's and at the end of the book is a huge sequence of appendices and this is an illustration from that so this isn't actually the map 2111 it's the map 1112 okay very similar and what Arnold did was very similar to what I just animated here except he wanted to show what happens to a cat depicted in the lower left when you apply the map to the cat and then reassemble and that's a picture of what happens when you do that twice and what you're seeing is that the cat is getting really really stretched out and really skinny in this direction and it's basically moving everywhere in this torus and that's a property called mixing it implies a property called ergodicity and that property is what Hannah is going to talk about in the next lecture by the way I'm deliberately going over a little bit because of the activity at the beginning but we'll just stretch things out a little bit in this lecture so that's the example we want to just understand let me give you some exercises so I gave you some kind of beginning warm-up exercises you might have seen it last week remind yourself show that all three of the examples to the extent that you can exercises to summarize one is to show all examples that I gave are measure preserving transformations secondly the board so I shouldn't give exercises yet at 2pm so I'll just give you a quick preview because I don't want to thank you so I don't want to waste too much time but it'll give you something to think about and then we'll state them in more detail so the second exercise is to prove that the set of periodic points for 2111 so that comes back to itself prove it that set so 00 something that set prove it that set is dense in T2 another exercise is to prove that create a puzzle the puzzle will consist of five right triangles and you can assemble them in two different ways one is as two squares and the other is as one square think about the Taurus and try to construct such an example think about 2111 it might be helpful next exercise and I will give hints and guidance for some of these next exercise is quite a bit harder so construct two measures on the circle that are both invariant under multiplication by two they both are fully supported meaning that the measure of any open set is positive okay so they're both and yet they are singular with respect to each other meaning that there's a set of full measure a set that has measure one for one of the measures that has measure zero for the other construct two of those and if you're really feeling ambitious construct infinitely many exercise three or exercise N think of the middle thirds canner set as a subset of the circle and show that that's invariant under multiplication by two and that's enough I will give more details so I will artfully curate which ones we will talk about in the session for now those are just things to think about over lunch or something but I will narrow it down to three questions or two questions that we will talk about in the session okay so I should stop now