 Okay, can everyone hear me? Okay, so let's start now You know, I've always been sitting out there. This is my first time on this side of the room and it's quite scary Anyway, I guess I can't say welcome anymore because by now hopefully you're settled in but it's really been a pleasure to meet Most of you I still haven't met all of you So hopefully in the next week I have the chance to talk to all of you Yesterday you had a presentation about ICTP by Stefano If you have any questions about ICTP or you'd like to chat about the diploma program, especially I'm more than happy to talk to you about it I am not not from the perspective of a student But as I'm a postdoc here and I at least can help a little bit with maybe the application Process or some of the things that I know about it and the experience of students with whom I've talked anyway Okay, so today I'm going to do an introduction to Ergodicity so probably many of you have seen most of what I'm going to say Tomorrow Davide will do ergodic theorems but ergodicity is something Extremely important in dynamics and also in other fields like for example even in probability theory. So Okay, but I will do it mostly from a dynamics perspective. So I would like to a little bit motivate Why we care about this concept of ergodicity And then I would like to define it and hopefully work through some examples to show Some various techniques that are used to prove whether a system is ergodic or not. So Okay, so just to I guess it's almost happened at the beginning of every class but to remind you our setup We have some space some set X We have the Borel sigma algebra the measurable sets of X We have a transformation from X to X and we have a probability measure and I'm going to assume that Mu is T invariant so I'm going to assume now that we're always talking about a measure preserving Transformation and I'm going to write NPT so that I don't have to write measure preserving transformation every time Okay So, okay, so let's suppose suppose. Suppose. There's a set Suppose there's a set a Such that a is a T invariant set so such that T inverse of a equals a and Suppose that it has positive measure, but strictly less than one so Something like this X to X Let's say I have some so a that's T invariant so I can decompose X like a union a compliment Okay, let me consider a conditional measure mu a defined by mu a set intersection a over mu of a and I can do the same thing with respect to a compliment side mu a compliment the set Intersect a compliment here and one minus mu the measure of I so I can do this right Yes, okay, it's clear to everyone that these are probability measures still Maybe it's not so obvious, but they're also still T invariant measures So let me show that I'm assuming that a is T invariant so I can replace a by T inverse of a So I have mu T inverse of a and I can pull out T inverse inverse of the intersection and Mu is T invariant So this is the same as mu of b intersection a over mu of a Which is exactly mu a of b Okay, and this is the same for mu Restricted to a compliment. Yes So now I have decomposed mu into two P invariant probability measures Yes, so if I think of restricting T to a and T to a compliment Then I have it can decompose this measure preserving system into two separate measure preserving systems this Characterization that we would like to find is one when a system cannot be decomposed So there's an analogy drawn sometimes between Ergodic systems and prime numbers So you decompose every integer into the prime numbers, right? It's factorization because the prime numbers are things that you can't really decompose further So we think of these measure preserving ergodic systems as the units the building blocks of other things the things that can Not be reduced anymore Okay, so that's one motivation So we want to find a characterization for when this cannot happen And maybe you already have in mind which property here, but before I actually Introduce the definition. Let me give a The same but a slightly different characterization of this so I'm going to denote by M Xt the space of T invariant probability measures on X Okay, so one thing about this space is that it's convex So Mxt is convex mean so that means that if I take lambda between zero and one Any two elements, so if I take mu one mu two and M Their linear combination is also still an M So mu one lambda mu one plus one minus lambda mu two is still an element of that so this space has this very nice property and Extremal points of a convex space. So an extremal point. So mu and be Written as a non-trivial linear combination of other elements in the space, right? so whenever Mu equals lambda mu one plus one minus lambda mu two For lambda strictly between zero and one then mu equals mu one equals you too Okay, so there is a Characterization as well as the extremal points of the space of T invariant probability measures are the ergodic measures Okay, so I haven't proven that it's probably clear from this decomposition that extremal points are Ergodic measures, but the other way needs to be shown too I think the proof goes if you're not an extremal point then you're not ergodic But the proof requires a little bit more than I want to discuss here. However, I'm happy to talk about it later in the exercise session okay, so if I would like to Prevent this decomposition from happening. What is the property here that I would have to change or what is the property here? That allows me to write these conditional these conditional probabilities So I I want to keep I will have invariant sets so I I mean I want to keep a Comment about invariant sets here, but what's the property of the invariant set here that allowed me? Yes, right okay, so that it's between zero and one so if the measure of a were one then the measure of its complement would be zero and I would have a problem here, right If the measure of a were zero then I would have had a prop problem here. Yes Okay, so this is kind of the motivation for the first definition So a measure preserving Transformation is ergodic Reset so if for every set that's T invariant so for every set a First of a equals a The measure of a is zero Or one Okay, so There are other ways of Characterizing rigidity or perhaps using this definition to come up with equivalent statements for different objects So let me Introduce that well, let me give you a claim as you're preserving so sorry MPT Doddick every am I writing large enough? Can you guys see on the back? Yes, okay, thanks if for every Measurable from X to R such that F composed with T equals F F is constant almost everywhere Okay So you can see that we have gone from a characterization in terms of sets to a characterizations in terms of functions Okay, I'm going to prove this claim as Written, but just to make a comment you can replace measurable or L2 here I should put this is me almost everywhere And this invariance Can also be me almost everywhere Then you can extend it Pretty easily, okay, but for now the statement in white is what I would like to prove So we go from a characterization of sets to a characterization of functions These are kind of the two that I'm going to work with today But I would like to say that you can also go further to a characterizations in terms of operators So if you look at this composition as an operator acting on L2 It's a linear operator. It's an isometry. It preserves the L2 norm and the characterization of ergodicity is that Any eigenfunction corresponding to the eigenvalue one is a constant function almost everywhere so it may be unclear what you gain from that but Once you introduce the setting where you can use functional analysis. It becomes extremely useful So actually the third characterization is also very important, but I won't go more into that today Okay, so let me prove this claim Okay, so let's go this direction So I suppose my measure preserving transformation is ergodic And I want to show that any t invariant function is constant almost everywhere. Okay So let me take a Function f composed with t that equals f and I want to show that this is constant. Yes Okay, so let me consider the following sets T is the set of all x's and x Such that f of x first of all, is this a measurable set? Why do I know it's a measurable set? Yes, it's a pre-image of the interval t to infinity, right? I've been assuming is measurable. So this is a measurable set Now just a comment if f is not constant over at almost everywhere then it takes at least two different values Right on some positive measure Sets right so for some t if the function is not constant For some t this is going to be a positive measure set that doesn't have full measure, right? But if I want to show that f is constant almost everywhere I need to show that every such set a t has either measure zero or measure one Okay, good because I'm assuming ergodicity. So the only thing I have to show is that these sets are t invariant sets And I you know I use this definition one So let's do t inverse of a of t Well, this is such that t of x I can write this also x and x such that f composed with t of x is greater than t Well, I'm assuming that f is t invariant. So this is just The set of x such that f of x is greater than t And now we've returned to it just being a t Okay, so these are indeed invariant such which is good news for us because that means that they have only So mu of a which implies f. So we have proved the first direction. Is it clear? Let's prove the other direction The other direction being I'm going to assume that any at t invariant function is constant almost everywhere And I want to show ergodicity Yes, okay So let me take it t invariant set a so let now a good way always to go between Set and function is to consider Characteristic functions right of the set. So let me think about composing the characteristic function of a with the transformation t I get the characteristic function of which set the inverse of a right I want all the points that would be mapped into a under t Under one iterative t right so this is the indicator function of t inverse of a now I chose a to be a t to be t invariant set So this is just the indicator function of a now I'm assuming that any t invariant function is constant almost everywhere the indicator function can only take two values right zero or one So either so either it's Equal to zero almost everywhere Which means that the measure of a Which I can get by integrating over the space the characteristic function of the set d mu is going to be zero or I get that This is one almost everywhere Which then implies that the measure of the set for the same reason? So that proves ergodicity So We have now just proven this claim is it clear so let's do a few examples to actually see how we use these Characterizations and definitions in practice okay, so The first example. I'm going to do is an example. You've been seeing a lot about so I'm just going to do circle rotations So example one of the only example. I will work through fully So rotations so recall that the rotation we I guess we say it maps from our mod z to our mod z and For now, let me assume that we are talking about the levain measure and I'm going to use I'll use additive notation for now. So I have x plus data mod one Where we can have e to the 2 pi i x x plus data Okay, and so let me consider first the case where theta is rational and to do an absolutely explicit Example, let me take theta to be one fourth. Let me construct this set a in the following way So I'm going to take zero to one eighth going to take one fourth to three eight going to take one half To five eighths, and I'm going to take three fourths to seven eight If I draw this on my circle Okay, so under the rotation with theta equals one fourth What happens to this first interval? Where does it get sent? So the next one right and this one gets sent to the next one and this one gets sent to the next one Okay, so what can I say about a under t? Yes, in terms of in terms of the of how I'm saying a set x under t What does that mean? For the whole set It's invariant right because I'm Staying within here at the iterate. It's a t invariant set. Yes. So t inverse Of a equals a Now what's the measure of a One half right i'm considering the big measure It's one half. Yes. What does that mean? Is this rational rotation ergodic with respect to the big? I found a t invariant set that was had measured One half so neither zero nor one so it's not it's not ergodic and in fact for any Theta that's rational so any theta that's p over q you can always think of the set a i goes from q minus one i over q One i over q plus one over two something like this So you can always construct the construct the analogous set for the appropriate rational number And uh, this will be invariant, but it will have Positive measure and not measure one. Okay, so rational rotations are not ergodic with respect to the big I have a question though Can you think of a measure for which this transformation is ergodic? Delta measure on what? You're close. I so I want to okay Think about so think about what you are saying with periodicity instead of considering sets here Let me consider. I don't know for example points in these sets. Let me take zero Let me take one fourth. Let me take one half. Let me take three fourths, right This is an invariant set right what if I take a measure That's supported on this orbit I need to ensure that every point has positive measure in here, but let's say the Full support of the measures here What do you think do you think that's ergodic? Yes? Yeah, maybe you could I don't know you could put a measure one fourth one fourth one fourth one fourth Okay, I want the full measure to be just be supported on these points, but I want them to have positive measure each Okay, so what about this example? Yes, right and invariant set must contain all of these points or none of these points If you have only two of these points in a set or three of these points in a set Then its pre-image is going to have different points. It's not going to be invariant Okay, so in fact you can have you can think of rotations as ergodic with respect to singular measures measures that put positive measure on sets that are measure zero in the bank okay, so You wouldn't be losing that much in the rational case in general though if you think of a system That has dense orbits As well as many periodic orbits and you consider such a measure Yes, you might have ergodicity, but you also will be losing a lot of the interesting dynamics that are going on because you're kind of Giving the rest measure zero So I guess my point is that first are considering probability measures Then we say okay, but important measures for us will be invariant with respect to the transformation And then we say okay, but we don't want to reduce we won't we don't want to be able to decompose our system further So then we're considering ergodic invariant measures But even ergodic invariant measures aren't necessarily all important for us So there are still further characterizations of measures that are more relevant or less relevant for the system that you're considering I don't know if anyone will talk about this in the rest of the school, but Anyway, just to keep that in mind Okay All right, so let me just say That I could show this using the claim by thinking of a function that's not T invariant I mean a function that is tan variant that's not constant. So for example, let me take the function f Of x equals e to the 2 pi i x q Okay, so if I do f composed with t of x I get e to the 2 pi i x plus theta Which is p and this is e to the 2 pi i q of x and e to the 2 pi i p which is 1 So I just get f of x back So I can also find an allegesly to a t invariant set That has positive measure between zero and one I can find a function That's t invariant. That's not constant. I guess also I should mention Sometimes we oscillate between saying a measure preserving transformation is ergodic or a measure is ergodic, right? So this is not a measure of preserving transformation if I consider the vague if it's rational, right? But it could be if I considered the measure that we discussed so if you have If the Transformation is clear and you're trying to choose a measure sometimes people just talk about ergodic measures So you'll hear it phrased in different ways Okay being irrational so theta not being in q Yes, so I was talking about the vague measure Good. Thank you. Yes But almost everywhere here is large, right? I mean it's small So yeah, so anyway, but yes, I don't want this right now. That was just a comment. So let's forget about this now. I'm back to the vague Okay so Let's consider the irrational case I would like to use the invariant function characterization. I'm going to need the Fourier expansion of a function Does anyone want a quick explanation of Fourier series? Or can I just use it? Okay, I'll just use it. Let me then just say Briefly a few words just in case it's the first time you hear it. So I have here a function from let's say R mod z to r or I consider consider any function from r to r if it's periodic let's say one period one for now And I want to I can decompose this function into its harmonics. So I write it in terms of Basic oscillatory functions like cosine 2 pi and x sine 2 pi and x and I look I write it as a linear combination of these Right and they have varying frequencies and you add these up and you can represent your function in this way Okay, and if you write these using in in the complex way, you get a very nice Concise way of writing your function as a sum So I'm going to take write it as n and z a n G to the 2 pi i and x so I'll look something like that Okay, so this is a way that I can represent my function so um And something that you're going to need in one of the exercises exercises I give you today is the Riemann Lebesgue lima So if you have an l1 function l1 periodic function on r Then uh the Fourier coefficients go to zero and modulus as n goes to infinity Okay, so these guys the norm of these guys go to zero as n goes to infinity Will be useful. I think for one of the exercises later, but I'll write Riemann Lebesgue lima to remind you guys Okay, okay now suppose That this f came from choosing so I want to take an f That's tn variance right I'm taking an f. That's tn variant under my irrational circle rotation And then I write its Fourier expansion like this So let me write then what f composed with t of x looks like e to the 2 pi i and x plus theta This is just e to the 2 pi i and theta e to the 2 pi i and x And I chose a t invariant function. So this must be equal to f which means that this must be equal to e to the 2 pi i and x So I have a property of Fourier series That if these are equal then the coefficients are the same for every n. It's uniqueness Okay, so I have that this Must be the same as this For every n Okay, so that means a n e to the 2 pi i and theta has to be equal to a n for all n equals zero. I'm in good shape Trivially yes, that equals zero Well, unfortunately e to the 2 pi i and theta It would have to be one for this to hold. Well, that's only true if theta is rational Can only be true if theta is rational So if I'm restricting theta to be irrational, this is never one. So that means that all of my Coefficients for n not equal to zero are identically zero. So that means that my function is really just A not so that means that's constant, right? Okay, so then by the claim I've shown that Irrational circle rotations are ergodic with respect to the Lebesgue measure So Fourier series are really a crucial technique for proving ergodicity So today in the exercise session I will ask you to prove that the doubling map that you've also been seeing a lot about is ergodic with respect to Lebesgue And I would like for you to do it Showing that invariant sets have measure zero or one and I would also like for you to try to do it showing that Invariant functions are constant almost everywhere and you will use The Fourier expansion and you will use Riemann Lebesgue lemma Try to do it in two different ways so you get used to the different techniques Okay, so if f is in l1 I'll say of x mu yes then I look at then I can write it like this so I want this let me It has to be periodic in r or it has to be from r mod z to r. So I I'm taking x Let's just say for our purposes. It's going to be zero one Okay So I function l1 Let me just put that here since this is what our example will be to x mod one is zero one Lebesgue measure mu then I write its Fourier expansion and Then you have the property that the modulus of the Fourier coefficients go to zero as n goes to infinity Not you not always Maybe I mentioned things about lp spaces at the end and containment and things Yes, I'm mixing between using measurable l1 and l2. This is a claim. That's true for l1 functions, right? Sorry say again. I didn't hear you. Yes. I am Yes, okay. Yes. I am using a stronger claim, right? So let me introduce a third type of example So I'm going to talk about the shift space that you've already seen defined in a particular setting by hana. I think yes You remind your example to Full shift Okay, so I like this example And this proving our goodicity for this will be the harder of the two exercises. I think it's really tricky Maybe for some of you it's easy But I like it because the invariant measure that we consider is not Lebesgue So we leave a little bit the world of Lebesgue and we consider a different type of measure So consider the seek the set of sequences at sequences everything I say Is also true for one sided, but I'm just going to assume we have two sided sequences And each coordinate of the sequence comes from some alphabet of k letters I'm just going to denote them by numbers one through k And remember the shift map Sigma little sigma moves everything to the left, right? So the image under sigma of a sequence I have A decimal or something marking at the place here This will look like so in the case of one sided sequences I Kind of don't I don't have a left hand side before the decimal and I get rid of the left most coordinate. Yes Okay, so you're familiar more or less with this map If I want to define a measure on this space, so I think you saw a metric For this space. I want to just define a measure on this space So I'm going to go back to what Oliver said and to what Irene said You know we have these sets that we care about we have the braille sets And we really can because of we have all these nice extension theorems We can really just say things about the generators of the braille sets And define things on the generators and then it nicely extends to the whole space, right? So I want to Do this because it will be easier for me to define the measure on just the generators of the of the of all the sets But I need to find what these are so in r or in zero one The generators are intervals, right? But I need to find the analogous concept here. So what are my generators here? So generators in this space are cylinder sets. So these will be generators of the sigma algebra And what do they look like? Well cylinder sets are Sets of sequences that are formed by fixing finite strings of symbols. Okay So you say from this point to this point. I want it to look like this and then you're free outside of that Okay, so let me write this I have y m This is the set of all x in sigma such that x i equals y i for all i Between m and n. So these are my cylinder sets. These are my generators. I'm going to think of everything in terms of this So if I asked you to prove ergodicity of the measure I defined Then you're going to have to look at sigma invariant sets Is that well The things we have to work with that we can say a lot about are the cylinder sets So try to always approximate sets by cylinder sets or unions of cylinder sets Okay, okay, so now I need to define a measure on these So let me take a probability vector okay So think of this like the probability I get one the probability I get two the probability I get k So you can if this is on two symbols You can think of this as flipping a coin if it's on six symbols. Maybe it's like throwing a die So these are independent, right? It doesn't the probability of getting One of these symbols in a place doesn't depend on the place Each of these are positive and their sum is one So with this I'm going to define the product measure. So the measure of a cylinder will just be given By the product of the probabilities of each of these symbols, right? So this is going to be p of ym Multiply it all the way to p of ym So this is my probability. This is my measure. Sorry And Let me start off by at least showing That it's sigma invariant. So claim sigma invariant So let me consider mu of sigma inverse of ym Ym what is the inverse image under sigma of this cylinder set? Yes, so let me just write it as a sequence. So I have ym To ym fixed and you're saying okay since sigma moved to the left sigma inverse is going to move it to the right So then I'm going to have an open spot where I had ym and everything was shifted to the right And what can I put here anything in the alphabet, right? So this inverse image Is sigma as mu of the union from i goes from 1 to k Of now the new cylinder sets i Ym I have full freedom once I shift it over I can have anything there All of those sets are in the pre-image of this cylinder set Now what can I say about each one of these cylinder sets? Do they have overlapping elements right because If I put if I consider the sequences where I put one here And then I consider the sequences where I put two here Well, they can't be on the same set none of none sequences with one here are going to have two here So all of these cylinder sets are in fact disjoint So I can write this as the sum of the measures Of each of these new cylinder sets have a probability For each one of these they're they're independent so I can pull this probability out. I get the sum i goes from 1 to k of p i mu times the probability of my original cylinder set Now because I have a probability vector I said all of these guys add up to 1 So when I actually do this sum this part goes away And I'm just left with mu of my original cylinder set So it's indeed sigma invariant Yeah, but it's a union of more cylinders other cylinder I mean is the union of two disjoint intervals an interval necessarily I mean I'm considering I'm considering Maybe so maybe so I Need to think if there's something that pops up I really want The cylinder sets to be I want you to think of the cylinder sets of strings that are Connected strings that are fixed Not that you have one string fixed here and one string fixed here So I just it's it's enough to consider this probability on The cylinders that have this the the string fixed in one place I'm not I mean and then I can extend it for sure But so maybe there's another way to do this but I wanted to do it with the basic building blocks It's just M can be negative M and N are integers So M could be minus 100 and N could be two Yes, so And so cylinder sets have fixed strings of different lengths As long as you want as short as you want Okay Okay, so your task today is to prove. Yes. Ah, well that So the measure of that will be the probability of y m Times the probability of y m plus one blah blah blah all the way to the probability Is the product So my claim and what I didn't show you but what it follows from all these extension theorems That we're saying if I define the probability the measure on these generators This extends to a measure on the space. Okay, I'm not showing this but I'm claiming this from things that we've heard before as well so This is all this is the information you need to know to show ergodicity in the exercise session It's not an easy task. I think but hopefully it's fun Okay, that's all I have for ergodicity. Let me very briefly just tell you what LP space is because it was We realized that we're using these functions and we haven't defined them The spaces at all So I think davide will use l1 tomorrow certainly So I promise I will be very quick So I have a measurable space here and I take a measurable function for now Let's say it's a real value function and I take p between one and infinity So I define a norm on this function Which looks like the integral over the space the absolute value of the function raised to the p Integrated with respect to the measure mu to the one over p So this is a norm And the space of functions the LP space of functions are all the measurable functions on x Such that the LP norm is finite But this is what we mean by an l1 function or an l2 function. In fact, they think really all we're going to need is p equals one or two in general uh I don't think we'll need Okay, I don't think we'll need l infinity functions um One thing I think that's important. Well, these are vector spaces and one property Nice property that we have because we're dealing with probability measures. So if mu is finite And for one less than or equal to p less than q less than or equal to infinity Lq is contained in LP We have containment. So This is nice for us. Okay Want to proof of this? Okay Okay, that's all so those are LP functions. I'm done and I will see you at two o'clock. So thank you