 Greetings. Let us start our morning session with the talk of Arthur Avila from CNRS, Poincaré and renormalization series. Poincaré series and renormalization. Thanks for the opportunity to talk in this special occasion. I was a bit concerned about how to make the presentation and I hesitated a little bit finally. I waited a little bit to mention to show something of Poincaré and just kind of it would be easier to talk a little bit about renormalization thoughts so that you are not too tired when I get to it and in a few minutes, not longer Poincaré would come in. So the basic objective that I am going to consider, I am not going to introduce renormalization in general but just the concept of renormalization fixed point will be necessary in this talk. So for us, it should be just an all-morph function, f, so f is the all-morph function satisfying a function equation which is, so you have some all-morph function f in some place that will come to that. So some iterate of it applied to some, there is some kind of scaling factor lambda. But it's given just as f is lambda. So if you look at, so lambda is some factor less than one. So if you want to look at, you have some iterate which is kind of in a smaller scale of that. It is the same as the first iterate in the original scale then you rescaled it. And then, so this is just some all-morph function. Of course you have to define the domain. It's not going to be considered the maximum analytic extension that you could have of such a function but we assume that it has some specific configuration somewhere. So there is a special point that's zero since the scaling is around zero. So there's a critical point that's zero. And there is a, the transformation takes some disk, some topological disk around zero and goes to a bigger disk as a degree two map. So there's a critical point of degree two here and then there is a degree two proper map. So you have discovery. And so if you have such a transformation like this, then satisfying such an equation, then you get this, so p is some iterate bigger equal than two as an integral. And this would be a real normalization fixed point. So you are going to be, you may ask whether such a real normalization fixed point exists and it's possible to show that for every p bigger equal than two, there is at least one real normalization fixed point. For two, there is a p equals two, there is exactly one. And for a larger p model conform to a fine change of coordinates, for larger ones, for all the z. So this is a functional equation that lambda is some scaling factor that's fixed for some. So the transformation in a small scale behaves exactly as the same under this kind of thing. So real normalization is the operation that makes this rescaling thing and this iteration. This says that the real normalization is exactly the same as the original map. So what's interesting, those are very interesting for several reasons. We can try to understand what's the repercussion of all these kind of copies of the own dynamics of large scales that appear in smaller scales and this gives a very rich picture. Traditionally, everything started, white people started analyzing this was because they just discovered the arising connection with some universality phenomena that was first seen in real dynamics. And those are going to be complex extensions of those real dynamics, which is the Fegenbaum-Coulet-Resset phenomena. That's kind of, is explained in this picture that maybe some of you have seen, which depicts in a family, here is a family of quadratic maps with some parameterization. In the horizontal x you have the parameter and in the vertical x you have the actual phase space, the space where you iterate and it shows the, the attractor of the system, where the orbits of the system, as if you iterate they are going and originally in the left side you have a very simple attractor, it's just a fixed point. Then it gets kind of mildly more complicated, becomes a period of two. There is a tractor that has two points and then it gets more and more complicated and then it's very complicated after some, there's some initialists one, two, four and so on. And this is a finite set and suddenly it's kind of much more complicated. This moment where it's kind of becomes infinitely complicated for the first time is related, it's something that's corresponding to a dynamic, that's a deformation of a renormalization fixed point and the behavior, the quantitative features of how you approach this special parameter are related to precise the properties of the renormalization fixed point, in this case of renormalization fixed point of period two. Okay, so now, so this is in real dynamics, to understand those real dynamics you are led to understand the complex dynamical systems or you can be interested just by themselves, this would lead to some pictures of the Mandelbrot set that I'm not going to show, but if you look at the complex dynamics you are going to be interested on, so here you have map, so the domain is not the same as the range, so there are points that escape and then you cannot iterate anymore. So what you define are the set, the actual dynamics happens at the set of points which are some complicated set of points here that I'm going to show soon, are the points that never escape under iteration. So formally you have this kind of Julia set that's going to be the intersection of the pre-image of the, so you have domains u and v in the pre-image of u. So you have this Julia set that's where the actual dynamics is going to take place for all times. And so I have pictures of this kind of close to the Julia set for a renormalization fixed point or at least as close as I got from it. Here's one picture of this renormalization fixed point at some scale. So it looks like this, actually you only see a little bit from the outside. The actual thing should be in black in that picture, but the colors get kind of changed as you get closer to that. So this is one picture that you get and then it's some scale and then if you look at the next scale you get another picture in a smaller scale. You get a picture like this, maybe I don't do the whole procedure of getting a full slide but you get some picture like this with more complications and you see the colors getting very approaching there. And then you look at the next scale. Oops, I jumped from one. So this should be the intermediate scale, so there is a kind of still a little bit tame thing. And the other one was the slightly smaller scale, not this one, this is the final scale where it's already very, very complicated. So this behavior that you saw that as you kind of zoom in, you get more and more complications, it's normal because when you get closer to a critical point, you actually have interaction of the dynamics in several different scales. So you have copies of the dynamics, of the same dynamics in any scale because you can kind of repeat this equation for different lambda. So at every lambda n you are going to have a new copy of your dynamic. So you have some kind of interaction between the dynamic consistent at all scales. And this reflects, for instance, in a theorem of McMullen that says that when you get, you're going close to zero, it's getting more and more complicated, meaning that if you amplify the Julia set and try to see in a little bow around zero, you're going to get things that get more and more dense in a complex plane. So as you do amplification, you get essentially the Julia set starts spreading everywhere. So this kind of complication is the first hint of the complicated geometry near zero for this kind of dynamics and it comes from this interaction of dynamics in different scales. Now the basic questions that are going to be interesting here is kind of very natural, is what's going to be the geometry of the set, the set that comes from kind of non-scaping dynamics or which actually is a dynamic that contains several copies of itself, and the most basic questions that you get once you get this kind of fractal set is what's the back measure first? Is it possibly back measure? And what's its dimension? Is it possibly of dimension two since it's very complicated and still with the back measure zero or maybe it has dimension less than two? So those are kind of very basic questions once you have a fractal set like this and that's what we try to... And it's when we ask this question that you get to the other part of the talk that we get to have to understand what the Poincare series of those maps and that's where we come to. So let's see what's Poincare series but starting from how they were introduced. So I went to get... It's not very good quality, I took some scum that was read in the internet from the archives but so anyway, so this is the... I tried to look where it's defined, it's in the first paper that Poincare did about flux and functions. So it's a paper that precedes the paper about flux and groups that he did because he was interested in flux and groups because of those flux and functions and not the other way around. So anyway, here is... In this article he defined those Poincare series and you see here the motivation, so that's why I put here and you see that it is... that he's looking for functions that are analogous to elliptic functions with the goal of integrating linear differential equations with algebraic coefficients. And to do this he comes to the need to construct these function functions that he's going to define and what are those. So we are going to see... So here's the motivation, why he's going to be doing this and in particular why he's doing the Poincare series and I give you... In the same article, in the third page now, he's going to get... So in this third article, in this third page, he just comes to this, so those flux and functions that he wants to define, he defines quickly as being the functions. So you are going to have a group of actions, so you have the Poincare disk and you have a group, discrete group of symmetries. So you have the Poincare disk, it is a model for the hyperbolic plane and you consider hyperbolic motions in this plane and that generate a discrete group and he's looking, so you have this kind of gamma, a group of symmetries and what he's going to call flux and functions are functions in D that are invariant under gamma. So that's just what is a flux and function. So just f of gamma z is equal to f of z. That's what he's going to define. It's a little bit... It's in the first page of the article that he does that and here he defines something else which is a function that he calls a data function, data function function which is some kind of function that it will transform, it will not be invariant but it will transform according to some rule, right, which is going, well, the rule is written there, which is, we recognize the rule as being some rule, so actually the data function transforms as a differential form. So it involves some power of the derivative and it is what is required. So a flux and function is a function that's defined on the quotient of the, by the group and a data function function will be something that of weight, so the data function function has a weight that's an integer m and it should be something that transforms that defines a holomorphic form or meromorphic form in the quotient surface. And then this kind of interesting situation, natural, we would be interested perhaps forced on the geometry of such surface, his interest on the function that you can define there and just to recall that it's certainly very interesting to consider because this kind of functions because actually almost all Riemann's surface arise as quotient by a Fox young group, so that there's the uniformization theorem that tells us that and so he's describing actually something very general at the level of Riemann's surface over there. So the points that just after, so that's just the definition of data function and then in the next paragraph it says how to construct a function that has those properties and he writes that formula which is a Poincare series. So that's what appears as a Poincare series. It's kind of summing, he considers a sum over the objects of the group. So he starts with some function a, I write in a modern language and then you apply elements of the group. I write like this and then you divide by some power n and he sums over all elements of the group and when you apply some element of the group to z then you see how this will transform and immediately it comes the property of how it's changed under those coordinates. So that's the way he defines a Poincare series and a little bit after, so this kind of how to construct those kind of invariant forms and to recover the Fox young functions he takes the quotient of two Poincare series of the same weight. So that's a way to get from all this to finally get some functions also. So that's what it's doing. I'm not going to continue very much on how those things develop but he looked at then, of course has to show that things are actually converging. So he writes the series, the group's infinite there's a question of whether this converges or not and it turns out that it's easy to see that those d gamma of z it's a sequence that tends to increase so it's bigger than one except for finite many values of finite many gamma. So the question is the lower the n the easier the convergence and he considers the condition n bigger or equal than two and he's doing everything here allomorphic functions so everything's restricted to integers it's an integer and he proves convergence in this case he actually proves convergence of the absolute Poincare series which is what you get if you put absolute values here so this would be the absolute Poincare series same thing so we get something like this so he proves this what he gets when you do this at this level and let's put here m equals two in the other case that he considers it is in this case we're coming from a quadratic differential to an aria form and basically the argument for convergence is just saying that the Poincare disk has in the Euclidean metric it has finite aria so that's kind of not very difficult to do to establish this convergence so here is the absolute so here's basically what how we would understand this and okay so that much for his original then he did many things so he wrote several articles on this so that later on he decides that he actually has to understand a little bit better the groups in order to say a lot of things about the functions he writes a paper about the theory of the Teohidi group Fugtian and which he says explicitly there that motivation there is preparation for his next article that's about those function functions so he develops this quite lengthy but I'm going to jump over here well first just the point is that Poincare series was a way to get some kind of olimorph function and schizo-miromorph that transform in appropriate way so in this case by this kind of power of the derivative and then 90 years later let's keep all this a Patterson he considered the problem of creating other geometric some other not any more functions but measures that transform in appropriate way also and he wanted to construct this associate to a Fugtian group and in order to construct those functions he came with using again a Poincare series so you're not going to be any more interested on olimorph objects but only on so you're going to be interested now not on this kind of objects so you're going to consider absolute Poincare series only so what he considers is so we have a Fugtian group and you see let me show a picture of a Fugtian group I just take some of those pictures from Wikipedia they're not kind of fantastic so here's one simple one so here this very simple group that's acting on the Poincare disk and it gives a little bit notion of the orb so as you iterate you more or less follow this kind of complicated network and what he considers is so everything kind of is going to infinity as you iterate so he wants to put a measure on the circle at infinity and to do this he starts putting point masses at an orbit with certain weights so he puts point mass at a fixed orbit with weights that are related to the Poincare series I would take just the Poincare series weights he for some reason he considers a different weight because he's looking at a slightly different problem but let's put the Poincare series weight and well then you get a measure that's defined in the disk he wants a measure that's defined at infinity then he considers so of course you have a choice of weight and now since now you're considering point mass at a fixed orbit so weight of those will be some d-gamma of so weight at gamma z you put let's see 1 over d-gamma z to the m where now m can be real he allows real weights so you put this and he gets interest in in order to get something that escapes to infinity he considers the critical exponent so now so he considers the absolute Poincare series again there's one here so it's kind of more similar so you look at this Poincare series and you look at the moment where it's kind of barely infinite so it can be finite it can be infinite so let's look at the moment where it's kind of a little bit undecided so choose m critical so that for so let's call it let's call this delta so that for n bigger than delta you have convergence and for m less than delta you have divergence so then for m bigger than delta you get some kind of finite mass here that you can normalize so then you can normalize if the Poincare series the absolute Poincare series converges you have a measure on the disk now as you approach the undecided parameter those measures sometimes with a little bit of trickery you can make those measures escape to infinity and give rise to something that's supported at the boundary so at the critical exponent exponent you'll get something get measure at infinity that's how Paterson did there are lots of things to discuss here but the point is this was quickly generalized to the setting of Kleinian groups by Sullivan went to high dimensions considered so Fox and groups arise as kind of hyperbolic motions in an hyperbolic plane and now you can consider an hyperbolic space for instance so if you consider the same so high dimensional case so if you consider this what you get is, well you can do it in dimension but if you consider in dimension three hyperbolic space there's something that's called the Kleinian group which is this I took the opportunity to look here so here's the paper where Kleinian groups are kind of named so basically what you have is that you have Fox and groups that are kind of there are motions that have the form so the hyperbolic symmetries have always this form and he considered since it was the situation where it had to preserve the unit disk and now you remove this restriction the circle is no longer some condition relative and fundamental so now the circle is not anymore there and you consider all transformations of the Riemann sphere and you consider again all possible motions that you have and since the Riemann sphere is the boundary now of the hyperbolic space this corresponds precisely to that one dimension up from what is the Fox and case so he extended this theory of Patterson to some kind of considerable there's considerable and this is something that was developed through many years to kind of understand as well so basically you're going to be interested on how this kind sorry I should have read my notes that I didn't say why what you can do with those measures so basically the point was already in Patterson that by analyzing those measures you get to understand things like the dimension of the limit set so the orbits here they are going to some limit set at infinity and I saw some in this drawing it seemed to have the limit set that was everything but it could be something else and you get in general some kind of fractal set at infinity and you can ask exactly those questions of dimension and those questions they are going to be related to study this it's interesting to consider precisely those geometric measures they give you some information about the behavior there and particularly the critical exponent is a way to compute dimension at least in several cases so what he so but then what happens in the case of Kleinian groups how you actually say something about the critical exponent and about those measures turns out that it depends a lot on the geometry of the tree manifold so if you go one dimension up you are going to have now the quotient of the hyperbolic space by the Kleinian group will give rise to a tree manifold hyperbolic tree manifold and it is the theorems that are shown that depending on the geometry of the tree manifold you are going to get different properties of the critical exponent and of the geometric measures so the first results are kind of by Salivan and Takya which give an estimate on the point ahead so I am in the Kleinian group case well, separately and they show that the critical exponent was less than 2 when the surface the manifold of simple in simple case which are the geometrically finite case I can't give very much idea of how the geometry comes in but frequently it comes down to properties of Brownian motion in the recurrence properties of Brownian motion in this manifold that are going to be exploited in the kinds of results the one that I really looked at a little bit more which is kind of closer to what we have here is the theorem about what happened in the geometrically infinite case so this is kind of relatively recent because it combines many things but we know now so geometrically infinite case actually you know that delta is equal to 2 the critical exponent so this is kind of a very complicated result that's actually due to B-Shop Jones plus the Alphos Conjecture it's kind of a funny thing I said that the critical exponent is related to the dimension and basically it is but there is a problem when the Lebesgue measure is of the limit set is positive so the Alphos Conjecture is covered by I think and and this is from 6 years ago or something like this so this Alphos Conjecture is the statement that the Lebesgue measure of the limit set is zero so we have this fractal set and to establish first that the Lebesgue measure is zero and then B-Shop Jones had shown that in all cases where the Lebesgue measure was two and this kind of so here they covered this zero and basically the point of this argument where the Lebesgue measure equals zero appears is because it ensures that some well basically the problem that should be created by positive Lebesgue measure would be that the Brownian motion would run away through the geometrically infinite end of the manifold if this happens it should create and it should create some problems with the critical exponent so but they block this this possibility because of this Alphos Conjecture and you can say that you understand essentially everything about at least the critical exponent in the case of Kleinian groups so that's kind of the situation now you ask what about the renormalization fixate points that were in in the beginning so there is something that's called Sullivan's dictionary that tries to create a parallel between Kleinian groups and so Kleinian groups are some kind of homomorphic dynamics of course but of a group on one hand each element of the dynamics are relatively simple because they may be transformation so it's not very complicated when you iterate but you have a whole group of them now if you have a single homomorphic map that's actually really nonlinear then just by iterating it you already get lots of complexity in the beginning so the point was that there is this dictionary that kind of tells you some equivalence between both theories and that sometimes they send to the level of proofs you can kind of translate one result to the other and hope that you prove some result in one part and then you translate it back to the other case and now you can try to get lots of theorems just by filling lines in this dictionary and it works very well in some case like so we won't talk about he just translated got immediately several interesting parts but in some other case it's less clear so you can try to look at for normalization fixed points there is a line in the dictionary that answers for it and then you see that according to McMullen there are some that are going to be what McMullen was implementing kind of things and he wrote a book at least that was called renormalization and three manifolds which fiber over the circle so there is some class in it you get that there is some class of manifolds that are going to be geometrically infinite which so basically I'm saying that renormalization fixed point will correspond to some kind of geometrically not a general geometrically infinite manifold specific type geometrically infinite manifold and accordingly to this idea you can hope that you can get a proof that the geometric properties of the Julia set so the dictionary says that Julia set of course it corresponds to the limit set for a client group so this is a clear line in the dictionary so you can imagine that the geometric properties of the Julia set will kind of reflect will get the same answers that you get in the case of client groups so then you can go on and try to translate this thing but there is a problem because in client groups you have a three manifold and for rational maps or polynomials or whatever you don't have a three manifold so you cannot run those arguments in one dimension bigger and particularly where you're going to run your ground and motion in part of your arguments so there was some hope that you could construct three dimensional objects that would be so kind of pain that you'd allow to run those arguments that work on three manifolds those would be some laminations it kind of gets more complicated but you have laminations by three dimensional manifolds with three dimensional leaves so it construct those guys and with the hope of getting there but those have very complicated behavior and technically it was not possible to make the argument work you try to translate the proofs at least several years ago which was not developed enough to do that ok since this looks a little bit complicated we can try to do it a little bit differently so after all we don't need I kind of took you all for some kind of hide at this client group but now you forget them and look again at the point of high series so what's the point of high series here point of high series you just write the same thing instead of having all those gamma you just have a semi-group generated by a single map so you're just going to have now dfn so you run over all the math guys that go there fn of y equals w equals z and then sum over n and now you put w and then you put gamma delta here so look at this point of high series you get convergence again for delta equals 2 and you can try to show that you investigate first if it also converges for some delta less than 2 but you have a series here you can try to just kind of make it converge by hand problem is that as you have this kind of self similarity you have copies of the dynamic series things get very complicated as they get near 0 you don't have some properties that are kind of good like hyperbolist and so on more and more complications and it's unclear how to deal with what's happening on 0 if you kind of only had things that run away from 0 then you'd be fine there's some way to kind of put your hands around this and kind of say a little bit of things the fact that it's converging somewhere we like it to perturb a little bit and get to convergence and the point is that we have to use that self similarity here so the functional equation gives some kind of has a reflection on the internal structure of the Poincare series so you can see that the Poincare series if you start expanding terms using the chain rule you are going to see copies of the Poincare series inside itself because you just expand here and an orbit can be the composing two orbits so you have some kind of here that multiply another everything's multiplicative so everything looks fine so you can start expanding terms and the composing orbits in several ways and exploring the functional equation and you come up with some kind of Poincare series that sees itself which gives the recursion it's not an equality not exactly formal but it's an inequality so in that estimate you have a recursive estimate that you get from saying the Poincare series on both sides which has the form Poincare series of f let's say there are kind of inequalities in both sides but it has some coefficients for some reason it looks quadratic b delta p delta f squared so there are coefficients a delta b delta c delta which appears as truncations of this so it's some kind of partial information here that avoids a neighborhood of zero the copies of the Poincare series you can see that the whole structure comes to that so that's that's a formula that we computed in several years ago about 10 years ago me and Michal Ljubic and with this formula we were able immediately to get a first counter example first thing showing that the dictionary is not going so well at this moment because for instance we can so you have to say something about those coefficients and let's see okay so let me give a connection with probability those coefficients here also have some scale the way that I produce this recursive estimate is associated to consider some scales so coefficients coefficients involve truncation some scale so there are several possible scales of truncation as long as you truncate you get something that makes sense but you can get different recursive equations involve different things and so when you analyze those coefficients for delta equals 2 you see that the coefficients for several scales for different scales they are kind of computed basically and the way they behave it just reminds us some parameters in a random walk so you have a random walk on the line looking at things that behave like probabilities to jump from one side to the other side after many steps without passing through some other place so there are things that say that things are going that direction or things are going that direction and the coefficients for different scales are exactly the probabilities of doing certain things from one last point to the other so you get again some probabilities things for delta equals 2 and the idea to take delta equals 2 because you are only trying to decide the dimension 2 or not at first and you can try to compute perturbatively what's happening to this Poincare series at that value and then if you get some good control you are going to at least decide this question anyway, there is kind of reflections to this random mechanism that appears also in the Kleinian group case but anyway to govern this you need this kind of Poincare formula and you can ask them alright so there are those coefficients it's just an inequality what's actually happening it turns out that while in the Kleinian group there are some kind of mysterious reason that makes everything get a universal answer what happens in the in the case of renormalization fixed points is that the initial geometry here the initial coefficients corresponding to some truncation will determine the whole behavior of the random walk so you can see that as I understand there is some accidents that may happen corresponding to some scales I don't see why those things are very important because what matters really is what happened near the critical point but some things that happen in kind of outer scale far away from the critical point influence and determine what's going to happen as we start putting all scales together it becomes more complicated and change the behavior and it seems that here it is possible so we didn't get this right away what we proved was that it's possible so it was published in 2008 that it is possible to have house of dimension equal to the critical exponent less than 2 which says that this path is already not going through this line of the dictionary so this is done by actually checking that those coefficients can behave in a certain way and then we kind of said several things about geometric measures and so on but what we couldn't decide at that time was so this corresponds basically to this random walk that I mentioned be drifting in one direction there are two directions here you imagine that there is this tower of scales and there are points that live in different scales that move up or they can move down they can move close to the critical point or away and you have to see what kind of drift you have for this random walk and you check that it was possible to have drift away from the critical point and in this case this led to delta less than 2 you can imagine that a way is like the geometrically finite and inside the geometrically infinite and just I'm going to finish so we recently proved that it's possible so now we now know that it's possible to construct things so to get some kind of, there are only countably many parameters that you have because there are countably many fixed points but it's possible that there exists some case where the back measure is positive so also the analogous of the alpha conjecture for those manifolds it's not going to go through for the renormalization fixed point and just to say so this is done I'm going to just well maybe it's not necessary I have some pictures that show a little bit how this random walk changes as we change the parameters I just say that this is the actual proof of this we will involve another renormalization theory also that is a different one that allows us to control what happens to the parameters as we make some parameter go to infinity and you can see the generation of the drift in one direction so anyway so that's the conclusion but just so basically that's kind of completely central this absolute Poincare series to get all those results are there any questions or comments? the random walk well it's in our paper that you do this analysis of Poincare series it appears some formulas that are for those coefficients so you see clearly I don't know explicitly say this but you recognize that you see that the parameter we give the formulas relating the parameters at different scales so there's a sequence of parameters and and you see that just by the formulas some of them are kind of decay exponentially while the other is bounded or is the opposite or otherwise there is no drift and then they decay harmonically so you identify those those probabilities that I mentioned so there are three case actually I wanted to mention this that I mentioned that it's possible to have house of dimension less than two this corresponds to two kinds of drifts you may wonder whether it's possible to have house of dimension two and the back measure zero this would correspond to a random walk that has no drift and my opinion is that this is kind of a unknowable problem because in my view these kind of coefficients have some kind of accident on them so those are accidents that happen for the large scale that have nothing to do what's happening at the critical point of some universal behavior and it's like asking whether a given real number that comes there like I don't know the fine structure constant or whatever is it a rational number it's kind of completely so there is this open question but I'm quite convinced that it's not at all noble yeah yeah you can get it in my it has to be in our paper because it's the only place that's done so there's our 2008 paper where there is this but in the client group case there is more things you can see actually the Brownian motion here it behaves like a so you have points that are basically what is the space in which there is a random walk well we just have in the plane so in the plane you have a random walk on the plane but under the dynamics so the dynamics takes points from near zero so you have the back measure and now you have points that maybe close the critical point not asking how you define it but asking the space where it is defined it's on the plane because there is a random walk on the plane which has some symmetry well it's highly dependent on the condition because a random walk anything can be a random walk but you're asking what's the probability that you have points that are for instance close the critical point what the probability that they come away from the critical point after some iteration so that's kind of questions that are you say it's some symmetry associated to the dynamics so you would get a symmetry if you look at the limit of the dynamics so you can kind of consider there is a tower construction that you have this dynamical system here the upper scale here you can kind of minimalize the dynamics and make this kind of outer disk becomes larger and then you're going to have something that's defined on the whole that has a dynamics in all scales also up and now you have the towers of dynamics and you can ask for points moving in this tower as you iterate going up or down in this so this would correspond to the limit that you get closer and closer to the critical point but then you just have points with respect to the back measure and you ask what's the tendency of what they do do they have a tendency to go inner in the scales or do they have a tendency to go up or away from the critical point but to make it really a random walk on the line instead of just half line you have to do this procedure to convert this to a tower that said we don't kind of formally introduce random walk language because it's not necessary we can just estimate things but to remark that to understand what is the meaning of what you are doing I think in terms of those random walks Thank you the speaker again