 Okay, so welcome everybody to the final day of the workshop, Marco, of partitions and Yandt Towers. And well, today I believe I will be the chair for Vilton and then Stefano will be the chair for the last talk. So it is a great pleasure to have Vilton Pinheiro from UFBA, Federal University of Bahia. And Vilton will talk about the thermodynamic formalism for expanding measures. Please, Vilton. Okay, so let me thanks for the organizers for inviting me. It's a great pleasure to be here in the CTP again, at least virtually. Okay, so thank you very much. And indeed, for me, this workshop was very useful because they disconnected between these two kinds of point of view about non-formally hyperbolic dynamics, about thermodynamic formalism and so on. So for me, it was really a pleasure to be here. I tried to follow up the course and also the talk. So thank you for the invitation and thank you for the idea to mix these two branch of the study of non-formally hyperbolic systems. So this is based in a joint work with Paulo Varelandas. And first of all, I want to say that in this talk I'm going to focus on measures, not on set. And many of the talks and the minibus, the people have tried to get a set with good properties and studied all the measures in that set. Here I'm going to focus in measures, not on set. This is a point. So I'm going to fix the setting here. So for me, this double X is going to be a separate metric space for all the talk. But this calligraphy sees a close set with an area. This will be the critical or a singular set, something like that. And you are going to work with a map that is defined in this metric space up to this critical set. In practical, when you have, for instance, a Riemann main folding, you have a critical set. What I'm doing is take out the critical set and look only for the complement of it. So this is simplify the notation and I'm going to explain during the talk about it. And when we have some concrete application, so you have to translate the information by this kind of setting in the application. So we are going to assume that you have this map F is a local and it's believable in this sense. So for every point, let me begin with draw, I like to draw. So for every point outside the critical set, you have some neighborhood that in this neighbor, you're going to see a belief map. The constant of lips depends on the point of course. So let me define what is a strongly transitive. Strongly transitive is a map strongly transitive is the alpha limits or the pre-orbit of all points is dense in the space, the metric space or if you want, the alpha limits the whole space. Okay, for every point. Okay, outside the distributed region. Okay. This is equivalent to if you to take in an open set and you iterate the open set. And this is going to be the whole space. Okay, so but in this. So, I'm assuming that if you have a set inside here. This is. Okay, but it's not. Let me take out this. So, I'm going to assume that this map is strongly transitive, and also that this method sweet pathological miss mixing. This mean that if you take the product of the measure of the map, you're going to this map you're going to be transitive. Okay. This is. So, but let me explain how we can get these two products. It's not very difficult to get to get to get maps and sets with these properties for instance, if you have if you take a C one. Let me say that of the interval, for instance, and you take a cycle of interval. So, you have a finite number of interval inside here. This is a cycle of interval, and the end of every restrict to this cycle for interval is transitive. It doesn't imply that the map the restrict to this interval, it's not only a strongly transitive but also with topological mixing can. So, we need to put for be sure. So, this, this will be strongly transitive and weak topological mix. So, are you are you the drawing has gaps between these intervals. Are they meant to be there or not these gaps. Yeah, because this is a cycle for interval. It's a cycle of intervals. Okay, okay. So the dynamic something like that. Okay, okay, okay. They have critical points here. This is an example, a more general example you take any expanding. I'm going to explain carefully what is explained measures but you take a remand compact manifold and take a C one plus offer a map. And with no flat critical region. If you take a measure, I invited probability with all the level of X point positive. And if this measure is supposed that the support of this measure is not known and and if this measure is a mixing. You are going to get. So suppose that this is the support of some measure, and I got with all the positive is point positive, for instance, absolutely continue to measure and it's not difficult to prove that you can find a subset open subset inside this, this is a total business subset we're going to be invariant for our invariant and strongly suppose also that your measure is mixing. So, you have inside the support open set with full probability for this measure, and that is going to be strongly transit and the week topology. So we can get a lot of examples easy really easy to make to give example of this kind of session. Okay. Let's go. Okay. So can this is strong trans the strongly trans the be related to the natural extension. And it stated in terms of the natural extension. This, this condition. But if you take the network. Probably yes, but I didn't think about it but probably yes but probably this probably maybe. Because it's a strong properties is easy to find the sets satisfying the both of the both are hypothesis but I think this. Okay, probably yes. Okay. So, as I told you, I'm going to focus on measures. So, in didn't expand in measure but you have to flex to make things a little bit more general to to obtain our result. So, I'm going to talk about zooming, measuring zoom in contrast on a lot of something like that. The idea is to generalize a bit that the idea of hyperbolic times, and we, we are going to use not only exponential exponential or contraction by backyard. But also, for the normal one and some something like that. So why that because we needed to. And when you are working with expending measure, it expended has a daily upon the expending and the level of exponent can go very weak. And we want to put all this measure in the same setting the same picture. And to do that, we want to not only talk about exponential growth or contraction but also to the exponential ones. And so the definition is very simple. You, we're going to say that a collection of function is a zoom in contraction is this is a contraction. This is very important. It's a discontraction maybe very weak, but you have to get a contraction and. Okay, this is a sub additive. And you need also that the sum of these contracts. Okay, for example, if you take a alpha in this contract like that this is the exponential case the first one here, or a polynomial case. We will see that it seems to see that this satisfy all the conditions. So, so what is a zoom a zoom in time it's a generalization of hyperbolic time. And so the idea is the following, let me draw a picture first. And I want to say that any is a zoom in time. Time for he did zoom time it depends off constantly in this offer this is the zoom in contraction, contraction and delta. This is the rates of the hyperbolic ball. Okay. And also, this L is maybe this, this hyperbolic time is not really for F but for iterator for F. Sometimes I cannot see the, the back art contractional expansion in the first iterate to have to iterate more. And so this is as to get this information. So, in, in, so what is a zoom in time, you're going to have some. This is the zoom of the point that is the zoom in pre ball. Okay, that's going to be sent. And so, and this internet for a ball. This is a open set. This is going to be a male motions and. Okay, so, but they, they, the important information is, you're going to see contraction. So for one here and alpha two here and so on. Alpha three here. So you are going to see a background in the case of hyperbolic time this background contraction explanation, but you can analyze it. It's the only difference with tons so the difference is that you're relaxing the exponential contraction. This is, yeah, this is what this is about. Yeah, allowing, allowing some of all. Yes, general rates. Okay. This is very important because indeed, if you have for a good dynamics, you have exponential expansion or contract, sorry, a sub explanation, exponential contract, you're going to get indeed a exponential one. The problem is, if you suppose that if you get put up the problem if you want to put all your major expand the measure in the same zooming contraction. So if you try to do that exponentially, you're not able to do that. So the idea is, if you relax the contraction puts for instance something like, something like that, you're okay, something like sub exponential like that, you're going to get all the expanded measure with this contract. This information is important. So, I think that most of you are very familiar with this kind of object in the mini course of Z and many people talk about it so. So this is. This is a zoom in measure. So, let us denote this, this set of natural number, the collection of all zooming a time is for a point P. Okay. So this is a time for with respect to some fixed the zoom in contract from Delta man. Okay. And we are going to say that a measure is a zoom in measure with respect to these three constants. If you, we have positive friction of zooming time for almost every point. So they set of point that we don't see a positive positive frequency has zero measure. Okay, this is a zoom in measure. Okay. Let us denote the collection of all zooming probability with this notation here. I want to point out that this, this set here is not compact. Suppose that your, your metric space is a compact. And in this case, we are, we may think, ah, maybe this collection is is a compact subset of the measure but not because this constant here, you are not, we are not put the information, the frequency of a zoom in times only that is positive. So you can take, you can get out of this. We can take a sequence here. That the limit is not an element. This, if you take the sequence of low and low, low frequency of the zoom in time. Okay. So this is not a compact set, but this is a very good set. I'm going to show you why. So, let me talk about the expanding measure. So, what is the expanding measure with this notation is a zoom in measure with with the contraction with the zoom in contraction being exponential. Okay, but look at that now I'm going to consider all the expanding measure in the sense that I'm out. You can get an exponential zoom in contraction in any rate of the hyperbolic ball and in L. So we'll get all this. So, for example, for zooming, sorry, or for expanding measure. Okay, again, if you have a local, a C1 plus map, and suppose that this map don't have is a local the trauma things don't have critical set. So, being an expanded measure means exactly all the exponents are positive. Okay, if you have a critical set but you're in the context of C1 plus map, and suppose that your critical region is not flat that is look like a polynomial something like that. And so, again, you're expanding a measure the collection of one measure will be the collection they all the X invariant probability of all the exponent positive. So if you have a mix of singular as points and critical points that is the derivative bottom infinity the derivative is zero so you have to assume an additional condition that there's low reference to the critical critical bar or singular set. So this mean that this is a finite. So, in this case, if you all lia cannot pose a sponsor positive and you have this condition you're going to this is going to be expanded. Okay, so it's, what works, we expect to be an expanded measure. You see, in this case, we are not fixing the hate of hyperbolic ball not not that basically. So, as I told you, I'm going to focus on measure not on sex. So I'm going to define an expanded liberal stage for some potential potential fee by a measure that the pressure on the measure that is the entropy of the measure in the integrator for that is potential with respect to the measure is equal to the supremum of all pressure for every expanding measure. Okay, so I'm going to say that causes the set this number, the expanded pressure of the potential fee. And if your state is a measure that realize I expand the measure that realize the supreme. Okay, and if your potential is zero so you're going to call this, and you have a liberal state for zero potential going to say that this measure is expanding at maximum as a measure of expanding much more. Okay, so let me say, okay, so this is all to recall the the setting. This is not important. So the, the, the various of the potential is this thing everyone know about it. And so this is not a show is the upper conformal derivative for the mapping in a point as we are assuming that F is a belief sheets. I look on them often this is a finite number. Okay, this is only a little bit from here. So I'm going now to state that the main result, the main results is default. So suppose that F is strongly transitive with topological needs. And so the first information is F have at most one measure of maximum. Maybe the map. There are no measure of maximum. No expanded measure of. But if you this map have one have a measure of maximum. This is only one. Okay, so for this first item, we don't need the condition. This is the condition of the conformal derivative be bounded upper bound. Okay, so what if you impose this this is easy to be realized by by a C to C one maps. If you suppose this. Okay, so we'll get get more information. Not to you know that you have at most one measure of expanding maximum entropy, but we'll say that this you have at most one. Sorry. If you have a measure of maximum entropy suppose now as default. This is, we have a bond of this and suppose also that you have a measure with maximum expanding. Okay, so in this case, by the first item, you know, this is the only one. But you more, more, you know more that this is the only one that realize this is this is this is the pressure of the zero, the expanding pressure of zero potential. This is. Okay. In the closer of all expanding measure. So this mean that you cannot approximate some some measure by expanding measure. Okay, all the measure that is can be approximate by expanding measure. And it's not expand the measure. Sorry, what's this. Okay, it's not expand the measure. The entropy is going to be lower this is going to be not going to be the top of more than that, you will have some gap of the entropy. So, they read so in the boundary you see and see it here. No, I'm going to explain about the gap after that. Let me say something different here. It's related with it that if you take any samples of expanded measure with high entropy. It's related a complete point of the set must be a expand the measure. So we are going to get some compacted in for a high, they expand the measure to high input. Okay, this is the second. So we have this this proper that's a good one that all if you take in any other potential with small variation. So with sufficiently small variation, you are going to have a one and only one. So let me emphasize this that we are in this location as some true information that. So can you just go back to the definition of maximal. Again, so. So when you say it's an expanding equilibrium state. Yes, where, where is the expanding. Because, because we assume that first, you're saying that means expand the equilibrium state if me is expanding measure that's realized the supreme. So E of F is the set of expanding measures not a garlic measures. Usually this is the set of a. Oh, I see. Okay, so you're you're restricting yourself to the space of expanding measures but are you always assuming that this is not empty than this. Okay. Okay, that's a good question but the, the, in this context, if you have, or this same. So, cannot say nothing about this. But indeed, if you have one measure of one expand the measure, maybe a periodic point. You're going to you have in data. It's possible mainly expanded measure and their interface going to be a positive so on, because they have thoughts of a strongly transitive and the week topology mixing and so that. So it's a. That company, or you have empty set the date, no expanded measure or have a branch of it. Okay, so this is the, the main result. And so I'm going to explain, let me see how much time we have. Okay, I'm going to try to explain the main step to get there. So, and I'm going to be quick. I say, I know that the main of you know, most of these things so I'm going to try to be quickly to a point to emphasize the crucial points, but I'm going at the same time to try to put at least in this slide of information. What is a full in this map. It's a, you have a open set be, and you have a collection of open to the sets of this set be, and you're going to send this open sets. And then you send it to the whole be, and at the same time you can extend the extended for the border. So, this is the second. Okay, so here, and the, the diameter of the diameter of they also let me explain but you feel take a be this is going to be smaller and smaller remade of it and this the diameter of this pre mates go to zero for every point that is remain by the dynamics in this in this be. Okay, so, and also we are assume that this is induced map. So this is something like that. For some part. This is what I'm going to call a full induce the mark of that. So the result is we have 10 years ago. And so they say that if you have a zoom in measure. You can always get a full in this map associate to associate to it. And indeed, this zoom in matter the zoom in this sort of this full induce the mark of the map is going to force zoom in return time to the set. I'm not say that is the first return time to the set is the first zoom in return time to the set that you have to wait for a zoom in time you see that the zoom in time is go back to the first zoom in return time to the set this formation. I'm not going to use here but the information thing is interesting if you you want to. I'm going to talk about this. I don't want to give much information. Okay, so more than that, they, they, this is a good in this map, the number of element with the, the, the same. The time is finite and so on. And okay, this measures has a unique leaf to this, this map this leaf is that got a measure. Okay, so this is important because we are going to use this in this map to get a lot of a property of data expanding measure. So, this is for zooming. Okay, but we're going to focus on it. This is a definition. And we say that a measure is fat, if the measure see open set fat, but here we need more, we need what we call a fat to submit. This means that take a zooming, I've got probability and by the previous results, the zooming I've got a probability has some food in those maps of state to it. And we are asking that we if you if new is the leaf of this measure, the support of me is going to be the whole counter domain of the industry map. Okay. So, just mean that the, the, the, the lift see the lift is fat. So the, the, the lift see open sets. Okay, so the idea is, is the following. We are not going to look for all the expanded measure. Because all the expanded measures sometimes you don't know, I mean information, but we can show that you can, we can approximate all expanded measure but but the fat one, but I do say fat one. And this induced the fact measure is has a lot of good products. So we are going to get a new user the market map that kept all these fat in the city map. Not all the fact, all the expanding one but all the fact expand one. So this is the idea. And more than that. No, so, but let me let me see the time. Okay, I have time. So, so by this side you mean in a smooth context this will be a measure that has that has like an absolutely continuous measure or, or somehow you're talking about measures that are not supported on, on, on zero. You see, something like that. Absolutely, absolutely continuous environment. It's good for this. It's, you can show that this is a fat and do some math, but what we are looking not only for absolute to continue on but all the measures that have this property because they absolutely continue measure. It's a one or not. It's only one measure this is not that you cannot approximate all others. Explan the measure but by absolute to continue. So we need more that you will have to think in the following way affect measure is affect a measure that see open set a fat induced the map is a measure that when you leave it. In this induced the map. This measure is fat also. Okay. So, let me explain why this true process is important. So the first one is a, it's a lemur is it's not difficult to show that if you have a stronger trans people map. And at the same time this mapping week topological week. So, a, all iterator of F is going also to be stronger trans people. So this means that the alpha limit of all points is going to be this. Okay. So this is important. And so, what we can do with this information. Suppose that F, F is strongly thought, transitive for all iterates. For instance, if satisfy both condition this is true. And in this case, if you have. Ah, just one. I'm going to explain the measure that is a zoom in measure with a exponential, the contrast exponential. So we are going to prove that you have a, a, you have going to get a residual set of points that with positive fraction of the only measure you have to. We may lose some, some of the contractional a little finger of the contractional you begin with this. Now we have this, but this is going to be a residual set with positive freaks on frequency of zoom in times. We are going to use this several times. So, so a consequence of this is in this free arm here. We say that give a zoom in measure. We have a reduced map. That's this. I'm good in this matter, full induce mark of a map that this measure can be left to this to set. But now we know no more. We know that the, the, the domain of the map is going to be dense in the domain. So the, the picture is like that. Really. Okay, this is important to, to get a full induces a fat induces of the measures. So, let me put a notation, I'm going to write this for the, the set of all fat induces expanded measure. Of course, this is a subset of this, but now we know more that this is dense. So, more than that, you can approximate and not only expanded measure by a fat induces expanded measure but also the entropy and give any other potential, you're going to get the, the interval of this potential for the, the expanded measure for the, for a full effect induces expanded measure as close as we want to the, the integrator potential for your So what's the consequence of this, that you, if you have this condition so we can mix it so the pressure on the expanded measure is going to be the same of the pressure on the fat, expanded fat induces expanded Okay, for an important for the potential. So you can try. So, as you know that at the same time, this sets is this every exponents measure can be approximate also explain the fat one, but you can focus to analyze the modern formalism not for the, the expanding but for the expansion fat induces that is important because this measure, you can put all this measure in the same scale what we mean about that this the, the collection of expanding fats & boosts in the map is inside zooming the collection of zooming probability with the same constant here. Look, this is the zooming contraction, the zooming contraction not more exponential, but it is super exponential, but it's the same for every measure. So this is good because now we can construct an induced map that gets all this one and so all this one. Let me explain what I can do. So the theorem is the following. I'm going to assume this. If you don't assume this, you have to not only to all the expanding and or the fat expanding this map for, but you have to fix the iterate of the map. If you assume this condition, you don't care, you can do this for all the iterate. So, that that's the reason that you get more information if you assume this thing. Okay. The result is the following suppose suppose this, that you have a upper bound of the upper conformance derivative, and suppose that you have a expanding, I got this one induced probability. So you can show that for any scale that you want to construct your induced map. This is good for some reason if you want to population. And for any that exists at some absolute that for any absolute below this one. There is a zooming induced the map. Okay, all the good profits, the number of element with the same every time is finite. The domain is open and dense in the domain here and this measure will be liftable to this one, but more than that. Sorry. So, okay, this is a cop and space problem. As we are assuming here, we can this is true but not so interesting and so you have. Okay, sorry, but the case. So, the good thing is, all fatting, all expanding fat inducing map is going to be a lift, liftable to this in just now. The idea is the following suppose that you know that you have a measure that marks my the expanding entropy. Okay, a measure me zero. So this is this saying that you can construct induced the map. That is well adapt to this one this can lift this and not only to this one but all the the fat induces the expanding measure. And this this set of fat induces expanded measure. It can approximate all the expanded measure and it can approximate also the pressure your all the expanded measures. So we can study this is. Okay, so this is important because in principle, we don't know if you are a candidate of expanding or maximum expanded entropy measure if this is a fat one or not. So, so let me let me explain. Let me I'm talking about the gap that they, if you have a fat expand those maps. Okay. I do so the map that maximize the maximize the expanded entropy, you're going to get get some kind of gap in the measure that belongs to the closer of the expanded measure that measure, but not to expand one that is. If you get, if you take a, sorry, if you take a measure here. You're going to get that that's the intro of this measure here is going to be not only less than the maximum entropy here, but maximum entropy here, sorry, but you're going to have a gap in this. So who I. The first information about this is this lemma suppose that this is a population not small but not so difficult. You say suppose that look for this expression here. This is the, this is the entropy. So, and this we can think up as the integrate of sorry. Of the recording point. Something like that. Okay, so what I will say this expression that you, if you have. This is going to to infinite this, this, this, this original, this number here, going to zero. Let me explain. So we can define a gap of left like this. You take. What you're going to do here, you have a map. Okay, and you're looking in all the invariant matter for the use the map. Okay, and you calculate this number that the, this number here is limit. Okay, this is going to be the. It's the gap of fact. And it's not difficult to show that this is less than log of two. Okay, this is a number. Why want to do that. Let me say, this is a personal ingredient for this scenario here. Okay, so let F be a food user to markup map. In this context, I forgot about zoom. We are going to apply this for the X on the matter, but this is abstract. Suppose suppose that you have a food user markup map. Okay. And so the first information is that is that is only one F capital F invariant probability to me that's maximize. That's realized. Oh, sorry. Yes. So that's realized this, the entropy of all the list for invariant measures. Look that this, this is a finite number because this is less than a topological entropy of the map original method. And the, the, the, the, the pressure. Okay, the maximum. So why that. More than that. Okay. So, you have only one. Capital F induces capital F. So I invariant probability for the use of the map that's realized this. More than that. This, this F is a Bernoulli probability. Okay, I'm going to show the default of this app. This measure. And if this measure is, is can be project to a measure of the original map. Then this projection is going not only to maximize a is the entropy of the leaf ball measure but also on the closer of the leaf, leaf ball measures. But if you get any measure that's belong to the closer but don't belong to the closer of the, the leaf ball measure but don't is not a liftable one. So the entropy of this measure is going to be a you have indeed is going to be a you have to, you get a gap between this measure and the gap is going to be given by this one. So, the picture is, you have inside the good and the, all the leaf ball measure and you should get a point in the boundary. The entropy here is going to be definitely less than the measure of maximum entropy inside. If, if this measure of maximum entropy. Okay, so, of course, if you think about this proposition and know that you have now in the context now we're going back to the context of expanding measure, you knowing that this proposition and know that you can construct a do so the map that all the fat in expanding fat in this one, and you know that the expanding fat in this one approximate, not only the expanding measure much all must also the pressure of the expanding measure. So, if you put this information together, you're going to figure out okay this, the main result. So, let me go back to the main results on it to. Sorry. So, so this is going to be the reason that the main results. Okay, so this is the ingredients, let me explain why why this. This one works. So the idea is the following. Okay. Let's be okay. No, sorry. This is not. Let me be a f invariant probability. Okay. This is a copy paste and sorry again. This forget all these things. So, but, you know, if you if you have you, you take a invariant measure for the use the map. So we are going to have this. This is not quite this is easy to see that we are using here that it is, because they use map is a full one that this partition given by the element of the domain of this map is a generated partition so when you get. It's easy to show this one. So this, we can, we can split this. This formal but, but in two parts. Okay, this part is associated to the gap that the photo. And this is another part. You see, look that here you have. This is a relative measure. So the idea is the problem. Let me put a picture. Oh, sorry. I mean, let's solve. Let me on drawing picture. And so this is the Europe. You see that suppose that this are you are equal to one are equal to two and so on. And you know that here you have some. The idea is with split the information of the entropy in two parts, the parts associate to this big scale here that is the set and inside a relative entropy inside. It's of this block. And if you want to maximize the entropy, you can see that you, you need to do two things for maximize inside each element of this. So, but this is this is easy because these are a finite collection of elements so you can do the typical things that you do when the shift to a finite shift. And so you, you have to maximize each of these. And after that, you maximize the, the, the, the entropy for get inside and as they consider only these blocks. And this, this entropy in this block is related to the first part that is related to the gap. And the, the points here is the following. If you, you'll say that you have a measure. And you can see that you have a matching variance and probably that that realize this, the, the, the maximal interference complex. And with this finite year, you'll say that this measure is going to maximize inside. And at the same time that is going to maximize as the big blocks and but the big blocks and the entropy of the big blocks is zero if you have a measure that this is infinite. So, okay, so this is what provokes the gap, I don't have any time here. Let me say, let me show only to analyze the formula of the, the, the measure, the Bernoulli measure that maximize the, this relationship here, this relation here that maximize this, this, this one, the formula of this one is precise the following. You take C as the supreme of this. And the, you're going to construct a Bernoulli measure with mass distribution in the element of the partition, given by that. This is the problem. So we know, we know the face of the, the measure that satisfies that and so on. But it isn't so. Let me finish here because I think for. So, thank you, and thank you. Okay, so we have questions or comments for Newton. Yes, I have a question. Please. Yeah, so yeah. Could you explain, go back to the definition of strongly transitive. Yes. Okay. Okay. Yeah, so, so, so here is sorry. Yeah. Alpha limit set. So here your, your, your, your map is invertible globally or not. No, this map is not. It's, it's locally invertible because you see, yes. So what do you mean by an alpha limit set for when you're not properly invertible. Yes, you, you take the, you know that the pre-opt of a point. Okay, that's it. Okay. Yes, the pre-made and you look for the accumulate points of this pre-made, all the accumulate points. Yeah, the same definition of alpha limit for any case. So, if you know that you're so, so, so how can you check this? I mean, how does it compare with topological transitivity. Good, let me, let me explain in two ways. F is transitive. This means that you have a dense set of points. So that the alpha limits is everything. This is transitive. This is not only for this set, but all the, this set of points, but all the points. You can, this is the one way to think the other ways is transitive. If you take the union of the four interiors of any open set. No, no. Then this is going to be the, your whole space. Sorry. Sorry. The closer of this is going to be the whole space. And the strong and positive. Means that in the end of the, it's not necessary to get the closer. Okay. Open. Okay. So, so, so in your abstract, you mentioned Vienna maps. So, so it is satisfied for this. So, this is, when you going to, to get a complete example, you have to adapt it. So, you have to show that you can take out the region. And also, you, you get the same information. Let me explain what's the, the, the, the delicate point here. Suppose that you have some critical reason here. Okay. And you're, you're assuming that your map. You want to, to get the information of this, this other thing. So the point is, you can get a point X and, and look for the investor branch of this point. And at some moment to this invested branch. Reach the critical set. So, in this case, all the, the, the, the, the, the, the brands that's a console you have to take away. So the point is, give you a point here, you can, you can define the, let me say the alpha, the non critical alpha. As the, as the alpha limit of the points, as all, you take all the investor branch of this point. So the, the, the, the path of pre-made that don't hit the critical point. You want to know that this is also a, a, this pre-made is also dancing the space. The point is, in general, you, this is different. Okay, so you have to be sure that this, because I want to do this one, you have to be sure that this is going to be everything. So in this case of Vana maps, what you can do. Let me, let me give a simple context. Suppose that you have the to simplify the, the, the argument. Suppose that you are, you have the Vana map, the, the map is analytical. So what you have, this is the silent and you have the critical region here. And when you enter into the critical, critical region, this is going to be some like that. And after that, something like that, saying so on. But as in this case is the analyst, the intersection of this, any territory of this with is going to be, or a finite number of points and script collection, or is going to coincide. If this coincides, you can see that the entropy, because the entropy, in this case, the, the, the intersection of the fiber is going to be finite. So the entropy is going to be less than the entropy of the, of the base. Yes, yes. So, so you, you, this, you can forget for this point. So the point here is, you can show, because of this argument that the, the expanding measures. With a high entropy is going to have the profits that the alpha X of X for this case is going to be the whole space. For all points. Yes. For, for, for X, me typical me. Almost every point. We can expand with entropy. Big. So you can, because of that, and say, okay, so we can, for God's for this measure and focus on the, the expanding measure of high entropy and for this expanding measure of high entropy and going to get that. You can apply the abstract. Okay, thank you a lot. So, so in this case, you prove that these young maps, they have a unique measure of much more entropy. Yes, and also that if you have a potential of small, because in this case, you can apply that in this case. The gap is a C two C one C infinity. The, the, the derivatives bounded by above. And so you have that that's a gap that's playing. And so you can show also that every potential is more variation other potential of more variation is going to have one, one measure of. And one equilibrium states. Stay. Only one. Okay, does someone has any other question or comment. I had a quick one, but I think maybe we're out of time actually so maybe let me ask just only only if you have a very quick answer, Milton is this these examples, these these results really need your generalization I mean it's not is everything not exponentially zooming time in these examples, do you have some examples where it's not exponential. So the problem is, if you want to construct, induce the map that cats all this. This is good. Okay, this this fat one. You see, if you take a measure here, each measure here has the. It's not all exponential but not uniformly so you need something sub exponential to capture them all okay. Okay, I understood. Thank you. Okay, so as Stefan said we are a little bit out of time so let's thank you again. And I believe we can make a five minute break to start with near. The chair will be.