 Well, thank you very much to the organizers for the carnimitation. It's really a pleasure for me to be here, and let me talk to you about my work, to clarify what is the right notion of the limit set in the high-care dimensional setting. We are following, until now, the ideas of Professor Kurt Kearney, which was initiated by Alberto and Pepe a long time ago. Let me tell you how are we doing the things right now, yes? Let me outline briefly how it's going to go my talk. Maybe it's better for in order to get the right order. So it's going to have three parts. In the first part, I am going to talk about briefly, very briefly about some features of the limit set in the one-dimensional case. There are features, nothing very complicated. In the second part, I am going to talk about Juan Pablo Navarrete theorem in dimension two, which is the main theme of the talk. Basically, I am going to try to explain to you how this theorem inspires us to fully give a solution of the notion of the limit set in the two-dimensional case, and then I am going to proceed in the third part to speak about this same theorem, now in the high-care dimension setting, and I am going to comment briefly on the proof, and if I have time, maybe I can discuss how this theorem now can be improved to get a right notion of limit sets and its relation with the cool-cutting limit sets, right? So let's begin. So some elementary facts from the one-dimensional case. Cleaning groups were introduced by Poincare as the only groups of some second-order differential equations. To say something, to be more precise, a cleaning group is just discrete groups of moist transformations acting on the Riemann sphere. It is well known, as we have seen during these lectures, that a cleaning group, sorry, splits the sphere into two invariant sets, one namely the limit set and the other one, the ordinary set, yes? Then it will be more precise what these things are. We have the following theorem or definition, it depends on what you are thinking, but for example, one can see, if you pick up a group of moist transformations, then the limit set is simply the closure of the cluster points of orbit points, yes? You can think this very same set simply as a complement of the continuity set of the group. Yes? You can also prove that this set agrees with the complement of the largest open set on which the group acts discontinuously on the Riemann sphere. And also is a, sorry, here is a complement, is also the complement of the largest open set on which the group acts, but now, properly and discontinuously. This is a very special feature, because in general, properly discontinuously is not the very same thing that discontinues such, but this is a crucial point in the definition of Professor Kulkarni. And also you can show that for the non-elementary case, I am going to say later which means, what does mean elementary, that the limit set is simply the closure of the repelling fixed points as Karolin said, or tell us in his lecture, right? So as you can see, you can define the limit set in several ways. So we are interested about to try to generalize how we can generalize these notions in the higher dimensional setting, yes? Some of the properties of this limit set, very basic properties. For example, if you take a cloning group, then the set contains one, two, or infinitely many points. Also the limit set contains infinitely many points, if the, sorry, if the limit set contains infinitely many points, then the limit set is a noteworthy and perfect set, which is minimal for the action of the group, yes? Another remark I wanted to make is essential, is that the proof of these facts, of this theorem of the previous facts, is that the proofs are interrelated in two things. One is the one terms about normal families, and the other one is related to the convergence property that the Mobius groups enjoys. These are interesting features. We are going to try to generalize later. Some examples of these kind of groups we have seen a lot. Some trivial are Schottky groups, Kitten Schottky groups, Triangle groups, fundamental groups of Riemann surfaces, blah, blah, blah, blah, you can, you have a lot of interesting examples. These are just, it's just a taste. Now, passing to our team, the discrete groups of predictive transformations have also a background or a history which comes from the differential equations. For example, they can appear as the monodromy of certain partial differential equations as they work in Ophioshida in the 80s, they also appear as a monodromy of certain higher order differential equations, and also in certain cases as the monodromy of certain type of recalculations. So which this in mind, I'm thinking, to make or provide a generalization of the notion of clean group, Alberto and Pepe in the 90s start to develop a theory of discrete groups acting in the, of predictive transformation, acting in the predictive space, and they begin to make things, yes? So some question that we wanted to ask or that we are still working because are not very clear for us is a following, a very simple question, how are the various properties of the limits that are related in the Hacker dimension of sitting? How can we generalize this kind of properties, yes? Very simple question, we still don't have very fancy question, we are still in the very beginning. Let me give you some, some languages I, for sure, everybody knows, but just for sake of complex techniques. Remember that the complex predictive space is to be defined as this quotient, is simply the space of, of lines, and it will know that the group of your biomorphic automorphism of this, of this thing, which turns out to be a compact complex manifold, is simply this guy, is this quotient, where this, this, this, where C star, sorry, acts as, by the usual scalar multiplication. The elements in this, in this guy are going to be called predictive transformations, and we will say that a matrix is a leaf of this predictive transformation, simply if this matrix is a representative of the equivalence class of the, of the, of the predictive transformation, yes? So, we can think that we are working with matrices, but we need to be cautious, yes? What else? Let's start with a very, very nice example, which is given, was studied by Kulkarni in his original paper, and it's very simple, consider the following transformation. As you can see, this transformation has exactly three fixed points in the predictive plane. This one is attracting, you have one saddle, and you have a repelling fixed point, yes? So what about all this notion that I told you in the very beginning? So Kulkarni shows the following things, for example, that if you pick this, the line is panned by these two fixed points, and this point, which is this, and you take the other one, then these are the only maximum regions on which the group acts properly and discontinuously. Yes? So this means that when you work, or when you are considering actions different from most transformations acting in the Riemann sphere, it can happen that no longer exists a maximal open set where the action is properly discontinuous, can be at least two, yes? It can happen. Well, another thing that Professor Kulkarni has shown is, for example, that the Kulkarni region, sorry, the continuity region is very simple, he used to take the complement of these two lines, yes? Let me propose thinking about this very simple example. Let me propose two possible, just to start, two possible definitions of limit set. In order to talk about this possible definition of limit set, one is the Kulkarni limit set, let me introduce very simple tools, yes? One is the so-called Mid-Verge Compactification of the Projective Group, yes? This is very simple. For example, consider the space of linear transformation from Cn plus 1 to itself. Clearly, this is a linear complex spectral space where the group of invertible matrices is an open sense set, yes? So consider this question, again, it's a projective manifold. As you can see, now you can show that the projective transformations are a form, an open sense set, in this space of self-projective maps, yes? So this is a way to see the set embedded in this one. So the funny fact is that we can produce a map for every element here from the projective space, but we need to be careful. So start with a matrix and then take away the kernel, you need to put it away in order to avoid the difficulties, and then simply define the action of this thing in the very natural way, yes? So you have a very well-defined map which extends the notion of projective transformation, yes? So a very simple question is the following. You have a notion of convergence here, since this is a projective space, and now you also have a notion of convergence, but now from the point of view of function theory, yes? So the question is how are related each of these convergences, right? So we have a theorem that tells us how these are related, and they are related in a very nice way. So for example, if you pick up a sequence of projective transformations which is converging to this element as a point here, then you can say that these elements converge to tau, to this very same element, but now as a function in this set, in the complement of the kernel, yes? So everything goes well. This is telling us that for example, the continuity of this sequence is very simple to compute. Simply, the complement of this kernel, yes? So this lemma, as you can see, provides us a generalization of the convergence property of the Mobius groups, yes? So one definition I would like to work with is the complement of the key continuity. As you can see, the key continuity is very simple to be determined, and that's all I want to say for this. Again, let me tell you again the notion of the Kulkarnian limit set. As we have seen from the course of Pepe, the Kulkarnian limit set is constructed as follows. Maybe it's a little bit different, but in the end it's the same definition. Pick first the closure of the cluster points of orbits of points. Now the Kulkarnian limit set is to be defined as a union of this set lambda gamma and another set, L2. This set is simply the closure of cluster points of the family gamma k, where k runs over all compact sets lying in the complement of this lambda gamma, yes? So the ordinary set in the sense of Kulkarnian is simply the complement. It's the very same definition as Professor Quica, yes? We have the Kulkarnian limit set has the following features. Maybe they are a bit, not very impressive, but they are useful in the following. For example, we have the following to Juan Pablo Navarrete, a former student of Pepe, which said that, for example, you pick up a group of project transformation and you have a closed set which is invariant on the reaction of the group, and you can ensure that the cluster points of orbits of compact sets lying outside the set that you take, and you can ensure that the cluster points of these orbits lie exactly in this intersection, then you can ensure that the Kulkarnian limit set is going to be contained in this set, C. So you can ensure that the Kulkarnian limit set has this kind of a quasi-minimality. It's something like minimal, but very close, yes? We have also these properties, the first part is to Professor Kulkarnian, the second part is to myself and Pepe, and said that, for example, the group, if you pick up a complex landing group, then the group acts properly and is continuously on the discontinuity region, and also we can show that the group acts properly and is continuously on the key-continuity set, and you have a precise relationship between these two sets. You can say always that the key-continuity set is going to be contained in the Kulkarnian limit set. So in this sense, we can have or provide a fair relationship between these two possible notions of limit set, right? We have many more things, but these things are the one I wanted to say. In order to show how complicated can be to determine the Kulkarnian limit set, let me present you some examples. So for example, if you consider this group, as we have seen from the previous talk, we have that the quotient is a hop surface, and in this case it's not very hard to show that the Kulkarnian limit set is simply aligned plus a point outside. Yes, in this case it's very simple and it's very fast to make the calculation. Also for example, if you pick four linearly independent vector points and you consider this, then when you consider the action of this group, of the group expanded by these elements, then the group is not hard to show that acts in the fine chart like the fundamental group of a theory. So it's not hard to show that in this case the Kulkarnian limit set is simply aligned at the infinity. So this is very tricky, let me just put the slide, please read it, and let me explain what does it say, or what I want to say. This example was due to INUE, this example comes from the fundamental group of the INUE surfaces, and it's very special, and the reason is because it's special is the following. In this case you can show that the Kulkarnian limit set is going to be a pencil of lines over a circle, but in this case you can show that the equicontinuity region is empty, which means for example that you can find subgroups of this guy such that the Kulkarnian limit set is not contained in the Kulkarnian limit set of the larger group. So this appears as bad because this is saying that the Kulkarnian limit set is not monotone. So this is why I want to talk about this example, forget about all this stuff. Let me recall some things that Professor Parker have said in his course. Consider the Fogone emitting matrix, then it's a mission matrix of signature 1n, define this group, which is simply the set of matrices which preserve the respective admission matrix and consider the complex ball which is this, the same thing that John did in his course. So as John said, we can classify the elements in this group simply by saying that the elements is loxodromic if it has two fixed points in the boundary of this ball and it's parabolic if it has one, exactly one fixed point in the boundary and it's going to be elliptic if it has at least one fixed point inside the ball. Yes? So we are going to use this a little bit of this language. Well, I'm going to need also this, recall this definition, about the chain-grim limit set. Remember that given a group that preserves the complex ball, we define the chain-grim limit set simply as a set in the boundary which corresponds to the accumulation points of orbits, of points given in the interior of the ball. Yes? And this says, as I think somebody has said or speak about this, has or enjoys the same very properties of the limit set in the one-dimensional case. I mean the limit set contains one, two or infinitely many points. The limit set is a nowhere dense perfect closest set and blah, blah, blah, blah, blah. It behaves very nice. Yes? So a very nice question is that how can you determine the Kulkarni limit set of the group but when you consider the action now in the full projective space, not only in the ball. Yes? And this question was answered by Juan Pablo Navarrete and he said basically that the Kulkarni limit set is very simple. Yes? It's very simple in the following way. Let me make a picture, a big draw, a little draw. So Kulkarni Navarrete said that pick the points given by the chain-grim limit set and then you take simply the tangent lines to the ball at the points of the chain-grim limit set. So and this is the Kulkarni limit set. Yes? Let me give you a different proof in order to show you how the things work in this case and the proof is different from the original one. The original one uses a strongly complex hyperbolic geometry. I don't like it by myself. I like a complex hyperbolic geometry but the problem is that when you use complex hyperbolic geometry you are not able to extend to another kind of groups. So we're going to try to avoid but we are going to try to keep the philosophy, the philosophy is the correct one. So the philosophy is very simple. First, the very free script we need is to get control about the continuity rate, the key-continuity rate. Yes? So in this case we have the following theorem that said that the key-continuity region is precisely, here is the complement, the key-continuity region is precisely the complement of these lines. Yes? Blah, blah, blah. And the proof, the idea of the proof is very simple. Take a sequence of distinct elements such that the sequence converges to some quasi-projective, pseudo-projective transformations and also the sequence of inverses. So in this case you can show that the image of this guy is a point in the boundary and the respective kernel is going to be exactly the orthogonal complement of the image of this guy. Yes? Because if this is not true you can have a figure like this and then you only need to compose the sequence with the sequence of the inverses and you get a contradiction. Yes? So here we avoid to describe the key-continuity set of the, here we are avoiding the use of the complexity property geometry to show that, to describe the key-continuity region. We are avoiding this thing. So this is very strong, a very useful story. A second fact that we need is the following that given the group here we can always find, and only elementary of course, we can find always an alexodromic element. And in this case, as Professor Parker did, we can ensure that the normal Jordan form of such elements have this far. Yes? So we have proofs in the high-heat dimensional setting where we can ensure, just by using algebra, that in any kind of group you can find alexodromic element where alexodromic means something. Yes? So as you can see, we have a very special thing here. We have, and that the alexodromic elements look exactly as the one in the Kulkarni example. Yes? You have an attractive point, you have a subtle point, and you have a repelling point. So the analysis that Professor Kulkarni did in his example can help us in order to understand the things in another general setting. Yes? So we have the following lemma, which is called the lambda lemma. It's due to Juan Pablo. And this reflects that I said to you. Yes? It says the following. Pick up an element of this kind. Assume that you have an open set on which this cyclic group acts properly. And this continuously, then you can ensure always that either this line or this line is contained in the complement of such set. Yes? So this is saying to you that in particular, if you propose some notion of limit set, you can expect that this limit set is going to contain lines. Yes? So let me give you a brief proof. It's in the very same spirit of the ideas of Kulkarni. Assume that the lemma fails, then you can find a point here, for example in this repelling line, such that this neighborhood, this red neighborhood is contained entirely in our set omega. Then just pick another point in this line. And then just consider the iterated of this line. As you can see, this is going to converge to this one, since this is repelling, and this is going to converge to this one, since this is attractive. So this means that this line is going to converge to the attracting one. Yes? So this means, in terms of the convergence, that you can find a sequence converging to this point, since the continuity of this group is the complement of these two lines, such that this guy converges to the blue, where this blue is any given point here. Yes? So if you're playing something like you can find, for every point here you can find a sequence here, such that the sequence converges to P, and this sequence is going to converge to W. Yes? So it's the very same idea as Kulkarni did in his article. So as you can see, a very simple fact is that in this case we are traducing the dynamic of the group, not in the, we are working the dynamic of the group, sorry, not in the ambient space. We are working the dynamic now in the dual space, I mean in the first grass line, yes? So this is a very first important fact. Now the proof of Navarrete is very simple, and we don't need a complex hyperbolic geometry a lot. You only apply the lambda lemma, the existence of the auxiliary elements, so by these two lemmas you can ensure that the set of lines is contained in Kulkarni. So in order to conclude the proof, remember that I have shown that this, we have this contention, yes? So very simple. So as you can see, before I continue, as you can see, we have three ingredients in order to prove the Navarrete's theorem in dimension two. One ingredient is lambda lemma. The second ingredient is the existence of lexodermic elements, and now the normal Jordan form. And the third ingredient is no precise, in a precise way, the key continuity region. As you can see, by the theorem of Jose Popepe, sorry, myself, we have a full description of the key continuity set. So this part is done. By the course of Professor Parker, we know that the lexodermic elements exist, and the normal Jordan form is fully determinated. So the second part is done. So the only ingredient that we need is to... the only ingredient that we are missing is the lambda lemma. So in the high-heat dimension of sitting is going to be our task. So let me tell you how this kind of theorem helps us to understand things, or helps us to understand the role of Kulkarni limit set in this two-dimensional sitting. Well, this was an idea of Alberto, and Alberto taught us that in the following thing. If you want to understand the relationship between the dynamic, you need to understand Montel theorem in high-heat dimension setting. So work for a Montel theorem, yes? And he told us you need to come lines, and that's what we did in this case. So this is a theorem of Varrera and Navarrete, and it basically said that if you pick up a domain of C2, and you pick up a family of projective transformations, and the projective transformations omit three lines in general position, then the family is normal. So a pick up, like a pick up. So we own these ideas to Alberto. Yes, this is an improvement of the Cartel-Montel theorem in the high-heat dimension of sitting. But now the improvement is that we set lines. The Cartel theorem said about eight-percent surfaces. The original one set five lines in general position, and the improvement of these guys is three. So it's really... So, as you can see, now the task, since we have a normal family criterion, is to count lines simply. So, the very first fact is that we have at least one line, that is a simple consequence of the lambda lemma. Now if you can, if you put yourself to count things, it's very simple or it's not very complicated to show that the number of lines is one, two, three, or infinitely many, but since you want lines in general positions, you need to count lines now in general position, and now the number of lines is one, two, three, four, or infinitely many, but in general position. So, you can set in a way that groups with infinitely many lines in general positions are generic in some sense. So, now we have the solution of the problem of the limit set is done by the group, and sets basically that accepting the case of Professor Kulkarni, for every group the Kulkarni discontinuity region is the largest open set on which the group acts properly and discontinuously. It's accepting, for example, Professor Kulkarni. And you can say more, you can say if you have three lines in general position in this set, then everything goes well. The Kulkarni discontinuity region is a key discontinuity region. And the Kulkarni limit set is a closure of the loxodromic repelling lines. Again, everything goes well. And if you have only two, well, don't worry, it's just a pencil of lines over a circle or maybe over a limit set of a Mobius transformation group. And in other cases, the dynamic is in the sense of the lines. It's interesting, he just said that the number of lines is exactly one or one line and one point, which correspond exactly to the fundamental groups of hub surfaces. So, this is part of worrying the two-dimensional case, which come from ideas from Alberto, from Pepe. We can say things about the geometry of the domains and blah, blah, blah, but I wanted to go to the higher-dimensional case. Yes? So, as you can see, the role of the Navarrete's examples is to provide those things to imagine things about another context. Yes? I mean, the theorem of Juan Pablo is important because it tells us first, you can expect that in order to understand the dynamic of these groups of projective transformations, you don't need to work in the ambient space. You need to work in the grass mania. Yes? And the other thing is that in order to understand that dynamic, you need a very few things, a lambda lemma and a description of the key continuities. And this avoids to use, in the very beginning, complex hyperbolic geometry. But, well, just in this stage, let me provide you a very simple example. Consider this very simple example, just like the one given by Professor Kulkarni. If you use the lambda lemma, which is very simple, it's not very complicated, you can ensure that if you have a maximal open set on which the group acts properly and discontinuously, then you can find two sets here, which corresponds to this space, which are these joints. And also that its closures give you this full line, such that this set is simply to be this set of lines. Sorry, maybe a picture is going to be better. So, the lambda lemma said that if you want to construct a maximal open set for the action of this guy, you only have to take here something like this. And then you draw the pencil of lines of this guy, of this distance. And then you only need to draw something like this. Something like, sorry, I don't know how to say in Spanish, en ramada, in English, something like this. So, in this way you can, this shows that if you try to understand the limits set now in the higher dimensional setting, again the problem is the problem of how to choose the lines. But now the problem looks more complicated. Now you can have not one or two maximal open sets on which the action is properly discontinuously, maybe it can happen that you have infinity limit. So, how can we expect to solve this problem? Let's see, the following, the previous example was interesting. It's interesting. Pica, an element in P1M, so by the lecture of Professor Parker, it's not very hard to ensure that whenever you pick a loxodromic element, the loxodromic element has this normal Jordan form. Yes? So, this means that if you wanted to understand maximal open sets on which the cyclical group is generated by this element, how are these things, you only need to understand the previous one and use only the lambda lemma. Yes? Some simple consequences. Pick a loxodromic element and take its fixed points, then for every element you can ensure that either this line or the other line in the complement, as in the lambda lemma, appear in any complement of region of discontinuity. This is for points which are the base of loxodromic elements. But now if you believe in this, then you can provide the following. You can ensure that you pick just two any points in the chain green limit set, then the very same thing happens. Yes? This is just because this is the very same area. Yes? The use of the lambda lemma. So, we have the following theorem of the way to myself, Mink and Marlon, which is a generalization of the Navarrete theorem. And the generalization is exactly the same. And it says the following. You pick up a group, a discrete group of preserving the complex unitary bond, then the Kulkarni limit set is simply the set of hyper planes passing through the chain green limit points, which are tangent to the bottom. And that's all. So, very simple. Let's see how can we prove this. Just remember, in order to prove this, we need three ingredients and we have two. So, we only need to provide the proof for the lambda lemma. Yes? So, let me remind you a little bit of projective, sorry about, let me remind you the Cartan invariant, which was discussed by Professor Parker, and it appears. Remember that the Cartan invariant for a triplet is given by this formula and is interesting at the very beginning because if you need to ensure that there is a transformation that takes three points into another three points, it's enough to ensure that they have the same Cartan invariant, at this moment. So, let us construct a map, which we call the return first atom map, because it's going to have the very same spirit as the first atom map of Poincaré. Yes? This is the expression. Maybe Adro is going to be better. Yes? We pick two points in the chain remember limit set. For example, you can think that your points are very special once in order to make easy computations, and then pick another here. Yes? So, what happens if you have a line here? Yes? So, you can form, or you can imagine that the other different planes look like this. So, you have a line here, and you want to make that this line travel to this way. So, this map is doing the following. You start with two points, they can be the fixed points of a loxatronic element. Now, change to these ones. Now, by the previous proposition, you can think that these are the fixed points of a loxatronic element. You can think they are not. Yes? So, you can now iterate and ensure that now this line is going to travel now into this one. Yes? The only thing is that you are thinking that here you have a loxatronic element. It's like a virtual loxatronic element. It's not real. Yes? You only pass from here to here. Yes? And now you can see that it can happen that this line does not arrive to this point. Yes? If you arrive to the same point, then the problem is solved. It can happen that this line moves, and this movement is simply that what we are calling is fizzy. Yes? Some things that we can say about this return map is the following. Why is this useful? For example, if you pick an open set on which the group acts properly and discontinuously, and you pick a point precisely in this intersection and consider the respective line, this one, then the phenomena that is happening is exactly what I told you. If this line is in the complement of this guy, then this line, the one that travels, is going to be contained in the complement of the continuity region. Yes? This is a very first thing. A second fact which is now very interesting is that this kind of movement is not any movement. It needs to be or it needs to belong to this group. So it's very special. So in fact, we can say, if you need to make some easy computations, that the normal Jordan form of this movement, of this return map, is simply this guy. Yes? And as you can see, the proper values of this matrix is simply given in terms of the carton invariant. Yes? So this return map encodes in a way the geometry of the chain ring limit set. Well, here is the draw that I just made here. So how can we use these things? Well, as you can see, the transformations by themselves don't help because the movements can or not necessarily are right to the very same point. Yes? And we can say, well, maybe if you move one point, the point, and you approximate to one of the extremes, maybe it can happen that in the limit, you can get the line. You can get to this guy. I mean, it can happen maybe in the limit that you are approximately hidden. You move to here or as you move from to here. Yes? But as you can see from our formula, you cannot expect this. There is not a kind of continuity when you move to the extremes. So you need to be or you need to do this carefully. Yes? So what we did to solve this problem is consider all the finite products of this, all these possible return marks, and consider the closure. Yes? So this is a compactly group. It has a very nice feature. Whenever you pick an open set on which the group acts properly and discontinuously, and you pick up a point here, and you can ensure that this line is contained in the complement, then you can ensure that the full orbit under this group, but now of this line is going to be contained in the opposite hyperplane. Yes? So this means that if you can ensure that this is a group, you can ensure that eventually you are going to arrive to this point and you can ensure that this line and this line both belong to the complement, and then you are going to ensure that the land dilemma is true. Yes? So this is the construction. This is the key idea. So we have two steps to prove the land dilemma. We need all return maps, and we have this control group. Yes? As you can see, we are saying that in order to understand the dynamic of these kind of groups, you need to produce a control group, but not in the ambient space. You need to produce a new group now acting in the grass mania. And now, generically, this control group turns out to be a group. So you are controlling the script dynamic through a continuous dynamic, which is in the same spirit of Poincaré as Alberto always is teaching us. Yes? Yes? So same philosophy. How is the idea of the proof? So it is just enough to prove that this part is true but only for phi squared. Well, here are the ideas. Basically, we have used the tools from the higher complex variables. Yes? As you can see, the problem we are dealing with is a problem of how is behaving the group of automorphism when we approach to the border. Yes? So if you see the arguments, we are treating with sequences which are called parabolic sequences, which are very useful in the study of the geometry of domains in the several complex variables. Yes? So we use these tools from several complex analysis in order to ensure that our things give or produce a group which controls the dynamics. So this is the very first time we use complex analysis to understand the dynamic. Yes? So these are the general ideas of how we construct the sequences. The philosophy is that in this kind of groups, you need to go to the Grossmanians and construct a group which controls the dynamic. Yes? I have five minutes maybe. I thought I wanted to talk. Maybe if you have any questions.