 Okay, thank you very much for the invitation first. Okay, so today we are going to talk about dynamics. So what is it that I'm studying today so we are studying map from a project space to project space to itself, which is a regular map. So it means that, essentially, if you want to consider it on a complex perspective, it's a holomorphic map. So it means that to triple. Z not the K you associate P one of the not the K or P not of the not the K, etc. P K of the not the K and the P is they are homogeneous polynomial with the same degree. And so the degree has to be two. So we'll see in a minute why at least two, and we require that it's really a mapping, it's not a rational mapping. So the intersection of the PI is zero is reduced to zero. So this is the kind of object we are studying. So what are we studying we are studying that dynamics. The dynamics of the map. It's the study of the orbit. So you study the sequence of it rates of a given of a given point here. Z is any given point in your project space. After the end is F that you compose and times. So the aim here is to describe the sequence and its dependence to see. It's quite simple question actually you just want to understand if you can describe completely the sequence fn of Z. Just knowing fmz. So, unfortunately, this is the dynamical system in question is chaotic. So you cannot expect to do that explicitly. And so you have to focus on on different questions, maybe a little less precise or questions of measurable nature, for example. One simpler question that you can ask is, can, can I say where are the periodic points. So what is a periodic point. So you would say that Z is periodic. If so there exists an integer for which the orbit comes back to the after a certain time. And the minimal and satisfying this property is the period. Okay, so first question that you could ask is, you can try to understand how are such points distributed. How are periodic points distributed. So distributed can be understood in many ways. Okay. So here I'm, I'm going to give three different answers in three different contexts. Okay, so here we are overseas. Okay, but it doesn't change everything but okay. You can ask how they are distributed in the Euclidean topology. So what is the closure Euclidean one. There is a small variation of this question but which actually is not that small variation. What is the closure in the size gate apology. Okay. And in the sense of measures. So we are going to make this question a little more precise. And by that, you can divide by, you can look at the measure one over D to the n times K, some of direct masses at points, which are very dividing and. And you can wonder if this measure converges this sequence converges and to what objects. So these three questions are related actually. I'm not going to discuss it very precisely, but there is an answer to question three. So there exists a measure probability measure. We generally denote by new sub S such that the following holes. So the first thing is that new sub F does not give mass to Zaris key close subsets of PK. Of C. So if you take a sub variety, it has measure zero for this measure. And the second important information is that one of the D to the NK some of direct masses at those points converges to this measure. Okay, in the in the weak sense of measures. So what does it mean it means that for all five, which is continuous with compact support PK of C, then the average of the values of the function fire. On those points converges to the integral of this function five against this probability measure. So. So, okay, I forget to mention that this is an important result by Jean Yvrien and Julian Duval. In 2000. Okay. And whenever F is defined a number field. There exists a proof, which is based on arithmetic expression statement by Cindy you want. And it's more recent it's 2008. Okay. So why this question is related to the two previous ones. So, because I mentioned that this measure does not give mass. So, in particular, those points cannot be contained in a series key close subsets. Okay, so this implies that that is always the case. Okay, so we have an answer to question to this is the first thing, and this also gives an answer to question one, which is that the cluster. This is the accumulation set. Not only the closure but the accumulation locus. Of predicaments in the Euclidean topology is the support of this measure. So they may not be in the support of this measure but they cannot accumulate somewhere else. Okay, so this gives a very fast and very unprecise description of what is known in this setting. And now I want to, to give you. It was a kind of appetizer to what is going to be the main topic of interest here that I will focus on. So here we're going to study things in parameter spaces of on the morphisms of PK. So what is the context. So we are studying families of maps and the idea is to try to give some equivalent statement to this one either in a fiber space on which your family will act or directly in parameter spaces of such families. So what is the family then of of such on the morphisms. So it will be a map from here a quasi projective variety times PK to the same quasi projective variety times PK. So it will be a map which is fiber, which to lambda Z maps, lambda, F lambda of Z and F lambda has to greedy. So what will play the role of our periodic points in the parameter spaces, they will be what we call post critically finite parameters. So before that I need to define what is the critical set. So what is the critical set parameter F sub lambda. It will be the sets of points in PK such that the determinant of the differential is zero. So this is the set of points where your map is not locally invertible. So you will say that the parameter lambda is PCF for post critically finite if so it's a dynamical condition. So, so if you look at the union of the images of this critical set. And this is algebraic. So a priori this set is just a countable union of hypersurfaces and you impose that it's not a countable union but just a finite unit of hypersurfaces. Okay, so this is. So, this is equivalent of lambda in lambda is equivalent to the fact that, in fact, there are case already taken I think no. In the images, M and M said that the critical sets of F lambda is contained the M image is contained in the M image. Okay. In some sense, you impose that your server though your critical set which is a sub variety is prepared. This is what you impose as parameter. So we have many reasons to, to, to have those parameters to be considered as the equivalent ones in parameter spaces to periodic points. So, there is one simple case where we can describe quite precisely what happens. This is a case that you may have encountered in your life without knowing it. This is a family. So this is a family of one dimensional polynomials, which is the form F lambda of Z is Z square plus lambda. And lambda lives in the affine line. Whatever is the field you want to consider. Okay. So here what is the critical set of F lambda. So this is the place where the critical point the derivative vanishes. So here you have zero and infinity. Okay. So for a polynomial. Infinity is always a fixed point. So, there is no real condition here. The condition is that, okay, lambda is post critically finite if and only if there exists integers such that F lambda m of zero is F lambda m of zero. So these are the parameters for which zero is prepared. Okay. So here same question. So same question, meaning that I want to understand. How are PCF parameters distributed. So this is my motivation for what's coming next. Okay. So again you have in the Euclidean topology in the Zaris key topology and in the sense of measures. So what is the main. So we know the answer to all of these three questions. And what is the main object that we need to, to define to study this. So this is the model board set. So this is the sets. So here again. You could do it or any complete algebraically close field. It could be non-archimaging. Okay the geometric picture will be different but the answer will be somehow similar. Okay. Take the sets of complex parameters for which the orbit of zero at this precise parameter is bounded in C. Okay. This is the model board set. Okay. So it's a, it's a complicated set. It's a fractal set. It's boundary as out of dimension two. So it's not very easy to draw that let me try to do a rough picture. Okay. So there is a main category here. And you have a, you have kind of don't write decorations everywhere coming around. Okay. And you have small copies of itself everywhere. So the model board set is this set that you have filled in. Okay. All in there. It's a full set. Okay. So the answer here is quite precise actually. So again, the PCF parameters are the risky dance. So this is the main thing which will be of interest for me later. Okay. And so what is the Euclidean closure of PCF parameters. Okay. You see, inside the model board set. This sequence, it defines a, so it defines a sequence of allomorphic functions. So this defines a sequence of allomorphic functions of the parameter. And inside the model board set. So in the interior, they form a normal family. So it's a quick continuous up to extraction. So everyone converges outside the model board set here. The sequence diverged to infinity locally uniformly also. So where can we have a different type of behaviors. It's exactly at the boundary of the model board set. And so in the Euclidean topology. You can see that PCF parameters accumulate exactly on the boundary of the model board set. Okay. It's a simple application of Montel theorem to see that. Okay. And I will give an answer to question three again. So for question three. Let me give you a more precise answer. So the initial answer is due to live in in the nineties. And myself and Gabrielle Vinnie, we gave a, I think it's in 2014, we gave a speed of convergence for this, which is there exists a constant one. So we set that for all and given one and for all functions, which is C to with compact support in C. So one of the two to the n minus one. So, so you look at the average at parameters lambda for which zero is and periodic. The equivalent of periodic points is periodic parameters. And you compare it to what is what can be described in several ways, but you can compare it to the equilibrium measure of in the blue in the potential sense of the model board set. So it's less than this constant and over two to the end. And you have the C to know which is getting out. Okay. So it converges and it converges fast to this measure. Okay. So what about other families. So let's stick to the case where maps acts on P one. In this case, there is a big literature on this question. And the one family which has been studied a lot is the modular space of those of those rational maps. So it's the equivalent of the modular space of curves of a given genius. You want to study those objects here you want to study those objects, but up to isomorphism, and you want to study how they deform up to isomorphism here the isomorphism is not just an isomorphism between P one and P one because well, there is just one complex structure on P one. And then there are other isomorphisms which conjugate your maps. Okay, so there is a natural action on the space of maps from P one to P one, which all have degree D equals D by P GL of two. And the action is that it conjugates your map. So this is the action that you, that you get. Okay, so actually this is a good quotient this is a geometric quotient, and this is a fine modular space. So really exactly as in the case of curves. And in this family, the answer to the same three questions. The answers are known. And so, there is a third and that you can attribute to many people. So the Marco gave a simple proof of the Zaris kid on city of PCF parameters. Myself and Gabrielle Vinnie will prove the convergence in the sense of measures. Okay. And, oh, and there is just okay. Okay. Okay, so what is the theorem is that PCF parameters are risky dance in this modular space. MD one. Okay, this is the first point. And the second point is that. So any generic sequence of finite sets of PCF parameters. Okay, let's just go and find finance sets converges to what we call the bifurcation measure. So, in some sense here again. We have a completely precise description of what could happen. And well, there is no surprise compared to what's going on in in the space of degree two polymorphs. Okay. And then ask the question which is what about other families, for example sub varieties of this modular space. This is the first question you can ask. And what about when K is larger than one. I'm not going to focus on the first question. So there is a beautiful conjecture by Baker and the Marco. Which is somehow kind of equivalent to the on the earth conjecture but in this dynamical context. But it's a very difficult problem as the under its conjecture. And. Okay, I solved this conjecture for one parameter families. When all maps are polynomials with shall favor a couple of years ago. And this is essentially the only non trivial case which is known, of course, many people were working on the way we did not do the only contribution. And they are very beautiful contributions by Baker and the Marco. So by Kyoka. And, yeah. And, okay, they also gave a variation on this problem with the NBN and figure, which is very interesting. Okay. But I will not spend more time on this program. So we are now going to focus more on on the case when we focus when we study dynamics in higher dimension. Okay, so about that. And there is this conjecture by Ingram, Ramadas and Silverman. Oh, sorry. Ramadas and Silverman, where the conjecture that PCF maps PCF parameters are not. There is a difference in incidence in this modular space. When K is larger than one. Okay, so this is a very different situation is, okay, there is a complete contrast with this situation with the previous situation. Okay, so what is the motivation for, for this conjecture. Let's say the philosophic ID. Well, assume that the critical set is smooth and irreducible. So if it is smooth and irreducible. And assume that the NK is different from two. And for degree reason, CF is of general type. So it's a hypersurface of general type. Sorry. So if it is of general type, it has no non invertible. Okay, this implies in particular that. But what if CF is periodic, meaning that fn of CF is the critical set. What happened in this case. Well, it has to have a non invertible on the morphism. Fn restricted to Sierra. Okay, so, so it's not possible. Okay. So based on this observation and of course making much deeper study using this kind of ideas. So the Ramadas and Silverman proved the following theorem. I think it's in 2019. So in grammar. Ramadas and Silverman in 2019. They prove that the set of maps. Such that FJ of CF is periodic with J at most two. This is not the risk it has. Okay, so this is the first step. So in grammar for each day, you will have to a priori increase the sets of parameters that you authorize the risk closure may grow each time. So it doesn't guarantee that PCF maps are not risky. Okay. So the main theorem here. This is a joint work with you and taflin and Gabriel Vinnie. So 2022. PCF maps are not our students in this modular space. Okay. So the G is drastically different from the one suggested by Patrick Roigny and Joe. And so, okay, let me say if we want to see there is a hand which is left. Sorry, can you hear. Can you realize the study the above in the above theorem. In the above theorem in the previous theorem F is a function on peak or F. And what is the parameter space here in this one. Yeah. Yeah, it's so yeah it's okay the classic polynomial map. It's the map from a PK to PK. It's just an under morphism. So you take all the space of all on the morphisms up to conjugacy. All of a given degree. So where is the parameter space. What is the problem in this case. So the parameter space here in is again the same guy. Okay, it's this parameter space, but you the only. So they say that if you look at parameters for which either the critical set is periodic, or it's first image is periodic, or it's second image is periodic. And this set is not the risky dance. This is what they prove. Is it more clear. Okay. You for the question. So the strategy is drastically different. And I must say that one of the motivations, the initial motivations for the strategy was for myself the study of series of paper by Jean Gao and Philip Habegar. These are the strategies of Habelian varieties. And uniform bogomol of type statements in these in these families. So this is one of the motivations for the strategy it was not necessarily aimed at the starting point at proving this kind of results. Okay, so this is where we are essentially now. The strategy is to use again bifurcation measure to give fundamental height inequality in the spirit of the work of Gao and Habegar and Dimitrov go up again. So this is the first step to use. Okay, I may describe a little more precisely what what I mean by that. The second is to use an arithmetic distribution result on quasi projective varieties. Okay, here it's under the assumption. So that PCF is risky dance. Okay, I have to make the hypothesis that the other students for them to distribute. So the third step has nothing to do with that it's a step which comes from a complex dynamics is to say well, this actually cannot happen. Okay, so the third step is to build knocking the cleaning open set you, which is non empty, which is contained in the support of this measure. Which contains no PCF parameters. Okay, so if your PCF parameters equidistribute this measure. So if they could distribute the measure mu in particular the accumulate on the support. However, if you have an open set, which is contained there which contains no PCF maps you have a contradiction. So this will give you the contradiction. So what is the the ingredient. I want to focus on now for the, the remaining time is how we obtain the fundamental height inequality. Okay, so first we have to define what is the bifurcation measure and why does. Why is it non vanishing so the non vanishing could be seen as a consequence of this number three, so let's, let's keep it as a, as a black box, okay. So the bifurcation measure. It's, let's let me describe it in a, in a simple, in a simple context. So let's say there is a good adaptation of that. So I assume you have a family of abelian varieties, family of abelian varieties of a human base. Always policy. You have on a you have a betty for the Asian. Okay, and it carries a natural form differential form. And this differential form is, so it carries a natural form. I like to write as t to the K. So this is the form which in your family. So you have your family of abelian varieties here. So you have your family of abelian varieties of a human base. If you look at the base, the base on the, you look at the fiber. This restricts there as the arm measure. So let's take here. It's hypersurface. In the case of hypersurface is it's a little simple to, to, to define but well, it doesn't make a big difference. Which projects subjectively so you can assume it's flat. And let's call it, I don't know, each. Then you can look at the restriction of this guy to age, and you can push it down to be so it's an object. It's an object on the. So it's a one one here it's a one one form. Okay, and the bifurcation so let's call it the current T associated to H will be this guy. So the measure mu sub H would be T sub H that you take. So dimensional be times. So there in this context there is a condition of non degeneracy. I don't know who introduced it, but I'm sure I saw it in Phillips papers and in subsequent papers. And this is equivalent to. So this is a condition which will give you a height inequality and this is equivalent to the non vanishing of these guys. So here instead in in coming back to the dynamical context. Instead of having here a family of a billion varieties, you have a so, you know, space times pk. And here, instead of having a smooth object. You have something which is somehow glued. New F lambdas. So which is the fiber green measure of this guy. Each fiber you have an object global object which is restricted to each fiber gives you this measure mu sub F lambda. Okay, and you can do the same construction here. It will give you the bifurcation. If you look at if you replace here, H with the critical set of your full family. Instead. Okay, this is the object. The same, which I will not spend time on it is that the measure is non zero in our context. Okay. Okay, so now what do I call the fundamental height inequality so it's not my naming of it. You have an on the morphism of pk, which is defined of a Cuba. You can define its canonical height. How do you do it. You look at dates argument, you take the limit of one of a d to the n. Here is the knife height. So it's the height associated to. With the standard metrization. And you composite with F to the end. So what is the. So it's a height function. pk of Cuba. With non negative real values. And it has some invariance property. So pre-composed by F, you just multiply by D and the Functorial properties of the height machine tells you that actually it differs from the knife height by just a bounded function. Okay, and these two properties characterize your height. Okay, so these two properties characterize. The function. Okay. Okay, I don't have much time so here what is the fundamental height inequality. The fundamental inequality is that. So, there exists an open set. In one. Okay. Let me be a little more precise so. Let me make a family. Define of a cubar. And you look at. It would be the set of lambda Z such that lambda is in the L sorry Z is in the critical set. And so this is a sub variety of lambda times pk. And so there exists you in this variety, which is open. Set that. Oh, then there exists a constant or so. There exists two constants. One positive. And one real. Set that for all Z in you. For all lambda Z in you, sorry. The canonical height of F lambda at Z is bounded by this constant. My height of lambda. And so this is any uncle height. Define about you. Okay, so in recent breakthrough in arithmetic geometry. Family in a about uniformity results in arithmetic geometry. It appeared to me that this was one of the important ingredients. So the method used in geometry where really relying on the geometric structure and the group structure of the ability and variety here. The idea is to replace these arguments by complex dynamical arguments. So, for example, what we are going able to do is to give an idea of what could be the constant see one it depends only. On the volume of this bifurcation measure on the modular space. Okay, for example, you can have a complete explicit description of it and the approach is to go through complex dynamical arguments to prove this kind of inequality. Okay, am I running out of time or should I stop. Okay, you have a few more minutes. Okay, a few moments. Okay, thank you. Okay, so then there is a question of a quick distribution arithmetic a quick distribution. Okay, there is this beautiful result by Chinese Yuan, which is back to to the 2008. There is an equity distribution general arithmetic equity distribution result, which is on projective varieties. And recently there's been progress is on the topic. Yeah, I will forget someone, but I hope not. So, you know, and you won and John gave a general. So, here it's on families of a billion varieties. And here it's a general equity distribution result on quasi projective variety. Okay, so, okay, so in QNAS case, this type of inequality appeared to be really the keystone. You want and John have a very beautiful theory about Adelie Klein burdens on quite a projective varieties. However, I'm not really confident with that with that machinery. So I cooked up by myself. Another result. This is of the same flavor. And again, for which I can rely only on complex analytic arguments for most of the parts of the proof of course you have to do some geometry at some point, but okay, my aim in all this was to have a quite consistent approach at every step. And this measure by the bifurcation measure will play the key role at each of the three steps of the proof. Okay, and as a conclusion, what I can say that in my title there was this uniformity. Question. So uniformity. It's meant in the sense of the result of, for example, which is that when once you have your open set, which contains no PCF maps, the question, the natural question is that there exists an upper bound. On the number of periodic points in the critical set such that for all class in this open set. The cardinality of the critical set intersected with periodic points of that is bounded by this constant. So, hopefully we will manage to reach that at some point. In Cunez strategy, and I think also in Philip and the young strategy. One of the key points is that you have a group structure, it's that your family of ability and varieties. It's a group scheme. So it's one of the important thing they use, which you don't have here. So you cannot use it. So, instead, you have to push a little further, the techniques from complex dynamics to guarantee that you have a height gap. Okay, so we managed to do that in some cases for now. For example, maps. If you take maps, all your maps in your family are from P2 to P2 and have the line at infinity, which is totally invariance. So essentially polynomial mapping of the affine plane, which extend automatically to P2. And then use, again, pluripotential theory to conclude. So, but here there is still a lot of work to be done. And so I think I'm going to stop there. Thank you very much for your attention.