 Čakaj. Zdaj je to vse. To je pravda, da sem tukaj tukaj na ACTP. Zdaj imam zelo, da imam tukaj. Zdaj, kaj imam tukaj na tukaj, je tukaj... imam tukaj, da sem tukaj, da sem tukaj tukaj, da sem tukaj tukaj, da sem tukaj, da sem tukaj, da sem tukaj, Vsi nekaj rizultov vsi splahte izmah, poloč te rizulta, vsak pa vžilega rizulta, vse, in z дажеvolj nesel, če na vsebe je zelo, če se bo zelo v del v rošu. Zdaj vsebe sem, kaj imam. Zdaj je tukaj prohač, da sem je vse, da si tukaj imam... zelo, in Reissi v Toulouse in Tamara Servi v Paris Saint-Pierre. Ok, nekaj je tukaj? Zelo je... Tukaj zelo je tukaj, nekaj nekaj nekaj nekaj nekaj nekaj nekaj nekaj nekaj nekaj nekaj nekaj nekaj nekaj nekaj nekaj nekaj nekaj Je zelo zelo, je f frema cn 0. Moli je to kolega najpovrduznoj stvari, sez njena tukaj zelo, če je tukaj zelo, vzelo zelo iz tebe zelo. To je zelo, da se počutite v tem formu, f of z je zelo, z plus p nu plus 1 of z plus eigenordnje term, kaj je P nu plus 1 je n-tuplj, ozvrčenjez, polinomjez, ozvrčenju nu plus 1. In, če se početimo, ne je to nezavrčen. In tudi je to nezavrčen. To je v Cn. Poživajte, kako je zelo na dimensiju 1, kako je tudi lofa-2-flava teorem, ki je vsev teore, ki je vsev teorem, ki je vsev teorem, ki je vsev teorem, ki je zelo vsev teorem, ki je zelo vsev teorem, ki je zelo vsev teorem, ki je vsev teorem, ki je aset, kaj je petal, oz. 4. Zato, da, kaj si počeš v petal, v petal in petal, zato objevno se konveriruje tengent v vsej vsej atratin. Tengent vsej tengent vsej atratin. If you consider the inverse, the inverse is still tengent to the identity, And the attracting direction becomes repelling in conversely. So you also have petals for the, for the inmens, which are these blue repelling petals, where for the map the dynamics is escaping and considering the union of attracting repelling petals, you get full pointed neighborhood of the origin. Vse je imač, da mi je danes... ... in nekaj nekaj. To je nekaj. To je fakt, da sem počukal nekaj, da sem pričal nekaj, da sem pričal... ... in tače... ... z naših počukov... ... nekaj... ... in kako si počukaj... ... z plus z kaj... ... z naših počukov... ... nekaj sebe... I to. One basin, which is the inverse image of a petal. And well, a typical question is what happens when you start in the intersection between an attracting and repelling petal, and the answer is that the orbit starts going away until it leaves the repelling petal, and then comes back to the origin staying inside the attracting petal, going tangent to this area. Tako je, a zelo to je poslutito, da je tudi začinila, da počutite vrati na prinské izgledanje. Počutite češnje vse ljudi, v zelo točnoj klasifikaciji iz toga, in pa je jeznja Karbonis modulaj, načinje analitice klasifikacije, tudi prizvačati, svoji inštje lohne in katrtino in narzitev. Však sem ne zelo pošličaj s njem. Prepečiš, da bi ne sp examinevzati v speciju. Ne, tega je pravidem inbega. Mi je quatreetil in svojo, njihoj slijet tebranje, kako je najfajno odzajati z zelofnice. Tebe imaš dokonjučenju poličnice. Vse patelje so vsezajne od zelofnice vsezajne plene. Vsezaj, da jih počet na stvari, zelofnice imaš vsezajne plene, zelofnice bilo ne odzaj. Zelofnice je to tako zelofnice. If you want to consider the set of all points, whose orbit is going to the origin tangent to this direction, you need to take the inverse images of this, and you get then something with fractal boundary as you can imagine. But you can define the formal definition of petal is a simple connected set with the origin in the boundary, which is f invariant, and such that if you have an orbit going to the origin tangent to this real direction, then it is eventually into the set. And with this definition, you can get exactly something with real analytic boundary set at this point, and with this set. You get a whole panel. Yeah. Yes, you can. It's a local thing, so you decide the neighborhood you're working, you find the pattern inside the neighborhood. OK, so this is a situation in one variable, and now we are interested in to see what happens in several variables. OK, there is one case where everything is easy. It is if there exists one-dimensional complex curve in cn f invariant passing through zero, we get a fatou flower, this kind of set is called a fatou flower. And, well, this is clear. I mean, if you have a one-dimensional thing, and if you have a one-dimensional complex curve, which is f invariant, you just apply the one-dimensional statement restricted there, and you get this kind of configuration. At least one-dimensional, I'm not going to talk about the problem of describing what happens in a whole neighborhood of the origin in several variables, but it's a very important, very much open problem, that is we are immediately attacking a preliminary problem here. The point is that in most cases this cannot happen. This, in general, no. In general, there are no one-dimensional complex curves f invariant passing through the origin. And a king gave an expletive example, and then recently reborn showed that for a map of this form, and write it down correctly, is z plus w square w plus z square plus lambda z to the phi. This map here has no f invariant curves for almost every lambda. Actually, for every lambda outside of a polar set. So we cannot expect to use this approach to get f2 flower invariant set. That is one-dimensional invariant set in several complex variables, in several dimensions. So we have to do something different, even though as you'll see this idea of having some f invariant one-dimensional thing, it turns out to be important later on. It is actually what we are working on now. So what to do? What I'm going to describe now is part of a Kall-Akin's theory. This has been done by a Kall in 1985, using his approach, using resurgence functions, and mole theory, and so on. And I came recast part of this result, and then got something new in more classical terms in 1997. The result, even the idea of what to look is the following. Here you have that there is z0, such that the orbit of z0 goes to zero, and the direction of this orbit, I will tell you in a second what I mean by this square bracket, tends to a direction in pn-1 of c. So this square bracket is just the canonical projection from cn-the origin to pn-1 of c. So assume you have an orbit going to the origin tangent to some direction, complex. Exactly, that's exactly one of the points. This is a complex direction. If this happens, this direction here cannot be any direction. It is what is called a characteristic direction, that is, it is a sort eigen vector for pn-1. If you take pn-1 of v, this is equal to lambda v for some lambda in c. The actual value of lambda has almost no meaning, because since this is a homogeneous degree nu plus one, if you just multiply by a number here, you get a power of it over here. What it does mean is whether lambda is equal to zero or not. Indeed, this is a very important distinction. If lambda is equal to zero, we'll say that the direction is degenerate. If lambda is different from zero, we'll say that the direction is not degenerate. Coming back to your comment, this is a notion that you don't see in one variable, because in one variable you only have one complex direction, which is a whole plane. What's happening is that we are trying to understand what are the directions that might contain a strong word, but if we add such a one-dimensional complex curve, it will be tangent to something, and we would like to know what is this something. The answer is the same in characteristic directions. The point is that characteristic directions always exist. You have at least one point where to start looking. One remark is that this does not give the whole possibility for orbits going to the origin. In one-dimensional case, all orbits are of course tangent to the one-dimensional, they are only there, but they are also tangent to a real direction. But in several variables, the resist orbits go into the origin so that the direction is not going anywhere. They are going to the origin along spirals. There are examples in this kind, there are also examples of small cycles in several variables, that is, of periodic orbits accumulating at the origin. The situation is the following. In generic cases there are only a finite number of characteristic directions. The exception is one case, which is called the critic, essentially when... What it means is that all directions are characteristic. It means that this p nu plus one is a multiple of identity. In that case, all directions are characteristic and the dynamics are much easier to start with. Indeed, actually I will not consider the rest of that case, because they are much easier than the other cases. In generic cases there are only a finite number of them. You can count with multiplicity. It depends, of course, on nu and the dimension. Anyway, what Ekalian Keem were able to prove was following theorem, which is a sort of converse of this proposition. If V is a non-degenerate characteristic direction, then there is a fatu flower tangent of V. Now I have to tell you what it is fatu flower. Well, essentially fatu flower is one-dimensional version of a picture. First of all, we need to have the equivalent, analogopetals. The analogopetal is called a parabolic curve. Parabolic curve is a phi from some d into cn, injective, holomorphic, such that first d in c is simply connected with zero and the boundary. Actually, when I'm talking about fatu flower, usually d will be exactly one of these fetals, exactly that one. Two, the image of phi is an invariant, and three, f restricted the image of phi to the k converges to the origin. Moreover, we shall say that i is, of course, I didn't say it, phi of zero is zero. One can always assume that phi is continuous up to the boundary, that's not a problem. And we'll say that this phi is tangent to the direction d if its direction goes to the direction d for zeta going to zero. So, you have to imagine that a parabolic curve is just a holomorphic injected image of one petal in cn with origin going to the origin, it is f invariant, and so inside here you have exactly the same dynamics you have in one variable, including the factor it is conjugated to a translation. A fatou flower is a set of new parabolic curves tangent to the same characteristic direction, but including different real directions. So, in a fatou flower you get the same complex direction, but with each one corresponds to a different real direction, a different real attractive direction. And, of course, you have the same for inverse, so you also have the rappel in fatou flower, but the point is that these petals, these parabolic curves are not inside something going through the origin invariant. They can have pretty bad behavior at the origin, there is no way to extend them in any way. Only up to zero, after zero there is no real reason for them to extend. Oh, yes, yes, yes, you can actually, it depends on you, they go to the origin like one over k to the new or new plus one, something like that. So, this is very good, so this is a color scheme. It's very good because it works in any dimension, but it is not yet the end of the story, because this is a generic condition, but it is not always satisfied of our maps with no degenerative characteristic direction. So, one would like to know what happens for along degenerative characteristic directions. Well, there is one case where nothing happens, in the sense that if you have a fixed point set, which is not just the origin, there are direction tangent to this fixed point set, and this direction are characteristic degenerate, but in that case, nothing happens. We have just a fixed point set. So, we'll make a extended assumption that it's not really necessary, but it simplifies some statement. We'll assume from now on the zero is an isolated fixed point. Ok, to give you an idea of what one might try to do in along degenerative direction, let me give you a very quick description, very quick description of how to prove such a theorem. The first step is growing up origin. Growing up origin, you replace the origin by pn-1 of the tangent direction there, and you can lift the map so that it becomes the identity of this exceptional divisor, this pn-1, and then you can assume that our non-degenerative characteristic direction is 1,0, then we can write the map in this form f1 of z z prime is equal to z plus z1, sorry, a z1 nu plus 1 plus i-order terms of the form of z1 nu plus 1 z prime comma z1 nu plus 2 and the other coordinates are written this way for identity minus a z1 to the nu z prime plus i-order terms z1 nu z prime square z1 nu plus 1 z prime plus, this is important, z1 nu plus 1 psi 1 of z1 here I am writing z prime is z2 up to zn so this is a first coordinate and as soon as v is a degenerate is a characteristic direction you can write a blow up in this way and non-degenerate is equivalent to having this a different from zero this coefficient over here well if this term here was not present what we would have is that z prime equal to zero would be invariant and so we will have one curve one dimensional curve passing to the origin which is invariant you can apply one dimensional theorem to get a F2 flower but in general this term here is over there so a king's proof the idea is to change variables in a sector these changes of variables are not defined in a whole neighborhood but only in a sector so that you can push this exponent here as high as you want up to one million and then there is a way to rephrase the existence of parabolic curves as a fixed point theorem fixed point problem in a suitable functional space and it turns out that the operator given this fixed point problem is a contraction with respect to a suitable matrix so you get a fixed point and so you get a parabolic so the point here is that what a point we wanted to make is that it's anyway is looking for in some sense some curve which is invariant even though it is defined only on sectors so what happens for the general characteristic directions here the theory is well developed only for n equal to 2 ok the first main result which is something I proved in 2001 is that if n equal to 2 and or isolated then there always is this pattern flower tangent to some direction v actually I proved something more precise so the pattern flower it exists even if all the characteristic directions are degenerate I proved something more precise and to state it let me write down in a slightly better way the map in dimension 2 so after grow up I think this way f1 of zw is z plus z per nu plus 1 a0 of zw f2 zw is equal to w plus z to the nu v1 of zw when there are final number of characteristic directions you can always do this and I also assume one characteristic direction in 1,0 and so 1,0 is a characteristic direction if and only if in writing this way v1 of 0,0 is 0 and it is non degenerate if and only if has 0 of 0,0 it is different from 0 in under this condition you can write down a0 of 0,w will be of the form a 0,0 plus a0,1,w plus etcetera v 1 of 0,w will be something similar but with no constant so it will be b0,1,w plus etcetera and then we can introduce 3 important numbers actually 4 important numbers m which is the minimum k such that a0k is different from 0 tau which is the minimum k such that k is different from 0 then what we call the pure order nu,0 which is the minimum between the order of b1 and the order of 0,1 that is it is the degree of the first non vanishing term in the power of city dispension and at the end the last number is the index delta which is the residue at 0 of a0,0,w over b1,0,w ok these are 4 important numbers and the statement I have more general than that is that here if this index does not belong it is not a positive rational or 0 there is this f2 flower tangent to that characteristic direction that is I chosen a characteristic direction which I can assume to be 1,0 I blew up I wrote it in this coordinate if the index is different it is not a non-negative rational then we have a f2 flower tangent to that direction that fium is a consequence of this because from going to here to there you prove that if you take the sum of the indices over all characteristic direction you get minus 1 so you should have at least 1 which is not a non-negative rational so this opens one conjecture that the same holds if yota is just different from 0 and on this conjecture there is an important result by Molino in 2009 and she proved that the answer is yes if the poor order is 1 and and this seems to be a technical assumption not really sorry but I will discuss about it in a minute and of course large open problem here does it hold m greater or equal to 3 up to now this is completely open even though I opened 5 minutes to tell you a possible approach to that question ok recently there has been a different approach to this subject essentially formal approach which is due to several people Brušero-Martinec Lote Serna Raskardua Sanchez and clearly forgetting somebody the idea is to consider our germ as the time one map of a formal vector field not necessarily convergent and trying to apply results known for formal vector fields and from that to the use something for about our original germ and it turns out that with this approach one can get results which are not exactly in and one can get something more but something different with respect to approach we developed so there are sort of complementary for instance Lopets and Nancy Sanchez have been able to prove a full result a few months ago actually they prove that we are in dimension 2 they also have some results in higher dimension but it is stated for in dimension 2 if f admits a form and I will tell you in a second what it is in variant curve not completely fixed that is not contained at fixed point set then either f or f minus 1 parabolic curve asymptotic to this formal invariant curve formal invariant curve I mean something where there is a general definition but let's say which is something defined by a pair of formal power series can you imagine as if you have an homomorphic curve it will be parametrized by a pair of convergent power series so you just take a pair of formal power series in one variable ok, you can give a meaning to a fact that this pair of formal power series is f of course in variant essentially that if you apply composer with pair of power series with f you get a reparametrization of the power series and not completely fixed it means that this parametrization is not identity and asymptotic it means that it means that on a sector this formal power series gives you asymptotic expansion of this parabolic curve in a sector, not in a whole neighborhood and the proof is that you can if you assume this you can write your map in a particularly good normal form so you can essentially say something about a0 and b1 and then applying a variation of the quims idea one gets a convergent parabolic curve asymptotically defined in a sector this brought us to think about ok when there might be exist a formal invariant curve and whether power series are enough and this is not what we are doing in the last few months and the answer is that we have a number of cases where we have we can prove the existence of invariant curves but and this is the most interesting part power series are not enough there is one of the main difference between working with in discrete dynamics that is with germs of maps instead of continuous dynamics that is using vector fields by the way these two theorems are not over imposed in the sense that means a formal invariant curve the normal forms that I described as u0 is equal to 1 but in general the index will be 0 so this theorem gives parabolic curves even when the index is 0 but there are instances where you can apply this theorem and not that theorem because all the formal power invariant curves in this sense are completely fixed so there is still some space and just to give you a first idea of what kind of space you can get we have a following theorem this is in cn assume the first thing we ask was what happens if d is non-degenerative characteristic almost finished then it turns out then there is a formal invariant curve in z this is our power series we coefficient in ring r where the ring r is c and then the usual power series or the ring this is the first new part of polynomials in the logarithm and you actually need this you cannot just get usual power series and and this is the last statement this is the thing that I proved yesterday in let's say n is equal to 2 and let's consider the case new series is equal to 1 that is the case considered by Molino and what I introduce m and tau so what happens is the following if n is equal to 1 or m is strictly less than n minus 1 or m is equal to n minus 1 and the index is different from n then we have exactly the same statement but and this is a more interesting case if n is at least 2 m is equal to n minus 1 and the index is equal to n then we have a similar statement but a different ring as a ring we have to take the the following k is equal to minus infinity to some k naught ak z to the t to the k ok so this has Lorentz series but we only polynomial opposite the grid part and s t we have to take the logarithm of z so what's left what's left is the case when m is greater than or equal to n and this is what I was working yesterday I got invited to dinner so you know duty course and but I'm pretty confident that this should be the worst can happen and after if in this way we will get the complete description for a former point of view what happens where the poor order is 1 then there is a general strategy that I proved some years ago to reduce by blowing up any germ to germ with poor order 1 and so our hope is that in this way we can get the formal objects which are invariant and then we can apply a modification to get actually convergent parabolic curves there is some hope that under some condition this approach can work also in several variables giving a way to attack this problem ok my time is definitely over thank you very much for your attention