 So thanks to, I want to thank the organizers to be here and this is a really important place for my life as a mathematician because when I was a student, I come here several times and it was very important to me to participate to this school. So I'm going to talk about finiteness of attractors for piecewise, not only for piecewise, but for piecewise situ maps of the interval. So this is a joint work with Paul Brandon and Jaco Pallis and so let me draw a picture. So we want to work in the interval, we can assume that this is the interval 0, 1, it's not a problem and we want to study how many attractor maps like that can have. Here we are going to allow also map has discontinues. So first thing, we want to define what is attractor. For me in this talk, an attractor will be a compact set and we are going to assume that the map is transitive in this compact set. This is not the only one definition of attractor. Maybe it's very common to assume the definition, but in this set all the attractor will be transitive and the important things of an attractor is to have a big base of attraction. The base of attraction is the set of points which in the future is going to be approximated into the attractor, that is the omega limit of these points is going inside the attractor. The future of accumulates in the attractor. So a big base of attraction you can define it in several ways and the most common is to have a positive Lebesgue measure. But you can also define, say that a base of attraction is big in the topological sense, but I'm not going to talk much about it here. So what is our problem? The first one is how many attractor a map of the interval with some regularity can have. So it can be zero, we can have a map without attractors, it can be infinite. This is the kind of question we want to talk about. And also, even if we have attractors, maybe the union of the base of attractor cannot cover all the interval. So the second natural question is that the union of the base of attraction has full a big measure and there are also the problem of classified attractors. So a first result in this set, you can think about the result of Singer and it's a very old result. And he said that he gives conditions so that the map only can have a finite number of periodic attractor. So to do that, he introduced in this context of one-dimensional dynamics, the Schwarzend derivative, this operator comes from the complex dynamics. So this is the definition of what is the Schwarzend derivative. For look that we have to deal with the third derivative, so the map has to be at least three-three to use this definition. And he said that, okay, in this case it's a city map, if you look for, if your map has negative Schwarzend derivative, the number of critical points, okay, bound the number of periodic attractor. So let me talk about a little bit for the simplest case of one-dimensional dynamics. That is not linear, so you can think about, okay, linear maps very easy to study in one-dimensional dynamics, but the second case is a map with a critical point. For instance, product map, something like that, okay. So this, the unimodal maps, okay, it's a kind of generalize of this, the quadratic map. We assume this map has only one critical point, and we are going to assume also that the boundary of this, of the interval is invariant by F, that is, in this case. And this is not a problem, it's only, we can normalize the map like that, okay. And a very formal example of this kind of map is the logistic family. I write down the formula there, okay. So what we can talk about this, the number of attractor for this kind of map. Of course, single map, that's, say that this map can have at most, if you, we use direct to this formula, three periodic attract, but indeed because of the, the zero is a fixed point, that can conclude that this kind of map has one, typically at most one periodic attractor. So but the first, first big result in this area about say, about describe the, the, period, the attractor for the unimodal map, the S unimodal map, was this result for block and a little bit, okay. And it's a bit confused, the date of this result, because this, in that time the, the, the Soviet Union don't talk much with the Occident, so it's, okay. But the point is, they show that first, S unimodal map always have a single attractor, okay. This single attractor can be a periodic orbit, a cycle of interval, and also a counter set. If this map is the attractor is a counter set, so the, the attractor will be exactly the orbit, the omega limit of the critical point. Indeed, not only the omega limit, in this case indeed the closure of the orbit, in this case the omega. So it's a, it's a very precise result. And more than that, the base of attractor, the base of attraction of this, the attractor cover, Lebesgue almost ever, Lebesgue almost all the interval, okay. So, but here I introduce the, the non-flatness property, and b-non-flat means that nearby the critical point, the maps look like a kind of polynomial. So what is, is let me show an example, an example of a non, of a flat map, this is a flat map. They should have came right down this in check that they, they trust in derivatives like that. So this is an example of a flat map. But here you are not going to deal with this kind of map, we are going to study the non-flat one. So look in the flat case, the names say that is near the critical point is almost a constant, okay. So, so what is a cycle of interval? A cycle of interval is a union of closed interval and the maps, the map restrict to the union is transitive. So you can go back to the theorem and say, so this is a union of transitive, or the attractor is a union of transitive interval, or the attractor is a periodic, or the attractor is a counter set that is the omega limit of the critical points is a very precise result. Okay. So to say the next result, I'm going to introduce the notion of multimodal map is a general, generalization of a multimodal map. It's a map of the interval and what we assume that all the critical, with a finite number of critical points, and we assume that all the critical points is a maximum, a local maximum, local minimum. In this case, we are not, that the maps has some criticality like that. This is forbidden. So maps like, we are not going to talk about map like that, okay. So S-unimodal map is a map with this kind of critical points, it's called turning point, and with negative trust and derivative. So block and a little bit also have a similar result about multimodal, similar to the union model one. In this case, we can decompose the interval in a finite number of sets. The set will be the basin of attraction of a finite number of attractor, and this has full aback measure. That is almost every point of the interval goes to one of the, is going to be attracted for one of these attractors, and, but I didn't mention in the one model case, but I'm going to mention here that we have for Lebesgue almost every point in the basin, these very strong properties, that not only the omega limits, the points is attracted to the attractor. This is the typical, but not only this, the omega limit will be exactly the attractor. So this is a very interesting result, and so the attractor again can be only a periodic attractor, a cycle of interval, and a counter set, and in this case also the counter set must be the omega limit of some critical point. So but now I'm going to talk a little bit about map with discontinuities. So we are going to for now, for now stay a little bit more with the condition of negative Schwarzman derivative, and one point is, okay you can think maps with any kind of discontinuities, but like that, but a very important type of discontinuity is when the discontinuities are at the same time a critical point. These kind of discontinuities appear naturally for flows, okay, and when you are, you can okay reduce some flows to one dimension, to start some flows, to start one dimension dynamics, and so this kind of discontinuities appear, and this indeed is the most difficult type of discontinuity, when the discontinuities are not associated to a critical point it's not so difficult to deal with. So and the simplest case of this kind of map is the contract in Lorentz map. It's a map like that, this map has only one discontinuity, and this discontinuities are associated to the critical point, this is a nearby C for the left and for the right the derivative is close and close to zero, okay, and let's also assume the negative Schwarzman derivative for this map. So last week in the Martens, Martens lectors, they talk about Wander intervals, and so the Wander intervals plays a very important role when you want to start one-dimensional maps, okay, and what is a Wander interval? It's an interval that is not, it's not, the interval does not intersect the base of attraction of a periodic attractor. I, let me, don't remember if I define, I can go back, okay, if not I define, so let me define this notation, let this double be the union, the union of the base of attraction for all periodic, okay. So Wander interval does not intersect the set, the double B, and does not intersect the orbit of it, does not intersect the critical point, so if you, if you restrict, restrict the iterate of for any J to the, the Wander interval, this is a homomorphism, okay, and the, the important properties, the, the, the iterate of the, the, a Wander interval does not intersect each other, okay, so it's not that the iterate of it does not intersect the, the interval itself, but does not intersect, 2 over 2 does not intersect, okay. One of the many ingredients to prove, to start the number of attractor in, in class, to make the classifications in several property of attract for one dimensional map is to prove, to understand the Wander interval, and most of case to try to prove that there are no Wander interval, but this I write down, I summarize some result about the existence of Wander interval against, the first one is very well known that, the Denjo result about no, the non-existence of Wander interval in the cycle, and after that for S unimodal map was a result of Gucca Hyde, and your cause also has a result for a map of the cycle with critical points, so this is different for this one, because this, the map is a different form of him, this node, and for, for S unimodal maps a little bit, so that the non-existence, existence of Wander interval for S unimodal map, and Sebastian in Wellington for C2 unimodal maps, Glock and Lubit for C2 multimodal map, and the general case for C2 maps was a result of Martens, Mellon and Van Stree. Look that the difference of the, this result of, and this one, this is for a multimodal, so in this case the, all the critical points have to be a turning point like that, and in the second one, okay, you can, this kind of map, this kind of critical can be happened. Okay, so for non-flats, C2 non-flats map, there are no Wander intervals, but the picture is comparatively different for maps with discontinued, so indeed we don't know much about today's things or not of Wander interval for map with discontinued, even for the simplest case that is the, the contract in Lawrence map. You know, we know that this map can have Wander interval, but the, the example of Wander interval that we have is associated to so-called cherry flows, cherry attractors, so this is a very particular one, in this case we understand very much, so but the question is, all the Wander interval that appears for this kind of map is a cherry attractor, so this is an open question, and several people try hard to prove or disprove this, but until now it's not, we cannot say almost, we can't say almost nothing about this. So even for the case of infinity renormalizable map, we don't know much about this, only a few results with a very strict combinator or something like that. So as we don't know if there are Wander interval or not, this, the instructions to prove the existence of the finiteness of attraction must be very different. So in 2000, Kelly and his student, Sampierre, proved that for the contract in Lawrence map, no flight contract in Lawrence map, we can have at most one non-periodic attractor. Indeed, the results say also that if you have a periodic attractor, so indeed we have three cases, one periodic attractor, in this case, if this map has only one periodic attractor, the base we attractor, we have full measure, Lebesgue full measure, so we attract almost every point. The second case is two periodic attractor, but in this case the base of these two periodic attractor covers almost all the interval, and the third case is no periodic attractor, and in this case, the theorem says that in this case there exists only one attractor, that's not periodic, and the base of attraction covers most all the intervals. This is the result of Kelly. For this, sorry, for this proof, they use some kind of induced map in the half of our Kelly tower, it's a very complicated stuff, and very particular for this kind of map. They use a lot of this result, a lot of the property that this map preserves orientation, and a lot of the combinatorial of this kind, this specific kind of map. So, two years ago, in a joint work with Paul, with Brandão and Bares, we generalized this kind of result with a totally different strategy. Our result is completely different for the Kelly and the sub-PiR1, and we show that if your map has negative Schwarzman derivative, it's not necessary to be in a flat in this case, and you can always prove that we have only an infinite number of attractors, and the attractor has these properties. We can also give a classification of the attractor, so I write down only a partial part of the result, so this was the former result, and yes, but in this case the bond is not very sharp in this work, it's bound by two, the number of critical values. This is the critical values of f, okay, you can refine it, but essentially it's this kind of model. This is the number of critical value, because now we have discontinued, so the number of critical points in critical value may be different, because now we have a sharper result, because maybe I can talk a little bit later about it, but let me talk about how we can get out the negative derivative condition. To do that, we have to be careful, because the Schwarzman derivative is very good to control the distortion, but also have some other implications, like negative Schwarzman derivative bound the number of periodic points, so if you don't have the condition of negative Schwarzman derivative, it's easy to construct a map with an infinite main attractor, like the periodic attractor, like that, so it's finite one, and it's easy also to construct a map that maybe the map has an attractor, like in this case a periodic one, but here there are no attractors in this part of the interval, because this is the identity and the map goes there, and so there are no attractors here, so the thing is a bit more complicated, so we have to deal with this kind of problem. So what's the sum? Okay, for instance if you say that the number of periodic points is with period N, Y is bounded. I'm going to show you. Okay, which one? This one? No, but this one is with negative Schwarzman derivative, so I'm going to show another one, so for this one it's finite, because the negative Schwarzman derivative, okay, but so let me only to emphasize the notation, this is the the union of the major attraction of periodic odds, and here is this, this is the prepariotic points, okay, and let me only comment that this is a periodic in your, in a little bit larger sense, it's a periodic like your, for instance you can have a map like that, so this is the critical point, and it is discontinued, but they made of this the critical points here, so this, this is really not a periodic, we have almost every point of the, every point in the this interval here goals is attracted to this point to see, but see itself is not invariant, so but in this case we can say that this is a periodic point for the left or for the right, so for me in this more general context, the context where this continues, we may say that a point is periodic, but if the point is periodic by the left or the right side, okay, so this is a result of Vargas, Edson Vargas and Sebastian von String for 2004, and they prove that for C3 maps, but here they are not assumed the negative stress and derivative is a C2 map, we can, we can have only a finite number of non-periodic attractors, you see, okay, and here is the, the base, the union of the base of attractor for periodic or periodic like orbits, and this is the set of periodic points, you know, you can have something like that, okay, this, this kind of point here is included in this set, and okay, if you take out these other things, what is remain for Lebesgue almost every point is a finite say, number of attractor and the base and attract are most every point that is not in this two set, more than that, as the result of blocking new bits, the omega limit is exactly the attractor from most of the points of the base, more, we have also classification of the attractor, the, the, this attractor can be okay, this is not, not good here because this is included in this one, so forget this first one, this in this case is a cycle off interval, and the account of sets with this type, so this and raise the first case here, the first case here is included in this part, okay, and so as we assume that we can have this continuity, we can say that a map is no flat by being no flat for one side and the other, in this case the degree of the criticality may be defined, this is only what I say here, so it's not important, let's go through that, and so I only want to define here the critical value, okay, we have this critical set because our result can, in our result we can have a point that is not, these points can be discontinued but the derivative may not be zero there, so this only talk about this, or that you can also have a point that the map is piecewise C2, but so we can have some points that derivative is not very good in that point, you can have the first derivative one but not the second one, so all this kind of stuff is put in as the label of a critical point, okay, and so but we assume that the number of critical or no regular points is finite, outside this is a C2 differential machine, so the critical value will be the limit from the right and for the left of this point, okay, so the result is this one, if we have a piecewise C2 map, okay, and the number of non-periodic attractor is bounded by the number of critical value, so this result, we bound by this one is far better than the first result, and again for this non-period, so we have the points that is attracted for periodic or periodic like ORB, the points that is pre-periodic, and okay, the points that is not pre-periodic and not for the attract in the base of periodic attractors go to one of these attractors and more than that, for almost every point here, the omega-mete is exactly the attractor and okay, again you can erase the first item, so what we have is that this attractor either is a cycle of interval or a contour set, in this case the contour set will be the omega-mete of a critical point for the left of the right is the same as the omega-mete of a critical value for some C in this set, the interval will be attracted to this kind of attractor, so if you have a wander interval and you may have a wander interval for this kind of set of maps, and what we can show that this wander interval, the omega-mete of this wander interval will be one of these attractors, will be, sorry, will be the omega-mete of some, exactly the omega-mete of some critical point, so let me talk a little bit about this step of this result and the first step is a kind of dichotomy, okay, so what we stood to prove this result is the place in the interval that is not visited by critical point, the idea is suppose that you look for your map and there are some interval that for every iterate of this map, the critical orbs don't go inside of this interval, so in this case if you look for the first return map to this interval, the brain will be big and the first return map is will look like some something like that, if you walk a little bit more and take this interval you can show that you take this interval as a nice interval, you can do that, so the first return map will be some kind of full induced remark of the map and if you have some good distortion, some good distortion property, you can show that this kind of dichotomy that for almost every point in the interval, or the omega-mete is outside the interval, go outside, or the omega-mete will be the whole interval, but if the omega-mete is the whole interval, the second option here, so this point is attracted to a cycle of interval, so it's easy to show that if the omega-mete, the interior of the omega-mete of a point is not empty, the omega-mete is exactly a cycle of interval, more than that the number of cycle of interval are bounded by the number of critical points because inside in the interior of each cycle of interval you must have a critical point there and two cycles of interval are disjoint or are the same, so if you get the, sorry, if you are in the second item here you'll say, ah, so this interval belongs to the basing of attractor of some cycle of interval, so what we want to study is the point that is not in the base of attractor of periodic points and the point that is not inside the base of attractor of cycle of interval, so the idea is to play with this dichotomy to show our result and the using the dichotomy it's not difficult to prove the second step that for almost every point that is not in the base of attraction of a periodic or it's not pre-periodic and it's not in the base of attractor of cycle of interval, okay, for almost every point like that the omega-mete must be inside the the the closure of the critical art or the critical values, so from step one to step two is easy, the the the difficult point is the step one, in the case that you have a negative schwarzen derivative the step one indeed is not so difficult because the schwarzen derivative helps you very much in this case, so the idea is to take a very small you can split your former interval in very small parts and look for the first return map to this interval so this will be like that and with a good distortion property so you can prove the step one for this kind of but in the case that you don't have a negative schwarzen derivative this is the the step one is the big problem, so for prove the step one in particular we show that you you may have a wander interval but first we have to show that the omega-mete of a wander interval don't intersect must be inside the the closure of the the critical art, so for prove in the in in the case that we don't have negative schwarzen derivative first we have to show that the omega limit of any wander interval is inside the closure of this so we have to have need some begin control of the wander interval and after that you play a lot of tricks to to get the first step so the first step with negative schwarzen derivative is not so difficult but we thought it is the most difficult part of the result the second step is comes easily for the first one and the third step it's not so difficult for the case of for c2 case the idea is you modify your maps and apply the the the second step to your former map and the modify one and compare this kind of things and get this third step but in this case for for if you your map have negative schwarzen derivative and you want to to make a deformation of it it's very hard so in this these two papers in the first case the the paper of negative schwarzen derivative the first step was very easy oh not so very easy but the third step was very hard and in the second case this change so this for map c2 maps the third step is not so difficult and here it's not the third one it's a fourth step and this is to to prove the transitivity and also to also to classify the the the type of attractor let me before I finish let me say two things that we don't know yet that they this let me come back that if this kind of attractor is minimal it's a minimal set this appears in the result of a little bit and blocking a little bit and also Vargas and the one string let me go back so this is a minimal set but to prove this they use very much this the kind of distortion control that they they something like a prior bond or something like that so this not and we bypass this kind of approach so it's it's an open question if in our case this this map and are minimal okay and okay I'm going to stop here