 Thank you. Well, first of all, thanks to the organizers for the invitation to be here. It's really nice to be in the video. So I probably, many of you have already heard me speak about these things. So you will probably be bored. I'm sorry. But basically what I will speak about is a couple of works with Patrice Le Calvé and Maisam Nasiri, which we are finishing the second part, like the first part was quite a few years ago. And the second part kind of completes, in a sense, the picture. So I'll try to give the whole picture this time. So the general problem that we are interested in is the following. We have a surface orientable and a homeomorphism, which preserves orientation. And we have an open connected set, which is invariant by the dynamics. And we assume some kind of, some nice property of this open set. For instance, that it has finitely many topological ends or something like this. But for the, usually the problem reduces to this simpler setting that will be the main focus, which is, we assume that the open set is simply connected. It's much easier to consider simply connected set. And the more general situations can be reduced to this case. So we will consider this case mostly. So the question is, we want to study the dynamics in the boundary of these kind of sets. So the dynamics in the boundary and the relationship between the topology of the boundary and the dynamical restrictions that you can have because of this. The idea is that the boundary of an open set of this kind is kind of like a one-dimensional object in a way. But on the other hand, as I will show later, it can be really bad topologically. So it could contain very complicated dynamics. So there is an interplay between the topology and the dynamics. The motivation for this problem comes in part from some problems in CR generic dynamics of area-preserving defiomorphism. For instance, when you have, I have a picture here. The typical picture that you see when you take an area-preserving defiomorphism is something like this. You have these invariant islands, which are produced by KAM phenomenon. You have elliptic periodic points that create a bunch of invariant circles. And then you have, in the complement of these islands, you have this region, which is called the instability region. Some region where the dynamics is chaotic in some sense, or supposedly at least. So the question, one of the problems was to try to understand better this kind of picture in the general setting. Because locally, when you have this kind of phenomenon, the elliptic islands, they are very well understood from the local point of view. It's KAM theory. But for instance, if you take bigger and bigger invariant circles here and you take the biggest one, the maximum invariant circle containing this point here, the topology of the boundary of this circle, well, we don't know much about this, the dynamics as well. Usually you know that near this point you have invariant circles with the rational rotation number and a bunch of things. But here, in the maximal invariant disk, we don't know very well what can happen. We don't even know a priori if this disk can be very large and bounded in a way. So this was one of the motivations. Well, and there are other related questions. For instance, when you take the stable manifold of a hyperbolic periodic point and you look at its closure, you have a compact connected set which is invariant. And then the connected components of its complement, well, they are open. If the maps are preserving, they are open periodic sets. And these sets have a nice boundary in the sense that I was saying there. So you can try to, if you want to understand the closure of the stable manifold of a periodic point, it makes sense to study the boundary of the complementary invariant sets. So that was also one of them. Okay, so let's consider the model case which is when you have this simply connected invariant set and its boundary is a circle. So it's a disk, my set. It's a closed disk. So in that case, everything is nice. We have dynamics, the boundary is a circle. So the dynamics in the boundary, well, you can define the rotation number and you can say as everybody knows that if the rotation number is irrational, then you have no periodic points in the boundary and you have a semi-conjugation to a rigid rotation on the circle and you have a unique minimal set, which is either the whole boundary or a counter set. In the case of the rotation number, it's a rational p over q. And then you know that, well, there are fixed points for fq. In fact, all periodic points are fixed points of fq. Moreover, all non-wandery points in the boundary are periodic points of fq, right? And in particular, this implies that when you take any point in the boundary, if you iterate it, it converges towards a periodic orbit of pq. In the future and in the past. So the dynamics in the boundary is very, very simple in the rational case and the irrational case is basically very similar to the rotation. So the question is, can we reproduce this for a general boundary when it's not a circle? And here, the point is that the boundary can be really complicated whatever I was saying at the beginning. So here is just a simple example. This is not so complicated, but it's already not a circle. This is not locally connected. But then you can have something like this, which, okay, it's still not so complicated, but a bit more, or something like this. So the open set here is the region, the complement of this spiral, right? So now all the points in this outer circle, they are very bad because you can't go towards them with an arc from the open set. But then you can have... The outer circle is part of the boundary of you, yes. It's just an example anyway. But here is a much better example, which is called a canister continuum. I think it's nowhere locally connected. And this can be the boundary of a disc. I mean, in this case, the disc would be the disc containing infinity in the plane. So it's the boundary of this disc. And it's very far from being a circle now. Now this really, I can't... Because this one, well, you can say, okay, but there's some circle there. But here, it's really nothing to do with a circle. And then you can have things like this. This is the water legs continuum. The continuum is the black part. The blue parts are one, two, three. Three invariant discs. Not invariant, there is no dynamics. There are three discs, topological discs, which they share a common boundary, which is the black part. So the boundary of any one of these three discs is the same set. And this set is, again, nowhere locally connected. It's really bad. So if you look at the complement of this set, there are three discs plus the unbounded disc. This set, in particular, appears very often just like this one appears. This one appears or something very similar when you have a homo-clinic intersection. You take the unstable manifold and you look where it accumulates, you get something like this. Here, this appears when you have something like a plucking attractor, a hyperbolic attractor. You always get pictures like this. If you are on the torus, you can have something like this where the open set is the black thing, the dark thing, and you have a virus accumulating on something which locally looks like a counter-set times an interval, also very far from being a circle. So that's the point. I mean, the point is you can't just say, OK, I will define a rotation number and reproduce what we do in the circle because the boundary can be very different. Here's one of the worst things that may happen in a way, which is the pseudo-circle, which is a compact connected set, which is the boundary of a disk, but it's not only nowhere locally connected, it doesn't contain any arcs. So whenever you have a continuous function from the interval to this set, it has to be constant. So it's really bad from the topological side locally, at least. When you zoom in, you always see things like this. You zoom in more. You see things like this. There's nowhere. You can't find an interval in there. OK. So what do we do? How do we define? Sorry? All of this. So this one, the one of the legs, it's a plucking attractor. So it appears as a hyperbolic attractor. This one also appears dynamically and this one can also be realized. There is a construction by Handel and also an example of Michel Hermann, where they do it like an attractor, or even as an hyper-serving environment set. Which, this thing? This one. The basin of a... Yeah, why not? Oh, yeah. You mean instead of three? Yeah. As many as you want. OK. The interesting thing about this kind of object is that they appear robustly because they appear as hyperbolic attractors, for instance. So they are not like pathologies. They are something that you can't avoid. And in any degree of smoothness. Sorry? This one? Yeah. It's an open disc. Yeah, it's an open disc. No, no, but this is the boundary of the disc. In the boundary of the disc, you cannot find any arc contained there. Any image of an interval. OK, so how do we define a rotation number? So the trick is to use a compactification which replaces the boundary. So you take the open disc and the boundary may be very bad, but you can just replace it by an artificial boundary which is nice, which is a circle. And if you do it right, the dynamics extends to this compactification and then you have a circle of a morphism and you can define a rotation number. And then you can try to reproduce the Poincaré theory on the circle to this new setting. So the way of doing this is using Caratt d'Odori's primance compactification. And it was done the first time I've seen it done dynamically. It's by Cartwright and Litterwood in the 50s. And so you can, if you do this, I will explain it here. So what you do is, so the idea is instead of looking at the boundary, you look at the directions. How can you approach the boundary from within the open set? Look at directions. You order these directions and you get something like a circle. And you put a topology there, that's roughly the idea. So there are two ways of defining this formally. There is a dirty way which is just using the uniformization of the disc. You just take the disc which can be very complicated and you have the nice disc. And you have a map here which is a Riemann map. So you have a dynamics here. And if you conjugate by this map, you get a dynamics here. And this dynamics, you can show that always, as long as this map is defined in the closure of you in this bad boundary, you can always extend this map to the circle here using a conjugation. So it's just a change of coordinates. So this gives you the nice boundary. Just forget about whatever is outside of you. And you look at it on this coordinate and you look at the closure. That's it. And the map extends. Here, yes. Yes, the primates will end up corresponding to these points here. But this is absolutely artificial. I am defining primates in this way. But there is a more topological way of defining the primates which is usually more useful. Just a remark is that of course this map doesn't extend to the closure here because this boundary here is maybe not locally connected and this is locally connected. But the point is that the dynamics does extend to the closure. So the topological way of defining primates, which is the more useful one, is to do it, I will choose a marked point here just to simplify. I mean, we don't usually do this, but it's easier if I do this. I just fix a point here. And I will define some things. So a cross cut of U is just an arc joining two points of the boundary of U which is otherwise completely inside of U. Just the end points are in the boundary and the arc is inside of U. So this is a cross cut. And then a cross section is one of the halves. This thing divides U into two halves. So I will denote by D of alpha the cross section which is the one not containing P. That's why I chose P just to have a reference. So this is the cross section D of alpha. And then you can order these cross sections. If I have another cross cut here and I have D, this is alpha prime, D of alpha prime, then you say that alpha prime is smaller than alpha because D of alpha prime is contained in D of alpha. So this gives you an order between cross cuts by looking at the corresponding cross sections. A partial order. So the nice thing is that of course cross cuts are mapped to cross cuts by the dynamics. Everything is nice. And so to define a prime main what we do is we look at nested sequences of cross cuts or cross sections. So you just take a cross cut here, a smaller one here, a smaller one here and so on. And you look at the corresponding cross sections. And you require that the diameter of the corresponding cross cuts goes to zero. And you look at this chain of cross cuts, modulus and relation because you will have many ways of representing the same object. I will show a picture soon. But the relation is the following. I will say that two chains of cross cuts are equivalent if they have the property that whenever I fix one element of one of the chains, the elements of the second chain are eventually inside of this one and vice versa. They go towards the same direction in some sense. So in a picture here, so this is a chain and this is an equivalent chain. They define the same thing. But of course this is a part where the boundary is nice. Here is a part where the boundary is not so nice. But here is a chain and this defines a prime. The diameter goes to zero, so it defines a prime end. But you can see here that prime ends don't correspond to points of the boundary necessarily because here what would be the point of the boundary corresponding to this chain. You have a whole segment here that you are accumulating in. And there are more complicated things but I can't show nice pictures of them. So this we have to do. So the point is you define the set of prime ends as the set of all chains, decreasing chains of cross cuts, modulo these equivalence relations. And you can show that this is a circle if you put the right topology there. So that's the topological way of finding and it's equivalent to this one. Okay. But what do you mean? What are the prime ends? It's a circle. I can't show you a prime end. So you just draw a sequence of things with diameter going to zero. No it's really, I mean there are no nice pictures that I can show you too. Maybe, maybe, okay let's, you mean here. Well here I can show you, here you have some channels. You see for instance the darker circle it has like a tongue that goes and spirals and accumulates everywhere. So this tongue it will correspond to one prime end in a way because you can choose a sequence of increasingly smaller cross cuts which goes away towards the tongue and then this defines one prime end. But this is an easy way. Here it's harder to find an image of what a prime end is. Okay. No, no, no it's just a whole meal. Yeah, I mean even if the dynamics here is smooth everything is zero here. But even, yes, yes, it's a whole meal. Yeah, in fact there's a nice thing that here if the dynamics is holomorphic then here it's holomorphic as well. That's a nice thing about the, but the diameter of alpha is that goes to zero. If you look at the cross section it can be really huge. It can be that for all the cross cuts, the corresponding cross sections are huge and you can't avoid it. The intersection of the cross sections. Well it will be empty but if you take the closures and you intersect it could be that you always get the whole boundary for instance. This is what happens in the pseudo-circle. So it's really, it's really bad. Yeah. So the problem is now that we defined the prime ends. So you have a dynamics on the prime ends because f maps sequences of cross cuts to sequences of cross cuts everything is nice. So you can do the dynamics in the prime ends. You can define a rotation number there. And the question is now how do we relate this rotation number with the dynamics in the boundary? The real boundary, not in the prime ends. So for instance the first basic question that you can ask is if the rotation number in the prime ends is zero then do you get a fixed point in the boundary? And well in general the answer is no. Here's a picture. So the disk is something that spirals towards another disk here in the future and in the past and the dynamics flows from one to the other. That's all. It's just like a translation but this is a rotating disk. This is a rotating disk and here the dynamics flows from one to the other. My disk here is this smiley thing. So what happens is the boundary of view doesn't contain any fixed points because this boundary side flows towards there, this flows towards there. Here everything rotates, here everything rotates. In the boundary of view there are no fixed points. But if you look at the prime ends, the dynamics in the prime ends is like this. The circle of prime ends is the dynamics. It's a north pole, south pole. And you can clearly see this because if you take a cross cut here its image goes inwards. So this must happen also in the circle of prime ends. So if you take a cross cut here it goes inwards. So everything flows from one side to the other. So in the prime ends you have fixed points. So the rotation number is zero. The prime end rotation number is zero. But in the boundary you don't have fixed points. So this is bad. But the observation here is that you have this situation where you have a cross cut which is mapped strictly inside, inwards. So it's kind of like an attracting region. So I will call this a trapping cross cut. When you have a cross cut which has the property that its image goes inwards. Either to the future or to the past. So what happens here is that you don't have a fixed point but you have a trapping cross cut. And in fact, Carter and Glitterwood in that paper I mentioned before they showed that it's actually a very simple argument that if you have a zero rotation number in the prime ends then either you have a fixed point in the boundary or you have a trapping cross cut. So the only way it can fail is that you have this situation where something is attracting things towards the boundary. And in particular, if you know that f is area preserving for instance you can't have this. You can't have trapping cross cuts if it's area preserving. So in the area preserving setting, things are nice. Zero prime end rotation number implies fixed point. Okay, so... Yeah? Sorry. Okay, so in your example... Yeah? Yes. So zero implies in the boundary there is a very unregional reason there is not a full boundary. So yeah, so the... One thing that we did with Rata here is we showed that precisely this is the only bad situation that you can have. If you have zero rotation number and you don't have a fixed point in the boundary then you must have some kind of sets which are attractors and repellers and they are rotating in a way and the boundary of you is contained in the basings of these things. So it's basically like this. Like both dynamically and topologically is very restrictive. So it's really an exception in this kind of situation. But okay. Was that the question? Yeah. Okay. So what about the converse? The point here, the point to remember is that in the area preserving setting things work nicely. So the converse... By the way, I don't know if I was clear enough at the beginning but I'm always assuming that you is on the plane and is a bounded invariant set by a home of the plane. I'm just simplifying. I don't know if I actually mentioned this but it was written there. So the converse is if the rotation number is not zero can we say that there are no fixed points? This is the other natural question because this relates to the question of whether when the rotation number is irrational you have no periodic points for instance. So this is much more difficult and in fact in general the answer is no but it's trickier to make examples. You have this example by Walker which is like you have some sort of Denjois example here with hairs spiraling towards an outer circle and in the outer circle you have everything is fixed point-wise. So these hairs are rotating with the combinatorics of an irrational rotation but the outer circle where they accumulate is all fixed points. When you look at the primates of these you will see the Denjois dynamics. So you have an irrational primate rotation number but you have a bunch of fixed points which are in the boundary of U. U is of course the white part here. So this is Val. They are not accessible, right? You can't reach them by means of an arc containing U. Yeah. In fact you can't... Yeah, I'm not sure if you can make an example where I don't think you can make an example with accessible fixed points. You can't. You can't. But there are other examples which are in fact in some sense more interesting. There's a bulk of attractor which is something where in the annulus you have an attracting set which contains like the annulus goes inwards and you have points there which rotate with different velocities in the annulus like the rotation along the... So this one for instance a fixed point and a fixed point which rotates one. And then when you look at the attractor you get something which is really topologically complicated. It's nowhere locally connected and so on. But the dynamics contains all sorts of periodic points of all different periods and different velocities and so on. So it's a very... And this appears as the boundary of a... It's a basing of attractor in a sense. So in this kind of example you get... You can get non-zero rotation number and fixed points and so on. The point is that all these examples they have the property that they are attracting. Like this example you can't... If you try to build it you end up getting... It's going to be an attractor basically. The boundary is going to be attracting. So again if you try to make it a preserving you can't. And the first... The first work we did with... Patrice and Maison was to prove that in fact this is the case that you can't do it unless you have some sort. It's a similar thing to the result I mentioned before. Either you have... You realize the zero rotation number of fixed points or you have a trapping cross cut. Well here it's similar. If you have a rotation number then either you have no fixed points in the boundary or you have boundary traps. I will define them just in a second. But it's something similar to what I said before. And in particular it's something that can't happen if the map is area preserving. So as a corollary if the map is area preserving when the rotation number is zero or more generally when it's rational you have a fixed point or a periodic point and when it's not you don't. So it's really... It's exactly as in the circle this time. So again the area preserving setting is... Everything is nice. So what is a boundary trap? So it's similar to the notion of trapping cross cut but of course you can't have a cross cut that is mapped inwards if the rotation number is not zero. Because of course if you have something here that is mapped inwards then you will have a fixed point in the boundary circle. But then... Boundary trap is just a union of cross cuts such that the region they bound is mapped strictly inwards either to the future or to the past. Basically it's that with some technicalities there. That's... In particular of course this is not compatible with being area preserving because it means that you are attracting something towards the boundary. You can't be non-wandering. This thing could be wandering, yes. Or not. But the point is it implies that arbitrarily close to the boundary there are wandering open sets for instance. So it's very... It's kind of restrictive in a way. So we say that the map satisfies the boundary condition in you if there are no boundary traps for any power of the map. In particular if f is area preserving it has the boundary condition. Or if it's non-wandering or something like this it always has the boundary condition. And another case... The nice thing about this is that the boundary condition also holds even if you don't assume any kind of area preservation or anything like this but if you assume that the map is for instance holomorphic as I was mentioning before and the rotation number is irrational then you automatically have this condition by completely different reasons. So it's not necessarily a condition that is guaranteed by recurrence or area preservation it could come from something. So the point is putting things together is that if you assume the boundary condition then the... Sorry? If you have a holomorphic map with an invariant disk it has to be invertible Yes? Yeah, but it was happening and we have a single disk So putting things together you have the nice picture for the area preserving setting for instance. You have that if the rotation number is rational you have a periodic point only of period Q exactly as in the previous case and if it's irrational then you have no periodic points. But of course in the circle we have a lot of extra information we know in the irrational case you have a semi-conjugation in the rational case you have this thing about all orbits converging to the dynamics being very simple and so you can ask whether the same things hold assuming this boundary condition or area preservation and in general you don't in the irrational case you can't expect to have a semi-conjugation to an irrational rotation for instance the pseudo the pseudo-circle the examples with the pseudo-circle with the irrational rotation number the hundreds example in particular it's not semi-conjugated to an irrational rotation but but in some cases you can with some topological condition which is a bit strong but not so strong you can you can get the semi-conjugation so some partial result in this direction was done by Tobias Jagger and myself but I want to get into this then you can ask whether when the rotation number is rational you know that the dynamics in the boundary is trivial in the sense that everything flows from periodic points to periodic points no no even in the holomorphic case you don't have that you don't have a semi-conjugation in general you don't if you do you mean are you talking about this thing this has nothing to do with being holomorphic it's a topological condition in the boundary semi-conjugation no even if it's holomorphic no in general no if you know something about the topology of the boundary you can sometimes show the semi-conjugation but even in the holomorphic case you don't have a semi-conjugation so the second question is whether when the rotation number is rational you have these trivial dynamics and this is nicer because the answer is yes and this is the second part of the work with my salmon patrice is basically showing that in the rational case everything behaves exactly as in the circle there's no exception if we assume the boundary condition it's a fixed point so let me state a theorem which is basically the important part that implies everything is that if the rotation number is zero assuming the boundary condition then every no wondering point of the map restricted to the boundary is a fixed point so if you know this in particular you know that every point of the boundary from fixed points to fixed points because the omega limit of an orbit is contained when there is set so it has to be all fixed point so the boundary dynamics is trivial in fact so that's the main theorem basically but actually the theorem is a the proof of the theorem goes through some topological result which is actually the sense of the proof which is a bit tricky to state but I will try to explain the the basically it says that so you have this open set on the plane it's bounded but it's complicated I don't know and then you have an unbounded component here that I called U infinity U infinity is the unbounded component of the complement and you have U and then what the lemma is saying is that if the rotation number in U is 0 and the map is has the boundary condition for instance if it's reperserving then what happens is that the boundary of U infinity is either all fixed points just the identity basically or U is compactly generated so compactly generated means that if you remove a point from U here and you look at the point of infinity you are on an annulus you can go to the you can leave this to the universal cover and look at the boundary of U in the universal cover of this annulus and you will see something like this I don't know this is the lift of U and this is the lift of the boundary and in general the lift of the boundary can be disconnected even though it's connected here when you lift it it can be disconnected for instance in the spiral example this example like this when you remove a point here and lift it to universal cover you will see a line which is this circle here and a bunch of lines accumulating like this this is the lifted boundary of this thing so it's not connected it has parts which go a bunch of lines so being compactly generated means that this doesn't happen it means that you can find here on the lift a compact set in the boundary that when you project it you get the whole boundary here you can't sorry a compact connected set here if you take a compact set that projects to the whole boundary it's not going to be connected so this is technical but it's really a strong thing to say that the boundary is compactly generated and it's actually the sense of why the other result holds because basically what you do is you use you take a point which is no wandering in the boundary and you use some recurrence properties to show that the boundary has to kind of twist and spiral a lot then it can't be compactly generated it's like this so basically the topological part is the important one okay and for instance a corollary of this topological result which I find really nice is that if you take another preserving map which leaves invariant a pseudo-circle like the Handel example if you have a fixed point in this pseudo-circle then the whole pseudo-circle is the identity the dynamics of the pseudo-circle is identity but to me that is surprising because in fact one thing is important is that it's not like the topology of the pseudo-circle implies that the dynamics has to be trivial because the Handel's example has a non-trivial dynamics it has an irrational rotation number and it's not semi-conjugate to a rotation so it's kind of complicated in a way but if you have a fixed, something is fixed then the dynamics has to be the identity so just to mention another consequence of this last result of the dynamical result one of the one of the motivations for our first work was previous work of Mater where he showed that if you take a diffeomorphism on a surface a CR-generic diffeomorphism which is that of preserving the prime and rotation number that disk is always irrational this is not a dynamical result because it's talking about the prime and rotation number there is no relationship a priori between the prime and rotation number and the dynamics of the map so it was kind of an abstract result in a way but then of course putting it together with the results I mentioned you get the dynamical information which will tell you for instance that in the boundary of this disk there are no periodic points if you take diffeo if you take an invariant disk and you look at the boundary there are no periodic points and the proof of this which is really useful for some things it relies on the following generic conditions you need to assume that the fixed points are all hyperbolic or elliptic and that there are no saddle connections these two are kind of standard conditions that you can require but then there is another condition which is that the elliptic periodic points have to be generic in a way which is either they are surrounded by invariant circles with the rational rotation number or they are surrounded by like homoclinic intersections of a periodic point which go around the point any of these two conditions works but both of the conditions are much harder to guarantee than the other ones in a way so the nice thing about the second result the one that I just mentioned is that you can get a more precise version of these which basically doesn't require any generic conditions you can explicitly state the conditions take another preserving map at least in the neighborhood of the boundary of an invariant disc so I have an invariant disc by another preserving map which is C1 and suppose you have a fixed point here this fixed point has to have real positive eigenvalues in particular it can't be elliptic you can't have elliptic points in the boundary of a disc and this maybe it sounds obvious because elliptic points rotate or something but it's not really obvious and the point is I am not assuming any generic condition the main reason to assume this condition here or this condition these things was to remove the elliptic points from the boundary but the point is you don't need any condition they can't be in the boundary this is not a generic thing and the proof is really easy I will I will explain why this is true so if you have if you have a point here of course I want to prove that there is a positive eigenvalue if there is a positive eigenvalue it can't be elliptic because by the way it could be that the eigenvalue is 1 this could happen but it can't be models 1 different from 1 so suppose this has non-real eigenvalues you can blow it up you can blow up the point to a disk and you will have some dynamics here which is defined by the derivative of the map and the open set here will accumulate on this circle in some way it depends on how the disk was accumulating in the point but it will accumulate at least at some points of this circle and so when you look at the boundary of view intersected with this new circle blown up you will get something which is an invariant set in this circle in particular you will get a recurrent point there there is a recurrent point in this circle right but the rotation number in U is 0 I'm assuming that sorry I'm not assuming that but okay I don't need to assume it because I had a fixed point here so the previous theorems tell me if you have a fixed point the rotation number must be 0 so the return number is 0 so the thing is you have a recurrent point here and there was this previous theorem which says every no wandering point in the boundary is fixed so what happens is that any recurrent point here in this circle which is in the boundary of view must be fixed and there is at least one so there is a fixed point there but this was the blown up dynamics so the dynamics in this circle is the projectivized the normalized dynamics of the derivative so you have a fixed point there so this means that there is an eigenvalue an eigenspace so you have a real eigenvalue and so so that's the proof it's really straightforward so as a consequence if you assume that I mean because remember that's why we have more general hypotheses you don't need area preserving you just need the boundary condition and because the map was area preserving here even if you blow up and you change things the inside of you you didn't change much it still preserves a measure which is positive on open sets in you so it has the boundary condition boundary yeah that is the reason that we do these more technical statements because when you try to make surgeries and modify things you lose the area of preservation sometimes or you lose the general condition but in technical conditions you can usually keep so as a consequence if you assume that the fixed points are all hyperbolic or elliptic when you take an invariant disk you either have no periodic points or if you do you can tell not only that the dynamics in the boundary is simple in the sense that everything goes from from periodic points to periodic points but also it's actually a union of saddle connections so it's really like you have here a saddle point wait saddle point whatever and so on a bunch of saddle connections maybe you have something like this also I don't know so this is the boundary of you but in particular in the rational case it has to be locally connected because it's an arc the union of saddle connections it's an arc and this is not a typical it's not very common to have this so it's telling you that if the rotation number is rational not only the dynamics is trivial but the topology is really very simple under this assumption that periodic points are hyperbolic or elliptic well it's fixed points here how much time do I have how much oh by the way this result this last result here was announced also by Fernando Oliveira 10 years 12 years ago but still a published but and the proof is completely different the proof he had so putting everything together we have this nice picture for the area preserving setting which is similar to the picture I showed for the circle even if the boundary of my disc is complicated I can tell that if the rotation number is irrational then you have no periodic points and here's a thing that I didn't mention but you can say something about the topology of the boundary you can say that it's annular which means it's a nested intersection of topological annular so it's some kind of topological condition and if the rotation number is rational it's the nicest situation because it's exactly like in the circle so you have fixed points in the boundary for fq every no wandering point goes from it's fixed and in particular every orbit goes from fixed points to fixed points and you have the additional thing that I mentioned that the boundary is compactly generated in the sense that I explained here unfortunately no basins are the problem because you see the counter examples I mentioned at the beginning where all the problem was that they were basins they were attracting or repelling so yeah you can make like the brick of a tractor it's a counter example to everything I'm saying and yeah and it's amazing so you can't improve this in this sense so I wanted to mention I will skip this but so the proof of these results they rely on one of the main things that we use is for the latest result is a result of Bégan-Crovisière and Leroux about the existence of maximal isotopes which is related to previous work of Patrice Le Calvé and we use this to precisely to relate what I said before that when you have a point in the boundary there is no wandering in the boundary this implies that the boundary has to be topologically complicated it has to spiral in a way to obtain this it's really tricky and we need to first choose specifically some maximal isotopes and then do some define some rotation numbers associated to the isotopes and work the way towards the topological results using some linking lemmas and so on so I will try to show I'm not going to say anything about this but I will try to show one of the ingredients of the proofs which I like that has to do with the shadows of arcs the idea is you can define the shadow if you have an arc if I have a my disk here and I have an arc let's say a loop I can look at the cross cuts defined by this arc here is one here you have some other one here alpha 3, alpha 4 when you look at the prime ends compactification of view these cross cuts will show up somehow here alpha 1 alpha 2 alpha 3 like this we have some fixed point here reference point so we define the shadow of this arc gamma the shadow of gamma is going to be a subset of the prime end circle which corresponds to the prime ends which are obstructed by gamma from P so is this interval together with this interval it's like if you put a light source here the prime ends that become dark are the shadow not exactly but the light bends yes so the basic lemma just 2 minutes is that if you take an arc like this which is this joint from its image by F so it's free by F then what happens is that the the shadow of the cross cut generated by gamma is contained in 2 intervals where the dynamics is just a translation this is really restrictive this is if we assume the boundary condition or if we assume that gamma doesn't contain any boundary traps so what this is telling you if you look at the dynamics of the prime ends for instance if the rotation number is 0 you have maybe you have some fixed points here and between the fixed points you have some dynamics like this which is translations and then what this is saying is that the shadow of any free arc which doesn't contain a boundary trap must be contained in the union of 2 intervals of this kind and they have to be intervals with different directions so this is maybe this is one and this is one so all the cross cuts must be here you can't have one here or here and this is really restrictive and this is the key to proving the topological results because when you have this there is another result which is the maximal cross cut lemma which tells you you can find certain cross cuts here which are maximally in a nice way and then what you do is you find one maximal cross cut here one maximal cross cut here and you get its image which flows this way and its image sorry I'm trying to explain the proof of this actually so I want I want because I'm out of time but the point is when you locate the shadow of their gamma you can then use this to prove something similar on the lift to universal cover and the compactly generated property what I mentioned before and with this and the linking lemmas and the maximal isotopes you get the the dynamical sorry I'm out of time so I will stop here you'll find an explication of fundamental attract for example plate and then its cross section on the Lovacevsky plane the connection the connection between the absorptors I'm not sure I don't think I've seen this for the plucking attractor I'm not really sure you can see the dowel of this the absorptors and then it's possible to find the rotation number on this absorptors some similar yes it's probably just a way of interpreting there are many ways you can do this so probably it's the same thing yes oh so it's not invertible okay she knows I don't know you have to ask Juliana or Leyva because I've never tried I mean but I know they have something yeah it's very difficult I'll be interested in realizing what's happening for a man you may happen that the impression of the big brain is the whole set and it's not that familiar but you don't know how to find the cross section okay that's a good question