 I am happy to introduce the last lecturer of this morning's sessions, Professor Marie-Claude Arnault from the University of Avignon. And she will come back to the topics we already heard about, but we can still hear much more. This is Poincaré's last geometric theory, please. Thank you and thank you for the invitation. It's true that there is some, there is a link with Barnet Bramham's talk, because we speak about the same theorem, but he has a simplistic approach of this theorem and I will speak of the dynamical aspects of the theorem. Okay. So before giving the statement, let me recall that so this, this, the article on a geometry theorem was published by Poincaré the year of his death. And at the beginning of the paper, he apologizes because he has no complete proof of the theorem, but he published it because firstly he thinks that it is true and secondly he thinks that it is very important for the possible application in celestial mechanics and I will speak about this application at the end of the talk. So before stating the theorem, let me recall what is continuous transformation of an annulus. So an annulus is a planar surface between two concentric circles and imagine that you draw on a table an annulus and you put exactly on this one annulus another annulus, a rubber annulus and you distort your rubber annulus in such a way that the rubber annulus is always in contact with the table at all its points and it's always exactly above the one annulus. So you always have the same general shape, but inside points are moving. So on the picture, you see what happens to some segments during the distortion of the annulus. You can have other distortions, for example, a rigid rotation, so it's not very distortion because everything wins together and to each such distortion you can associate continuous transformation of the annulus and in fact to each point of the annulus you associate T to each point X, T of X that is the final position after the distortion of the annulus, the final position of X. Okay, now I will use some coordinates on my annulus, so polar coordinates, the radial coordinates that is R and theta, polar coordinate, angular coordinate and the annulus is described by the equation R is between A and B and we consider a continuous transformation of the annulus and there are some conditions that were given by Poincaré for this continuous transformation and the first condition is what is called the twist condition. So when you do your distortion in fact you can notice that the two circles at the boundary of the annulus are preserved, but we ask a little more, we ask that during the whole distortion the two boundaries of the annulus moves in opposite directions and this means that one boundary will move clockwise and the other boundary will move counterclockwise and the second condition that was given by Poincaré is that the transformation preserves the area or more generally as a positive integral invariant. So for example you see the yellow domain is mapped on another domain that I hope has the same area but not the same shape. So if the two hypothesis are satisfied then Poincaré says that there are in the interior of the annulus, so not at the boundary, two points that are not modified by the transformations that are fixed. So be careful, it doesn't say that when you do your distortion the points are fixed. It just says that the initial and final position are the same for two points but maybe in between they move. Okay and you can prove that not necessarily exactly this condition but some conditions, two conditions are necessary. For example if you consider a rigid rotation then this transformation is a continuous transformation and it preserves the area but it's not a twist transformation and in fact there is no fixed point and it is easy to build an example of a transformation that is a twist transformation that doesn't preserve the area and so has no fixed point. Okay so let me comment on one point of the theorem that was a little problem. So Poincaré asserts that because of some degree argument in fact you have just to prove that there exists one fixed point because if you have one fixed point you automatically have two fixed points but this is not correct. The argument was that the sum of two degrees has to be zero but if you have one point with... I don't exactly explain now what is this degree but if you have one point with a degree minus one if you want that the sum is zero you have another point with some non-vanishing degree but in fact you can have one point with zero degree and so this argument was not correct and in fact it's very strange because well the argument is Poincaré of formula so Poincaré knew this formula so it's strange but it was not correct. So Poincaré introduced a third condition that is there is no invariant point in the rule annulus and he said if I can prove that any transformation that satisfies firstly the twist condition and secondly it has a positive integral and variant then it does not satisfy the third condition so it has one fixed point then he must have two fixed points and another way to say that is what is written here he says I will prove that any twist transformation that satisfies the first condition so there is no invariant point has no positive integral and variant so his strategy was the following one so he considers such a twist transformation with no fixed point and he tries to build a loop that wins once around the inner boundary of the annulus such that its image doesn't meet the initial loop and so you see here C is mapped for example on T of C and the domain between the inner boundary of the annulus and C is mapped onto the domain between the inner boundary of the annulus and T of C and this is a larger domain it has a larger area or a larger positive integral and variant and so there cannot be any positive integral and variant so it's done if you can find such a curve but it didn't succeed to prove the existence of such a curve in every case so just the year after the death of Poincaré Birkhoff succeeded in proving more or less the term in fact he succeeded in proving that there is one fixed point if you assume Poincaré hypothesis he used exactly the same argument for the existence of the second fixed point so this argument was not correct and at the end of his article Birkhoff speaks about Poincaré's strategy and what I explain to you is that maybe I will put the condition for if there is a ring that is mapped strictly inside or outside itself so a ring between two invariant curves then there is no positive integral and variant but it's not clear that if there is no positive integral and variant there must be a ring that is mapped on a subset of a superset of itself and that's what Birkhoff noticed and so he said that he didn't know if the strategy of Poincaré could work because Poincaré wanted to prove something that is stronger just the statement of the theorem and so that's what he did he won't I do not know whether the modification of Poincaré's theorem which resulted in the condition that an integral and variant exist is replaced by the weaker conditions that so these conditions there is no ring with a equal to i as in a boundary which is mapped on to either a proper subset or a proper superset of itself I do not know if this statement is true so 13 years after this first proof that was not completely complete then Poincaré published another proof and another statement in fact and it had an extension of Poincaré's last geometric theorem so maybe I begin to explain why it is an extension so Poincaré, excuse me, Birkhoff doesn't assume that you have an invariant annulus in fact he assumes that your initial annulus is mapped on to a distorted annulus that is between the inner circle and the curve here so here you have this annulus, a little annulus and there are other conditions I will speak about that just after and why does he ask weaker conditions in fact the motivation is to apply this theorem to what happens close to a fixed point of map that preserves for example the area and of course this happens for a lot of dynamical systems if you look what happens close to a periodic orbit for example for the three-body problem and when you are near a fixed point you can use polar coordinates and you have an invariant circle that corresponds to a radial coordinate that is zero so what you obtain for the fixed point when you use polar coordinates but you have no other invariant curve and Poincaré wanted to study what happens close to a fixed point and particularly what happens close to a elliptic fixed point and here you can have one invariant curve but no other invariant curve of course if you live in the 21st century you know that you have for example KAM theorem and you can find another invariant curve if you have good hypothesis but this was not the case when Birkoff was alive so he needed a statement with only one invariant curve okay and so this annulus, this white annulus there are some hypotheses of this annulus for example each radial half line meets the boundary of the annulus at one point but I won't be too precise okay so Birkoff assumes that there is an almost invariant annulus and he has a twist condition that is very similar to Poincaré twist condition he assumes that he has at most one fixed point so it's different from Poincaré hypothesis Poincaré assumes that there was no fixed point for the proof and he assumes that there are zero or one fixed point and the conclusion seems to be what Poincaré wanted to prove so there is a ring, so I will comment on what is a ring a little later there is some ring that has the inner circle as inner boundary which is mapped onto either a proper subset or a proper superset of itself in the proof so here the proof implies completely Poincaré theorem so you have two fixed points and in the proof they use a perturbative argument to compute some indices along some arcs, some paths and the perturbations of the map used is the composition of the initial map with some radial translation and he introduced an interesting notion of epsilon chain or delta chain epsilon chain and an epsilon chain is just so what you obtain, you take a point you take its image by T and you allow a small perturbation, radial perturbation in the upper direction and with a size less than epsilon and then you take the image, the same thing you allow a small perturbation and so on and with this ingredients he succeeded in proving the theorem but there is a small problem in fact when you read Birkhoff's article he said okay a wing is something that is between two closed curves and while the problem is that for me a closed curve is a loop something like a distorted circle but for Birkhoff this was not the case and what he called a closed curve can be something that is a little more complicated so here for example, excuse me here you have a curve that accumulates on a part of itself and in Birkhoff's statement so the one boundary of the ring can be something that is not very nice it's not just what I call a loop, what I call a closed curve and the final statement was proved by Patricia Carter in 82 and here he really proved that there is with the same hypothesis you can find a loop, a closed curve that it's mapped on another closed curve and that the curve meets its image at at most one point so you can have two curves so let me recall you have the twist condition plus at most one fixed point and you have a curve, a closed curve that it's mapped on another curve and there are two cases either they meet together at one point or they don't meet and when there is one fixed point in fact there are examples where you cannot avoid this case you cannot find two curves that is mapped on a different curve and when there is no fixed point you can prove that what Poincare wanted to prove and Poincare wanted to apply the theorem to prove the existence of periodic orbits with long periods in some classical problem the problem of the three body problems and well maybe we can ask ourselves why do we want to find some periodic orbits so the first reason in fact is that well maybe it's the only thing that we can do maybe it's the easier thing we can do but it's not the only reason because when you catch periodic orbits you can describe what happens to be a view close to the periodic orbit so for example if these periodic orbits are hyperbolic you can have what explain Rick Mockel you can have homoclinic intersection and curves and if the periodic orbits are elliptic this means that the differential the differential of the some differential is like a rotation then you can prove the existence of other periodic orbits and something more sometimes okay so Poincare wanted to study the three body problem and in his article he gives the program to prove the existence of periodic orbits with long periods but of course it's short and there is no, it's not an extended version and in fact that's Birkoff who did the job the complete job okay so you have three bodies and you want to study the motion and of course you know that we don't know how to compute the solution for three bodies we know to do the job for two bodies so we will use that and so what we, the assumption is that one of the bodies of the three bodies that I will call P, the particle has a very small mass and in the equation has a zero mass so this particle doesn't influence the two other bodies but it is influenced by the two big bodies that I call S and G I call them S and G because of Birkoff but I don't know exactly why they are S and G yes but what is the particle? so S and G, they have some well known motion and we assume that this motion are along ellipses and even along circles and we assume that the third body moves in the plane of the two circles of the big bodies and you assume that P is close to one of the two bodies the J body so you have J so I'm not sure that J is what you think because J has a very big mass and S is very small in Birkoff's article yes really so maybe I don't know astronomy very well but okay, Birkoff assumes that the mass of J is very big if you compare it with the mass of S and so this is not the modelling of the third problem Sun, Earth, Moon because the particle, the small body is close to the very big body so I don't know exactly what it is modelling and the true problem of Sun, Earth, Moon was treated by Charles Cornet but in the 20th century so to study the restricted so when one mass is zero we call this problem restricted problem and planar it because it's in a plane okay so to study the problem goes the rotating coordinate system that is attached to J and S and so you see that the motion of J and S is very simple in the rotating coordinate system because they don't move better the motion of P can become more complicated but there exist two families or simple periodic orbits for P and you can prove one family of direct orbit and one family of retrograde orbit and I will call them periodic orbits with small periods so if you imagine that S has zero mass this orbit has just circular excuse me circular periodic orbits but here S has a mass that is non-zero that is small but non-zero that's not exactly circular orbit so now let us think about the position of the particle we are in a plane so to determine the position we need two coordinates and we need two coordinates for the velocity and Wegmockel spoke about integral of the motion something that is constant along the orbit and that is called Jacobi integral and in fact it's not exactly maybe you think it's the energy the usual energy kinetic energy plus potential energy it's not that exactly something that is different so we were in a four dimensional space and we fixed a value for the Jacobi integral and now we are in a three dimensional space or manifold and what can be proved is that each level of the Jacobi integral contains exactly one periodic orbit of each family I described before one periodic orbit with small period and so Birkoff using this periodic orbit Birkoff was able to build one annulus one surface A that is an annulus not an annulus like at the beginning a planar surface between two circles and distorted annulus so it's distorted annulus and the boundary of this annulus is just the union of the two periodic orbits that are in the level of the Jacobi integral and this annulus is Poincare section so the future orbit of every point of A that is not at the boundary meet A in the first point that I denoted by T of x moreover you can extend continuously this transformation to the boundary of the annulus okay and well T is not exactly a twist map like in Poincare statement but when iterate of T so the ends return map if you iterate your map you find a twist map and you can prove that it preserves some integral positive integral and variant some area and so you can prove that there are for some N fixed point 40 N okay so you can find two other periodic orbits but in fact I didn't give this modified statement but if you assume that you have the twist condition and the preservation of the area you can prove not only that you have two fixed points but an infinity of periodic orbit with long period and so if you apply this result you find in your restricted three-body problem an infinity number of an infinite number of periodic orbit with long period let me just comment about something maybe you think that there is a direct and retrograde orbit of that is the twist condition it's not the case the twist condition comes from what happens close to the two periodic orbit and the the linear part of the dynamical system it's more subtle than that and the articles of Birkoff are in fact very long to read and very technical and let me notice that in fact you don't find periodic orbits you find periodic orbits in the rotating coordinate systems so maybe there are not periodic in the fixed okay I will stop here thank you very much for your nice talk we have time for questions or remarks if you want really to have a mapping from the annulus you need at the collisions because you go from direct heel motion to retrograde one and in fact he did that in a strange way because he introduced coordinates which were not regular at collisions I don't know what Birkoff does Poincaré does a regularization Conley does a Levy-Chevita regularization but I don't know what I don't remember what Birkoff did Birkoff used the Levy-Chevita regularization to yes yes, yes it is not the case we thank not only the last speaker but also all the speakers of this beautiful morning