 شكراً جزيلاً لكم شكراً لكم الأشياء شكراً لكم سيفانو أنت مخطب سأتحدث عن تبادي الإلبتك الإكوليبريا في حاملطونيو لذا سأكون always considering a frequency in RD سأكون considering a Hamiltonian function where the HKs are polynomials of degree K so here the origin O is an equilibrium or a fixed point for the flow, the Hamiltonian flow given by the Hamiltonian equations that I will call star and I am interested in stability of O let me immediately say that the same study would also apply in most of the cases to the stability of Torai for systems that now I will write in action angle coordinates as follows so omega bar is equal to omega 1, omega D and the flow and here we have the torus T0 which is equal to TD times 0 is invariant by the flow so same question we want stability of T and now the stability is of course understood in the action variables R we have three notions of stability the first one is Lyapunov stability by Lyapunov stability it's the classical topological stability where we say that the point that starts in the neighborhood of zero remains in some neighborhood of zero we have stability in probability which is also called Km stability we will be interested in this stability also and this is when zero is density or T point for invariant quasi periodic Torai there are Lagrangian etc and the third type of stability we will be interested in is called effective stability which is a quantitative approach to stability so this is stability in probability here effective stability is a quantitative approach it is just asking for how long a solution that starts near zero remains near zero nevertheless this is the most classical one the oldest approach and it's really natural so let me just go ahead and give a definition so we call Th of R it's the maximum time such that Phi T such that for any Z less than R Phi Th of Z is less than 2R for any T less than Big T it's clear it's just saying that during the time T of R the flow doesn't move from the ball of size R outside the ball of size 2R so give upper and lower bounds on Th of R that will be the exercise in effective stability what are the ingredients when one studies stability for Hamiltonian systems there are four ingredients mainly the first one is the regularity of H is it real analytic in which case this is a power series that is convergent in some with a positive rate of convergence when you complexify X and Y the other possibility at the other extreme is considered for example symplectic maps with piecewise affine twist and this situation is completely unknown whether this will yield will lead to instability or stability we expect instability for very low regularity but not much is done so between low regularity and real analyticity there is a lot of difference what is the second ingredient the second ingredient is non degeneracy conditions on the HIs the third ingredient very important is arithmetic so what are the arithmetic of omega what are the arithmetic of the Torah we are looking for all this is important let me just remind you what is a Diophantine vector omega is said to belong to a Diophantine condition with constant gamma positive and top larger than zero if for any K in ZD not zero we have that K omega the scalar product is lower bounded by gamma over K to the power D this is Euclidean norm when omega is resonant if you have a rational relation on the omega Is it is very easy to understand that diffusion instability is very likely to happen so we will go ahead and standing assumption it is interesting to study stability in the case of resonant frequencies but we will be only interested in absence of resonancy so omega bar is non-resonant which means that if sum of K i omega i is equal to zero then all the K i's are equal to zero for here if K is in ZD this is natural because this will give will be reason for stability for instance because of averaging so before I talk about non-degeneracy I probably have to introduce Birkhoff normal forms that are the most the powerful tool to study stability so Birkhoff normal forms so we will also assume that H is at least infinity so because omega is non-resonant and H is infinity then we know that for any K in N there exist symplectic local symplectic diffusion around zero such that H composed with 5K writes in a normal form which is omega bar which is omega bar I plus plus rest and here the rest is the polynomial of degree at least K plus 1 in the variables X, Y and I is equal to I1 Id it's equal to so these are the action variables and this is the classical Birkhoff normal forms for an elliptic fixed point and we have similar Birkhoff normal forms for Torai but under just the additional condition that omega is the often time omega bar otherwise we don't have Birkhoff normal forms for Torai so BNF for Torai if omega is in DC DC is just the union of DC gamma tau over all tau and gamma so now what are the usual non degeneracy conditions that one resorts to to have stability well first of all let me say that Birkhoff normal form of order K implies that TR is lower bounded by R to the minus K right because this is stable and this is the rest and the rest starts with a polynomial of degree K plus 1 and this means we have this nevertheless this will be true only for R bigger than some RK and RK is probably very very small as K increases if you don't have any control on the arithmetic so now if you put in arithmetic so now if omega bar is in DC gamma tau then one can use the Birkhoff normal forms and find out that TR is bigger than exponential minus R to the power 1 over tau plus 1 and this for R less than some R that depends on gamma and tau and the norm of H in some in some analytic norm but this is much better than this right because it's really giving you exponential effective stability how do you obtain this if I want to do it waving hands I will tell you just to do Birkhoff normal forms perform B and F up to K of order 1 over R to the power 1 over tau why is this so because when you perform Birkhoff normal forms you pay each time with a small divisor but the small divisor will give you after you perform K times will give you K factorial to the power gamma to the power tau divided by gamma to the power K and the rest so now I'm I'm estimating the rest after I've done this Birkhoff normal forms on a ball of size R or say 2 R okay so on a ball of size 2 R the rest I would have here a 2 R to the power to the power K because H I will now assume H here H is real analytic this is important in this conclusion it's really important in this conclusion okay so you optimize here choosing this K and you get by sterling formula something like this is like E to the minus R to the minus that's more or less the this is yeah so that's more or less the reasoning and we immediately see what I said so now if you want to go beyond this simple application of Birkhoff normal forms you need non-degeneracy conditions often so the non-degeneracy conditions you have the first condition is Kolmogorov condition it's when B is non-singular in this case because B is non-singular because B is non-singular the frequency map near zero that is given by the gradient here which will be omega bar plus Bi in the first approximation in the linear approximation the frequency map is difthomorphism from a neighborhood of zero onto neighborhood of omega bar and that's why you visit all the frequencies and therefore you visit in particular all the good the often time frequencies and by KM scheme you save the story and this gives you KM stability the other condition is known as the Arnold non-degenerate iso energetic non-degeneracy condition this one is can be stated like this this is the D plus one matrix and you want this matrix zero determinant and this will give you KM stability inside each energy surface which has a meaning because energy surfaces are invariant by the flow the third and most relaxed and on the generous condition you need to you need to first accept that there exist an infinite bulk of normal form so formally formally one can define phi infinity as a power series phi infinity as a power series and H composed with phi infinity it's a symplectic the filmorphism formally is equal to NH of I so there is no rest anymore and when this application as a formal application has so Rusman condition non-degenerate condition is that NH does not live in a subspace of positive co-dimension so it's also about visiting all frequencies or visiting many frequencies when the action varies finally the last condition I will be interested in is the NH steepness condition I'm not going to write it down explicitly because if I do so some people may want to spend 10 minutes thinking about it and I would lose them so I will just say that H so suppose the Birkoff normal form here the beginning of the Birkoff normal form some function H of I is steep if for any subspace for any plane so I call plane any linear subspace of RD for any plane lambda for any path gamma inside lambda of of length xi gradient of H on gamma gradient of H projected on on lambda will become larger than xi to some power A for some point on gamma so it's saying the following you have a plane here lambda and you're moving if you move by xi along some this length and dominant I mean a distance from the point you started with okay of that goes beyond xi okay then at some point inside the path the gradient will have a projection inside the plane and what is this for it is simple to see why this is useful for diffusion because look if you want to diffuse with Hamiltonian of this kind then obviously you need to start by diffusing in a direction that is orthogonal to omega right but when you are diffusing in a direction that is orthogonal to gamma to omega because the Hamiltonian must be constant you want to diffuse in a direction orthogonal to omega and then it's telling you that the gradient you will meet the gradient inside this plane which means you need to move inside the plane also you cannot be along a sub plane in this plane and therefore you reduce one more direction of possible diffusion and you are now in D-2 possibilities because now you have another omega prime and you must be in the orthogonal of this one and this is a lower dimensional plane and you do it D times and you cannot diffuse anymore so that's the idea that's really the type of proof that is behind nekhoroshev exponential stability results so what are the conjectures and results in this in this area well first of all first of all there are only only two known cases of Liapunov stability so what are these known cases the first case case one is when all omega i's are of the same sign right because here if all the omega i's are the same sign this is a convex function the energy levels look like ellipsoids that converge to zero and they trap the dynamics so you cannot escape from the neighborhoods of zero this this is true in any dimension and the second case so when I said there are three ingredients of stability I forgot to say that there is a fourth one which is dimension and topology and it works in low degrees of freedom so for D equal to two and for H that is are known known degenerate but we have km stability inside each energy surface but each energy surface is three dimensional so the two dimensional torai separate the surface and trap the dynamics so this gives isoamgyetik km stability gives Lyapunov stability now the common sentence that we often hear and use that says that km stability in two degrees of freedom implies Lyapunov stability is misleading as this example shows I have a theorem with Masha Saprikina that says the following that says attention attention if omega bar is equal to omega one omega two and omega one omega two is strictly negative so different signs then there exist H that is c infinity as in star that is k Kolmogorov km stable but not Lyapunov stable so I think this question Stefan Omarmy asked me this question in a talk so okay so this means that there is some energy surface inside which you can still escape in most of the energy surfaces you will have the km phenomenon but not in all of them okay now beside these two cases the non-conjecture of Arnold is that for d larger than three and omega i not of same sign I could say it a little bit more I mean in a sophisticated way I would say that the quadratic part here is sign definite if the quadratic part is not sign definite then generically H as in star O is Lyapunov unstable okay and we have Dwadi in the 90s gave examples gave c infinity examples of unstable elliptic equilibria Raphael and the first paper was with with Patrice Le Calves in dimension 3 and then he generalized it and the idea here is very nice because he can diffuse no matter what is the bulk of normal form but it's only smooth let me insist on the fact that although the conjecture is stated for generic Hamiltonians not a single example is known so question is every analytic equilibrium elliptic equilibrium stable so of course the answer is expected to be wrong but no no examples are available let me insist on the fact that it is easier to prove stability for fixed points and it's easier to construct diffusion for Torah so I'm insisting on seeing it this way it seems equivalent statements but I like to say it like this and for example here I can construct a Torah we can construct a Torah that is superleuville and that is analytic and that is unstable but not for fixed points so there is a problem for fixed points at this moment and there is another conjecture in the field central conjecture also is that Hermann conjecture is that even if you don't want to consider generic conditions then all analytic elliptic equilibria that are often time km stable so this also is not at all known and let me now mention the series of recent results that is related to these questions and try to discuss them with you and ask you some questions so the first theorem I'm going to mention is the joint work with Eliason and Cricorian from Paris and it says the following if if H is analytic and NH is non is Rusman non degenerate then all is km stable I must say that our study of stability of equilibria is related to the so-called fundamental problem of dynamics of studying the stability of perturbations of completely integrable systems and in this system if H is not Rusman non degenerate then easy to to preclude km stability to produce examples without km stability okay so this is should be the optimal condition even in our our situation is actually more difficult than this one because we are in a setting where the perturbation of the completely integrable part is a singular perturbation because it depends on the equilibrium itself when you will write the you cannot separate the perturbation from the integrable part because the integrable part will be the Birkhoff normal form and it is given by the perturbation while here you can think of small H as independent of big H think about the a priori unstable diffusive examples here it also works for Torai so if omega bar is the often time then and and H is non degenerate then we can have also for Torai km stability moreover in the case of Torai moreover if NH is degenerate then T0 is accumulated by manifolds foliated with km Torai so in the real analytic situation you always have accumulation of the torus at zero by other Torai but they don't have we could not prove no positive measure for the moment for the moment so we didn't prove positive measure so we didn't prove Hermann conjecture the first statements are true even for smooth Hamiltonian but this one really needs analytic condition at least at least Hermann conjecture needs analytic condition as the following theorem shows so for any omega in rd d larger than 4 there exist h in c infinity as in star such that 5gh is such that O or T is not km stable unfortunately we could not do it for d equal to 3 we don't know what happens for d equal to 3 for d equal to 2 the celebrated last geometric theorem of Hermann tells you that even in c infinity Hamiltonians the km stability will hold not Liapunov stability but km stability for d equal to 2 so d equal to 2 h c infinity and omega is dc then km stable this is Hermann's last geometric theorem so this construction we did it for for Torai we did it with Eliason and Raphael and for equilibria we are doing it with Marsha Saprikina so here again for equilibria you need it's a little bit more subtle the construction because you must take care of what happens in zero you cannot diffuse for example with negative r as you do here so think for example of the case where all the omega i's are of the same sign so here you are saying that you will not be km stable but you will be Lyapunov stable so these in particular will be Lyapunov stable and not km stable when all the omega i's have the same sign okay my third theorem I want to talk about is a joint work with so it's not a bit of normal form but almost with bunmura and nethermann and it says the following it's a result on effective stability so it says that if h is real analytic as in star and if omega bar is geofantine then t of r is bigger than exponential of exponential of a constant times r to the power 1 over tau plus 1 and this constant is explicit you can explicit it in terms of gamma tau the norm of h the dimension etcetera so this is called so we say that o is double exponentially stable or o is sticky okay let me say just comment about this theorem well first of all it's not exactly what I said I forgot to say that one second I need an extra assumption if well sorry here I forgot to say that there exist and d of omega bar it's a set in the space of polynomials of degree 2d and of 2d variables and degree 2m and here m is just d squared over 2 plus 2 and this is nd of full the bag measure open dense okay such that such that if so let's say this is say this is a definition we say we say that o is sticky if we have this okay such that if if sum of hk so the first terms in the Taylor expansion of h for k less than 2m are in these in this set then o is sticky so let me comment on this well first of all this result was known when the h when in the brick of normal form the second term you get the quadratic term so when you do brick of normal form you have h composed with 5 is omega i plus ti bi plus blah blah if this is sign definite we say that the Hamiltonian is convex so for convex Hamiltonian morbidelli and georgilli georgilli and morbidelli had a similar statement for tori for convex okay our condition is much more general because convex is a nice set it's far from being generic the second thing I want to say is that the theorem is still okay for UV frequencies but but you need to adapt the exponential here will be some function that depends on our explicit function on omega and r that depends on the arithmetic and the other thing I want to say is that the same result holds for tori and that in in a recent work we're trying to we prove some kind of the following result for general for general h we have that most km tori of h of i plus epsilon h i theta analytic are sticky okay so this is the result that is the counterpart that is the extension of morbidelli and georgilli they prove that if h is convex then the tori here will be double exponentially stable we show that for any h for very general h the very prevalent h the km tori all of them will be double exponentially stable now the other thing I want to say is that it is true for jevre and it's true for for symplectic maps and it's the only case where nothing is known is nothing is known for leuville tori why well because because you cannot do a bit of normal forms okay now there is one question that you can ask is do you really need non degeneracy conditions and you can conjecture exactly like in herman that for any analytic equilibrium you have stickiness if it is geofantine double exponentially stable so the conjecture will be that geofantine torus is double exponentially stable it is km stable and in general it is liapunov unstable that's the full that's the I would say the goal I mean the goal of the whole theory the other question natural question is whether this double exponential bound is optimal and then it's hard to answer it because it's hard to construct diffusive analytic equilibrium even super liuville diffusive equilibrium we don't know how to do it so let alone doing diffusion and computing the diffusion speed for geofantine equilibrium that is non degenerate nevertheless we have a clue to the fact that this double exponential is optimal since we proved the following thing there exist with David Sozin from PISA there exist there exist jeuvre alpha h bar equals h of i plus epsilon h i theta so this is a small perturbation of this integrable part and h is quasi convex so quasi convex it's convex in the orthogonal direction to the gradient to omega bar and as and h as in star ok and h is quasi convex but nevertheless all km tau r are not more than sticky in the sense that we proved that for any given torus t r in the neighborhood of the torus is going to be less than exponential of exponential of 1 over r to the power 1 over alpha here we regret not to have the factor tau that should appear 1 over tau but for the moment we cannot see it but we're working on this and it's it's a little bit technical I don't have time to explain any proof I'm sorry I thought I would explain some proofs but I don't have time