 Ok, merci beaucoup. J'aimerais premièrement tanker l'organisateur et le comité scientifique pour m'inviter. Je vais parler d'un travail récent. C'est mon étudiant, T-Ball of Refurb. Ce sera sur le spectrum marclé de l'anose of geodesic flow. Je vais expliquer le problème. Nous prenons un manifold, un manifold close. J'utilise une métriques, une rémanieuse métrique, la métriques G. Nous considérons le set de close geodesique. En fait, si c'est un terrain très vieux, que si la métriques G a une courbe négative, je vais toujours écrire la courbe sectionnelle avec une courbe K, et la courbe création sera une R. Mais ça va venir plus tard. Ensuite, dans chaque classe de freo-motopie, il y a d'autres classes de close geodesique, 4G. Nous allons dénoncer par L of G, c'est-à-dire gamma of C. La classe de freo-motopie est dénoncée par C. Il y a d'autres classes de close geodesique gamma C, 4G. Et je dénoncerai par L of G, c'est la rémanieuse métrique. Donc le set de freo-motopie class, vous pouvez identifier le set de classes de conjugation des groupes fondamentaux. Nous dénoncerons par C le set de classes de freo-motopie, qui vous pouvez identifier le set de classes de conjugation des groupes fondamentaux. Qu'est-ce que le spectrum de Mark-Land? Il y a un map que je vais appeler L of G, le big L, qui va de C à R+, et qui va associer juste le lendemain de la classe de freo-motopie. Donc il y a une folclore, c'est... Je ne sais pas si c'était la question que vous avez posé avant, mais je pense que c'est en papier de Keith Burns et Katoch en 85. Donc c'était le sort de conjecture. Vous pouvez le trouver dans le papier de Burns-Katoch. C'est que si vous avez deux métriques G et G' avec une courbe négative et si vous avez le même spectrum de Mark-Land alors il existe un diffumorphisme comme que l'une métrique est reposée par l'autre, donc l'autre métrique. Donc ça pourrait dire que ce spectrum est un paramètre d'un set de classes isométriques sur une courbe négative. Donc ce conjecture a été sorti dans dimension 2. C'était un théorème par Otal en 1990. Il y a aussi prouvé par Chris Croak qui a utilisé des résultats de Feldman et Einstein qui dit que dans dimension 2 le conjecture est en fait vrai. Et en fait c'est été prouvé, c'est aussi bien connu que si vous avez le même spectrum de Mark-Land c'est équivalent à dire que les deux géodésiques sont conjugées. Donc vous pouvez aussi voir cette question si vous avez deux géodésiques qui sont conjugées est-ce que le conjugé est en fait venu par une isométrie de conjugée de géodésiques en courbe négative. Donc dans dimension 2 le problème est toujours ouvert. Donc il y a un peu de résultats. Donc il y a un résultat de Catoch en 1990 je ne m'en souviens pas. Donc le terme de Catoch qui dit que dans dimension 2 dans dimension 3 si vous savez que G&G' ou un conformal de l'un à l'autre et que c'est négativement curé alors si vous aussi savez qu'ils ont le même spectrum de Mark-Land c'est le fait que G' en ce cas il n'y a pas d'amorphismes. La preuve est très courte très clé donc dans la dimension 3 nous savons quand les deux métriques sont conformés à l'un à l'autre mais nous ne le savons pas en général. Mais il y a un autre résultat dans la dimension 3 qui est parmi Besson-Courtois et Gallo en 1985 c'est une sorte de corollerie de Besson-Courtois et Gallo qui dit que si Mg est un espace local de négative curé et si vous avez un autre manifold ... c'est le même et si G' est comme que le spectrum de Mark-Land est equal à Lg alors G est le pull-back de l'amorphisme donc ce travail en utilisant l'entropie de rigidity a actually solved the problem when one of them essentially has constant curvature or is a local symmetric space and I think as far as I know at least that's the only case where one has result on this Mark-Land spectrum rigidity so we managed to do some in any dimension actually so let me state the result ah ok maybe I should first before I said the result I should say that in fact to pose the problem you don't really need to have a negative curvature so there is a natural class of metric such that this Mark-Land spectrum still makes good sense is for another metric if the geosic flow which is another this property is also true that you have one unique close geosic in free or motor peak class in fact the conjecture can be extended to geosic flow with another flow or a metric with another geosic flow sorry where it's not negatively curved you have some example I think by maybe Donnet or I think you can construct manifold which have a bit of positive curvature but still the flow is another sorry you don't need a negative curvature to have another in general it still makes sense and there are example this I know I think it's maybe Gabriel must know that I think it's Donnet yeah so let me so we have actually several theorem so let me say the first in this case we assume that mg0 so we fix a metric g0 is either of dimension 2 and g0 has another geosic flow or dimension n bigger than 2 and g0 has another geosic flow plus non positive curvature I should say actually that in the proof of croc this also work in non positive curvature hotel proved it for strictly negative so assume mg0 is either a surface with geosof geosic flow or in higher dimension we need to add this condition then there exist ck neighborhood for some k which is explicit of g0 such that for any metric so ck neighborhood u for any metric in this neighborhood the conjecture is true so in fact the result is even new in dimension 2 because in dimension 2 in the another case this was this is a global result we close but you need some assumption in the curvature so k in our proof it's really non optimal I think with a lot of work we could get down to something maybe like 2 or 3 or something but in what we do we get something like 3n over 2 plus 8 or something like this because it comes from analytic method and I'll explain a bit of the proof but likely c2 or c3 we hope could work so that's the first one so we have another one which is a bit related to this and which is actually related to a question of romoff about a minimal feeling conjecture and I think which was posed by croc was a question posed by croc croc and d'arbicof so this is the term 2 so take mg0 to be the same mg0 as before and assume well then there exist a ck neighborhood u of g0 such that for any metric in u so in this case actually k is a bit better in this case k has to be bigger than n over 2 plus 2 or something like this comes from the proof again such that if the length spectrum of g is bigger than the length spectrum of g0 meaning that for any freeromotopy class the length of this one is bigger than the length of this one then the volume for the first metric the first metric is bigger than the volume of the metric we have fixed and if this is an equality then g isometric to g0 so this is quite similar to this feeling minimal feeling conjecture of romoff at least in the closed setting so the length the marking spectrum sort of tell you something about the volume so that's the second result so let me comment a little bit about this so as I said before this really says that locally at least the length spectrum is really like a parameter for the isometric class of anus of surface for instance so in fact in the proof we managed to get some sort of stability estimate so we can also say so let me maybe mention this we can also say at least it seems hard to obtain from the proof of hotel or crook saying that if the marked length spectrum are closed one to each other can you say that the metric are actually closed so sort of stability so this we also managed to get from this method so we can control the distance of the isometric class as long as we know of course that the metric are close enough at the beginning we can control the distance between the isometric class and the term of the of the marked length spectrum and this is a holder stability estimate so let me write it here so again mg0 is as before so what we can see that for any s small as small as you like the statement might look a bit technical but you see there is a C it depends on g0 and on s there exist an epsilon and there exist nu of s which is a big o of s as c small it's kind of explicit actually so something of the order of s such that for any g where g-g0 in the norm of g0 is in cn or ck is less than epsilon so this means as before that g is close enough from g0 in some ck norm there exist a DFAO such that c star of g minus g0 in some sobo-left norm actually negative sobo-left norm so this is a kind of weak norm c and g prime and g and g0 are bounded in some ck norm you can interpolate it will change the exponent but this estimate will give you also estimate on some ck norm if you know a priori that cn norm for g is bounded for instance so this is controlled by c time lg minus 1 I'll explain what this means in l infinity of c and here you have a power which is 1 half minus nu of s so what is l of g so this is sobo-left negative sobo-left norm and l of g is just lg over lg0 and what is 1 1 is just 1 when you evaluate on the conjugacy class of c so this quantity just control the distance between the Marklin spectrum so it says that you get a stability estimate you know the isometric class of g and g0 controlled by some power of essentially the quotient of the Marklin spectrum and the last theorem so we use this local rigidity to get a global rigidity which is not of course, it does not solve the problem but at least it says something yes psi is close to entity it's actually you can express in term of epsilon so psi essentially epsilon close to entity and the last theorem before I explain a bit of the proof is the following so if we fix let m close manifold so we can show that if you fix a positive a and if you fix a seconds where bk or positive numbers for any seconds and any a there exists at most finitely many isometric class of metric with curvature less than minus a and so or g is just the Riemann tensor and the k bound if you differentiate the curvature tensor k time with respect to to g this is bounded by bk so in fact it says that if you want if you assume that the curvature the curvature is bounded a priori by some given bound in sinfinity oh and of course I forgot with the same with this and the same marclin spectrum so if you know it says that if you know that the curvature is negative and bounded in sinfinity by some fixed thing there is at most finitely many with the same marclin spectrum in fact from the proof it's it's clear that we don't need bound in sinfinity we just need bound in cn for some n depending on the dimension but it's not so explicit so how to obtain this basically it comes from theorem 1 which is a local rigidity plus a compactness result so you need to show I'll tell a word in the end but essentially the idea is just that if you have a family of metric with bounded curvature and and a radius of injectivity bounded below uniformly there is a theorem of Hamilton which said that you can extract a subseconds which converge so a subseconds but the subseconds of course you have a subseconds of dephomorphism involved in there and once you know you have a subseconds which converge you reduce to the local rigidity result so to get something better to remove this sort of assumption one would need some result which said that if you have a family of metric with the same markline spectrum can you extract a subseconds which converge up to pulling back by some dephomorphism and this is not so clear essentially the question is how from the markline spectrum one can get geometric information on the curvature or things like this so one way one can think is to go to the spectrum of the Laplacian because we know there are relations between spectrum of Laplacian and land spectrum so if one could relate to spectrum of Laplacian there are some result compactness result like this by Osgood Philippe Sarnac or other people later in higher dimensions so there is plenty of possible things to do with this but so far we just assume this condition so let me explain the proof now I want to do the proof of this result it's very similar to the TRM1 I'll just explain TRM1 and essentially the stability estimate comes from the proof so you'll see so the proof involves several aspects so of course it's a non linear it's a local result the problem is really non linear this map LG is non linear with respect to G non, not yet but that's a good question would be nice to to have something so let me explain the idea of the proof 25 minutes or 20 minutes 30 minutes so plenty of time so of course you have this map you consider this map G give map to LG which is the map and spectrum map this map is not so good it's actually better to consider this quotient because this is going to be bounded when G is close to G0 so you fix G0 is fixed and you consider this map which I will denote by LG and this one is going to belong to L infinity of C because if 2 metric are comparable it's not hard to see that the market spectrum are going to be bounded by the quotient of the 2 metric basically so this is going to be an infinity of this space so this G belong to metric you can say in CK for instance on M and you have a gauge invariance which is the diffomorphism invariance so the idea would be to you would like to mod out by the diffomorphism group and in fact because we are working locally we can actually take those which are isotopic to identity so G is near G0 and the idea for a local result would be to try to prove an implicit function theorem you would like one would like to prove un implicit function theorem and that would make the deal of course life is not as good as you might expect so in fact this doesn't work so one cannot really prove an implicit function theorem so this fail at least we cannot prove it but I think it's impossible to prove but it's wrong so what means proving an implicit function theorem it means you have to linearize to be an isomorphism to be an isomorphism here you are working in infinite dimension so to be an isomorphism you would need the range to be close and that's the difficult part I mean so let me explain a bit what is the linearization first so in fact if I look carefully it is maybe possible that Nash-Moser type theorem might work for this but we do something a bit more by hand so the usual implicit function theorem will fail I'm a certain but you might expect a Nash-Moser type so some sort of loss of regularity so first what you can try to do is make a Taylor expansion of this map so we do a Taylor expansion this you can do by stability result so it's a lemma which essentially is the corollary of the stability for another flow which is another for Moser and in fact the version we use is the Delayave-Marcomaurien so the stability result tell you that in fact if you fix C for C fixed so you fix conjugacy class in pi1 G map to LG of C which is just a length of the geodesic in this homotopic class this is going to be a regular this is going to be C2 at least so here you need G in the space of C3 metric so C2 map on this manner space so it says that you can actually expand LG L of G is going to be L of G0 which is just one because you remember it was LG or LG0 plus the linearization times the difference plus a term of order 2 so something which is controlled by the square in C3 now ok so this is fine so now if you assume that LG is equal to LG0 you get 0 if LG equals LG0 equal 1 which is your assumption what you get is that the linearization applied to this tensor, this is a tensor here in C3 of M a tensor of order 2 symmetric tensor of order 2 this term you know that this is going to be a big O of G minus G0 to the square so now one has to compute what is this linearization sorry, this is the linearization at G0 not at G I'm doing a Taylor expansion at G0 so we get this that's the first thing which is important to notice now what is the linearization so the linearization it has been studied a lot actually since the work of Gilmin and Kasdan from the 80s it's called the X-ray transform it appears in tensor tomography and it's something which is very natural, it's just integrating functions or tensor on closed geodesic so let me explain what is DLG DLG0 it's called the geodesic X-ray transform in fact we will denote it by I2 of G0 so why 2 I'll explain a bit but 2 correspond to 2 tons source because we're working with symmetric symmetric 2 tons source so what is this I2 of G0 so it's an operator which says map C0 or C3 but let's define it for C0 of M symmetric tonsor of order 2 to L infinity of C and it's just you take a symmetric tonsor H and to a conjugacy class C you map to 1 over the length of the integral of the tonsor so how to integrate a tonsor along a closed geodesic what you do is you contract it with the derivative of the geodesic so this is an element in unit tangent bundle so that's the natural so gamma dot means just the tangent vector to the geodesic so this is called the X-ray transform for 2 tonsor it's quite instructive to view you see integrating a function along closed geodesic this makes more sense when you view the function on the unit tangent bundle not on the manifold so 2 tonsors on M they are actually quadratic function in the unit tangent bundle so you can view symmetric tonsor as a special type of function on the unit tangent bundle so there is a natural map which I denote by a symmetric 2 tonsor associate a function on SM so SM is just the unit tangent bundle and it's just you take H and you map to the function which to so X is a point on the base on M and V is the tangent vector in the fiber so V here is in TXM with norm 1 and it's just the function H X and you contract with V so it's just the natural way to view a tonsor on M as a function on SM so I define this and then I can also define the X-ray transform as an operator on SM which is just integration on function on closed geodesic view viewed as closed orbit of the geodesic flow on SM so I denote by so here there is not 2 anymore it's just the map which to C0 of SM map to L infinity of C and which to a function F associate the operator the L infinity function which to C give the integral of F along from 0 G0 gamma C F gamma C of T gamma dot C of T DT 1 over the length so here this is really the this is the closed orbit of the geodesic flow geodesic flow on SM ok and what is I2 I2 is just I of G0 compose with this pullback from tone source to function on SM ok just flow from the definition essentially so now these 2 operators will be of major importance in what we do so this is the linearization so first you want to if you want a local result you need to show that this I2 is injective that's the first thing so this is actually well known so this the study of this type of operator goes back to at least in this setting some old work of Gilmin and Kasdan in dimension 2 and then they worked I think in the 80s study of I2 so let me state a theorem about this I2 so I2 is known to be injective as long as I assume that G0 satisfy the assumption of my theorem which means either the G0 has another flow and M is a surface or it is a higher dimensional another flow but with non positive curvature so that's a theorem so if MG0 is so let's see how to state it ok, sorry of course I want to mention that the kernel of I2 contain all the lead derivative of G0 where V is a infinitive vector field on M ok, because these are variations of diffomorphism and it's an easy thing to check that if you integrate over closed orbit something which is a leaderity of the metric you get 0, so this is just the gauge invariance so I2 will not be when you look at the action on tensors it will not be injective because of this gauge due to diffomorphism but once you mod out by this you may ask if it's injective and this is what the theorem say so we will say that I2 is injective say if the kernel is really equal so it's not really injective but people call it solenoidal solenoidal injective but I'll just say injective for me injectivity of I2 means that the kernel are equal if equality we say that I2 is injective the kernel is really just the leaderity so is injective in the following case so either dimension of M I2 and G0 and OZ an OZ flow has an OZ geosic flow or dimension M is bigger than 2 G0 has an OZ geosic flow and the curvature is non positive so this case was actually proved by Paternain Hello woman 2014 and I gave a kind of a related I also have a proof in some paper in 17 which is bit related to this idea and for this case this was proved by Krok and Sharifudinov in 98 and there is also a proof by Paternain, Solenoid and woman 2014 so it said that the linearized operator is really as good as we want it's really injective or at least the kernel is just the tangent space to the diffeomorphism so now to apply sort of to say something about the non linear problem as I said one would like to get stability estimate to prove that this has close range so this is a tricky question so here is what we do so first remarque it's a classical result in differential geometry first any metric any actually let's say infinity but it works with ck ck any metric close to g0 we can decompose it it is of the form g g equal g0 sorry psi star g0 g is of the form psi star g0 plus h where h is divergence free with respect to g0 so this is the divergence operator so the divergence operator is the adjoint of the map v gives the derivative so this is what I call dg0 is this map v so it goes from vector field to two tons source and if you take the adjoint this is a divergence essentially and it's a sort of slice theorem it says that you can always pull back the given metric by default such that the rest is divergence free and in fact when you look at this what I call injective it's really saying that it's injective actually on divergence free tensor so this thing is really in fact injective on the kernel of dg0 star if you mod out by this gauge if you're not familiar with this think about this as quotient thing by the gauge so the gauge is not really a serious problem in variance locally so it says that in fact we can always assume we can always reduce to the case where g-g0 is divergence free with respect to g0 it's sort of said that it's orthogonal to the slice the orbit of the different morphism action ok so now the whole difficulty in this problem is that this I2 operator it map to L infinity function on the discrete set so it's not very convenient to work with get estimate stability estimate on this operator is a really tricky thing so what the new I think the really new trick in what we do is by bringing a new operator which kind of replace this and has much better properties and I explain quickly how we define this operator so this operator it comes from dynamics actually from long time dynamics so you see difficulties that closed orbit or kind of discrete set so it's not so nice so you would like to replace the integral over closed orbit in terms of integral over all the orbits so what we can what we do is that we we define an operator as follow a denoted pi so apriori pi it goes from infinity function on sm to the dual which are just distribution and it's defined by this condition so the pairing is just a limit when lambda goes to 0 plus so what you do you integrate over or you put a dumping and you apply to the correlation so this is just the L2 pairing here so this is called a correlation function in dynamic and you integrate so of course for this I have to assume that f is orthogonal to constant so you know that so the correlation for geosic flow in negative curvature they are going to converge to equilibrium very fast actually exponentially fast by some result of Liveranie for instance so this thing if the average of f is 1 it's actually going to be a big O of t to minus infinity in fact exponentially decaying so it means that when you let lambda go to 0 this actually makes sense but for this if you hold you need f to have regularity and in fact this is this is true also it's actually hold if f is holder for instance an f prime holder for some alpha positive so if you look at correlation with a little bit of regularity you get strong decay and you can define this operator so let me say so roofly speaking what is this pi operator it's map function to distribution so not to function but you have to think of it as pi f at a point x of course it's not it's real distribution so you cannot really make sense of it it has it's really like the integration of f along the along the geodesic of course you cannot make sense of this because it's just plus infinity but this operator sort of this by but you have to view it as a map to distribution that's it and this operator is the key to solving the problem so this operator has very nice properties this we proved I proved in some previous work it has the following property so pi it's a bounded operator from for any s positive map to h minus s m so it's almost map L2 to L2 but not quite you need you need to apply to a little bit of sobof regularity and it map to a little bit of negative sobof regularity it's killed so if x is the vector field of the flow of g, g0 if you compose pi with x it's 0 on hs and also pi x is 0 on u in hs such that x is in hs so that's so this operator map to invariant distribution actually and in fact it's wrong if you if you take the closure you get all the invariant distribution so that's the first thing and now the second thing so for this for this result it's based on some on some work of for and Jostrand and Leverani-Butterley Butterley-Leverani I really use more for Jostrand but the first work in this setting was by Butterley-Leverani and this is based on the theory of anisotropic soboyspace adapted to the dynamic and the second property which is the key one I mean this is one key one but the one which is really important is that we get stability for this operator we get nice estimates if you apply this operator to function which are symmetric tensor so you take symmetric tensor you view it as a function on sm and you look at this norm this control so as long as f is divergent 3 this control the norm in h-s-1 on m so this was also proved this follow also from my paper so we get really a stability estimate for this guy in some certain norm while for i2 it was not possible to get you see so now the question to go to the non linear problem and use the linear problem is how to relate this i2, this linearized operator to this pi operator because you see this pi is defined by a correlation thing and this is done by some sort of leaf-sictarem and in fact this is related to the talk of gyro this is for this we use some work of Lopez and Thielen on positive leaf-sictarem which is related to this subordination manier lemma manier result so there is a positive leaf-sictarem so there is some in the work of Lopez and Thielen it said that if f is a herder function on sm and the integral of f along all closed orbit is positive non negative then there exist you herder and f prime herder such that f is a co-boundary plus f prime and f prime is positive and also you have bound on f prime so f prime the norm of f prime in c beta is controlled by the norm of f in c alpha with some c depending just on the flow so now just maybe one minute to conclude how to use these things so you what you can show is that if lg is bigger than lg0 then the integral over gamma c closed geodesic of g0 of the difference that's an easy thing to do is actually positive so this difference of metric I call it h I can always assume is divergence free by what I said before h is actually divergence free with respect to g0 so I have this positivity here so I can apply this low pestule so it said that h is actually x of u plus h prime and this guy is positive and controlled in some herder norm by the norm of h so now to conclude the result we can do as follow so we know so if you know that the integral over all closed orbit is positive you can always approximate if you integrate h along the Liouville measure this is going or pi2h so here I identify function and functions on sm and st having positive integral over closed orbit imply also that the average for the Liouville measure is positive because we can approximate Liouville measure by a very long integral over closed orbit so one has this so we get some estimate like this this is bounded by actually i2 g0 of h maybe twice or something again because the average over the Liouville measure you can approximate you can bound above by some integral very long orbit and this because we assume they have the same length spectrum when you do the Taylor expansion what you get is that this is bounded by a constant depending on g0 time the square here for some ck norm and now from this I'll just finish what you get is the following so you take the h-s-1 norm it's bounded by my estimate on the pi operator here I apply this with this h it's bounded by a constant time pi i2 star h h-s hs non, sorry, h-s bounded by c' another constant time ah, yes, then I use that this operator is bounded so it's pi 2 star of h now this guy I write as xu plus h' so because pi x is 0 I actually remove this guy so I replace now h by h' and this guy is positive that's the key now because this guy is positive here I have this in hs norm now cos pi is bounded from hs to h-s now I interpolate this hs norm I can interpolate between l2 and some h big so I get h' l2 time almost 1 so 1 minus some epsilon and h' hn some very big norm n here with some norm epsilon of n so something very close to 1 just interpolation and now this is bounded here a priori so this is small bounded by delta and here I just use a holder I just use Jensen so it's just the l1 norm to 1 half time the h' time the l infinity norm time some delta here which is just the difference between the two matrix and here because I have l1 norm of h' and h' is positive this is just the integral of h' over the nuville measure to the 1 half and this h' I can write it again as xu plus h or minus h but xu it's a co-boundary so again the nuville measure is 0 so I get back h here so I bound above by h l1 1 half time this and now now if you now I can use the bound with the i2 here I get an h square here so you know I have a 1 half and a square so I come back with h to the power 1 essentially here with some ck norm and essentially you get in the end you play a bit with this you do some interpolation and what you get is that h minus 1 minus s is bounded by h minus 1 minus s maybe to some power 1 plus alpha which is positive times delta so power bigger than 1 so it's said that if h is small enough it's 0 so that's the idea sorry it's a bit technical the end but the paper is online so if you want to really see the detail just the idea to have is based on this low-pestuline positive lift sector M and this new operator where we have a good estimate sorry now pay2star is just the operator which to a tensor on M a symmetric tensor on M associate a function on SM it was just this just by contracting with V you have a tensor I heard about this problem but so it's non generically it's not known that generically this is the case I don't know maybe yeah it could be actually I haven't thought about it but it's maybe something to look at I think maybe I don't know I don't quite see how how from length of jade-zique I don't know I don't know I don't quite see how how from length of jade-zique I don't know I don't know