 Okay, first of all, I would like to thank the organizers for the invitation. It's a pleasure to be here Again after I think last time I came here. It was something like 10 years ago Okay, I'm gonna talk about a metronome dynamics Let me introduce the base setup You have R2N with the Canon Cosme platform Given by this guy which which is exact is given the primitive is lambda is the Leoville form is lambda And you have a tip come a tonion right class come a tonion. It's a proper a metronome is Homogenous of the grid to such that the energy levels are compact a Compact the regular energy levels are compact some manifolds more precisely compact spheres so the regular energy levels we take K K is a regular Value so the previous of the pre-image the regular that the the regular the energy level is a star-shaped sphere The fact that it's homogeneous of the grid to implies that This level sigma is star-shaped is a star-shaped sphere Is a star-shaped sphere in our train what means star-shaped? It means that it's transversal to the radio vector field Okay, so you have the radio vector field here This is transversal to the radio vector field Okay, for instance in the convex case So you have these spheres which are the regular energy levels of the metronome and you want they are invariant by the metronome flow Okay, and you want to set the study dynamics of the metronome flow on sigma on these energy levels Okay, it turns out. It's easy to see that The study of this the these dynamics the the metronome flow of these star-shaped hypersurface the Dynamics is a key is equivalent to the study of the dynamics of brief flows on the standard contact sphere What it means if you take this fear the take for instance the round sphere S 2 n minus 1 in R 2 n Okay, and then you restrict lambda the levial form to this fear Okay, I take the restriction. Let me call alpha is lambda has trickled to this fear And then you take the kernel of alpha The kernel of alpha I call this psi the kernel of alpha This is what this is one dimension one distribution on the sphere. This is what we call a contact structure Okay, and now take any contact form one form on the sphere Let me call Alph beta Take any comp take any one form on the sphere such that the kernel of this one form is this psi Okay, so take this fear the round sphere and now take the restriction of the levial form to the sphere and then they take the the kernel of this of this restriction of alpha and You have this distribution is a contact distribution and now take any One form such that the kernel of this one form is psi of course lambda is such one form But you have a lot of possible one forms that satisfy this condition Okay, and now associated to this one form what this is called a contact form Okay, you have a canonical dynamical system is the rear vector field It's uniquely characterized by the following two conditions the first condition is that that make out is our beta is that the contraction the contraction of the differential of beta is Zero so our beta is in the kernel of differential of beta and the second condition is a normalization condition namely that beta apply it to the to the river to this vector field is constant equal to one This is called the rear vector field So associated to it to any contact form beta like this. That's fine this condition. You have this canonical dynamical system Okay, and the study of this Rib flows of the flows of these rear vector fields is equivalent to the study of Hamiltonian flows Like like this restricted to this energy levels Okay Is it clear? Okay, we have we want to study the dynamics of these rib flows after the Hamiltonian flows on sigma and if you want to study dynamics the dynamics or of course the Since the regular the energy levels regular you don't have singularities. So the most basic Dynamic objects are the PDH corbids So if you want to study dynamics, you look for the first thing you want to look for you want to Look for are the PDH corbids Okay, and the key point And the key point is that this periodic orbit in the Hamiltonian set up In general if you have a float find periodic orbit is something difficult, right? You don't have something to start with but in the Hamiltonian context. You have a something very useful which is the fact that Critical a periodic orbits are critical points of the action function. You have a variation of structure Behind the PDH corbids more precisely what you can you have the following. So you have this Hamiltonian And associated to this Hamiltonian you have the action functional. What is the action functional? You consider the space of Close curves with fixed period t Okay, you know to end and you associate to this space of cool of the You have a functional in the space of these curves, which is the action functional associated to the mettonian, which is given by this Is the integral of lambda lambda is the primitive of the simple form is a reveal form over the closed curve minus the integral of the mettonian Along the closed curve And it turns out it's a it's a classical fact that the PDH corbids are critical points of this function So we have as I said this variational structure to find PDH corbids Okay, so now denote by P the set of simple PDH corbids on Sigma simple means the following of course whenever you have a PDH corbids If have a PDH corbids Then you have actually infinitely many PDH corbids why because you can take the iterates of the orbit They are different orbits. Okay, but the simple means exactly that It's not it's not the iterate of some period corbids Okay, simple means it's not the iterate of some PDH corbids So they're not by P the set of simple PDH corbids on Sigma we say that Sigma is not degenerate if every PDH corbids including the iterates of simple ones are No, are not degenerate it know the general degenerate means the following means that if you take the first return map and Linearized the first it take you the linearizer punk ahead map the linearizer first return map. It does not have agent value one Okay, a pediatrics color is lipstick or stable if every agent value of its linearized punk ahead map has models one Okay under generic conditions why elliptic orbits are important because under generic conditions The existence of elliptic pediatrics orbits imply a rich dynamics. You have positive topological entropy and Well, you have a lot of you have very rich dynamics under generic conditions And they're not by P the set of simple elliptic pediatrics orbits Okay, so I have a P is a set of simple pediatrics orbits on Sigma Sigma is a regular energy level and P is a set of a simple elliptic pediatrics orbits on Sigma Okay, and the goal of this talk is to study the multiplicity and stability of pediatrics orbits on Sigma In other words, we want to get a lower bound for P for for the cardinal for the cardinality of P and P This is the go of the talk and there is a classical conjecture. I am a turn and dynamics I think it goes back to point ahead That states that That You have at least n pediatrics orbits simple pediatrics orbits on any Sigma and at least one elliptic pediatrics orbits One elliptic pediatrics orbit This is a classical conjecture This is very important here because in general dynamics you look you consider generic dynamic Well in dynamics here in this conference because most of the talks you consider generic The dynamics in under generic conditions, right? The point here is that we are not considering generic conditions We are taking any Sigma any I'm a Tonya as before Okay under generic conditions you can show You can show in fact that P you have that you have infinitely many elliptic pediatrics orbits under general under C to generic conditions So the problem is is that we are not assuming Generic conditions we are considering any star-shaped I press star-shaped sphere in our train And we want to get this lower bound for the number of simple pediatrics orbits and for the number of elliptic pediatrics orbits Any questions so far is no is no you can see actually what we what is known as the following See infinity generically you have infinitely many pediatrics orbits and see to Generically you have infinitely many elliptic pediatrics orbits But if you don't assume generic conditions, you do have examples with finally many pediatrics orbits and The easiest example is given by an irrational ellipsoid in our train It carries precisely any pediatrics orbits a little irrational ellipsoid is the following to take an ellipsoid take coordinates you You identify R2n with Cn take complex coordinates Z1 Zn and then a rational ellipsoid is Something given by this Where are E square these are our I square are rationally independent And these guys are rationally independent Then you take sigma given by this equation Then it's easy to see that you have precisely any close orbits Okay, the Hamiltonian force completely integrable is foliated by invariant or I and you have precisely any pediatrics orbits And it's known to generate Of course you can ask why why why why if they are rationally dependent then you have infinitely many Dependently have infinitely many yes For instance well for instance if it's one for the case of the wrong sphere all the orbits are closed But in general when they are rationally dependent you have infinitely many Of course you can ask why the low why you have this lower bound N Something magic why why where it comes from right right and pediatrics orbit has a topological meaning Which is the following? It turns out that you know so you say you have this standard contact sphere, right? This you have this sphere with this contact structure There is a general construction in simple geometry called the pre quantization bundle, which is the following I can explain you it's not so so important in this talk, but it can explain you in few minutes If you take it's a way to construct contact manifolds from simplatic manifolds in the following You take some simplatic manifold be omega And you assume that the carmology class of omega is an is an integral carmology class Okay, so you assume that the carmology class belongs to aid to be Z Okay, under this under this Assumption you can construct an S1 bundle over B. There is an S1 bundle over B Okay, which is a whose other class is the carmology class of omega and it's a contact It's a contact manifold Okay, and it turns out that the standard contact sphere this is called the pre quantization bundle of B the circle pre quantization bundle of B and it turns out that the sphere is The pre quantization bundle the sphere s 20 minus one is the pre quantization bundle of Cp any minus one Okay, is the this is the circle bond in this in this case. It's just a hop vibration Okay, and N is a total rank of Cp any minus of the homology of Cp any minus one This is the reason why you have this low the topological reason why you have this lower about N For the standard contact sphere of dimension of dimension 20 minus one in general You what what you expect is that so you have this general construction In general what you expect is that whenever you have this pre quantization bundle Then you have at least our Pediatric orbit simple pediatric orbits where are is the total rank of B with rational coefficients And at least one elliptic pediatrics one elliptic pediatrics. This is what you expect in general Okay, but unfortunately Sorry, it's far from being known. It's a very hard problem Even for the standard contact sphere. It's far from being known. Yes What so again no no no no actually you we expect that for the sphere you have at least any elliptic pediatrics There's a good point But it's a harder problem. It's a harder problem. Yeah, but we expect that you have at least actually we actually expect that if you have Finely many then all of the pediatrics orbits are elliptic like in the in the way Like in the rational ellipsoid in the rational ellipsoid they have precisely any pediatrics orbits and all of them are elliptic Okay, so let me give you a state of art of this problem a Survey of No results Now first support without any condition on Sigma Sigma just a star-shaped sphere Well without any condition the first result was obtained by Habinovitz in 1978 he proved that you have at least one pediatrics orbit and It was really not trivial because of the fall the the breakthrough of Habinovitz was a falling fact Well, we know that the closer orbits are a critical point of dysfunctional but this functional So it's like in the case of your desk flows right for when you want to find close your desk You take the you take the energy functional and you want to find a critical point of the energy functional In the case of the Jodesk flow is easy to find this Okay, essentially easy right because of why why because the energy functional in the case of the Jodesk flow is Is well behaved it satisfies palace made it's bottom from below The Morse index of the critical points is The Morse index are finite so you can you can use the classical Morse theory, but unfortunately this functional Is very ill behaved Okay, it's not bound from below the Morse index are typically infinity So you cannot use classical Morse theory like you do in for Jodesk flows So Habinovitz was the first one that showed that you can do you can use this Functional to find pediatrics orbits, but in a completely non-trivial way Using what the so-called mountain pass lemma It was the first one that showed that this guy is really useful Without assuming any condition on sigma That's the point more recently Christopher Gagini and hot chains and And then in a joint work with Victor Gainsville Doris high and umbata in evics We prove independently that for this fee that's three you have at least two and You long also prove this result using a result that in in our paper in this paper with Gainsburg high in in the abyss Okay, so for S3 Without any condition you have at least two close orbits When Sigmund is known to generate Namely that you remember that the linearized book a map does not have does not have agent value one Then it's easy to see that you have at least two Pediatric orbits in any dimension Okay, but you see in general the difficult cases that is that is the den is the the generate case And for the elliptic orbit no general no general result is no We don't know if you have at least one elliptic orbit in the general situation You just don't know without assuming a some generic condition. Okay See yes The point is that that's the point right because of course you can approximate But you you don't have control of the time of the pure of the of the orbit You have no control because you find this you have you find this periodic orbits using Sometimes you prove this using contradict an argument by contradiction So there is no hope to to control the the peel of the orbits to get something the limits in general Okay, now assume now, so now let's assume something on the on the on sigma suppose that segments convex What what does it mean? It means that it bounds a strict convex domain Okay, so it's more it's something more restrictive. So in in general you have a star-shaped Upper surface right but now suppose you have something convex Okay Then in this situation The the results are better The first result is due to echelon hofer They prove that in any dimension you have at least two periodic orbits two simple periodic orbits in 1987 And then a very important very remarkable results remarkable work of long zoo of 2002 They prove that you have at least the flour function of n over two plus one periodic orbits Wang which was a he was a student of Long improved this bounds when n is odd Getting this the sailing function of n over two plus one Okay, so improved by one when n is odd Okay, but you see any this is a very hard question. So if you improve this by one periodic orbit, it's a very good result This is very it's in principle. It says, oh come on. It's just one periodic orbit But you're not assuming this is generally you just assume that's convex. So any improvement just by one periodic orbit is very important So you will get this improvement When by one when n is odd and so you see when n is four The the previews the previous in a quality gives you three periodic orbits And Wang later in another paper published in the semi-year, but He improved this this this result in dimension when n is four Okay, getting the expected lower bound Okay, good question There are two two main reasons first of all The variational methods in the convex case in the convex case are the variational methods are easier because you see this Functional in general is as I said is you behave But in the convex case you can replace this function by some dual functional, which is much more tricktable It's much much much is much better Okay, and the critical points of this new functional correspond to the critical points of this Correspond to periodic orbits and the other thing is that convexity implies some conditions on the index of the periodic orbits Which are very useful to get periodic orbits These are the two main reasons why the convex case is Easier or less hard than the general case Well, see Convexity is a c2 open condition c2 open condition No, it's strict convexity Yes, I mean strictly convexity But that's that's the the points the variational methods and the the index of the periodic orbits And in the non-degenerate case in the convex case you can show that the conjecture is true. You have at least any close orbits But in the non-degenerate case, as I said the important the important case is the The generate case is the hardest case And what about elliptic orbits? They prove that if you have finally many periodic orbits, then you have at least one elliptic orbit In other words, if you have if you don't have elliptic orbits Then you must have infinitely many orbits It's not a good result, but that's what they prove Okay, now, okay, so we started with star shape it It's something more Something general and then we restricted to to to the convex case And now we impose another restriction. Namely We suppose that sigma is invariant by the tip of the map So it's convex and Symmetric in the sense that it's invariant by the tip of the map Then It turns out the result that the results are much much better You can prove that you have at least any close orbits Without any only assume that the symmetric and and and and and uh convex Without any non-degenerate assumption okay, and This is due to new long zoo. Okay the first result and the lantone don't offer Eklund proved they proved in 1995 That in this case you get at least one elliptic orbit Okay, so in this symmetric case in this convex symmetric case We do have a positive answer to the original question Okay, now, let's think about this convexity assumption convexity assumption, right? As I said Assuming convexity you use classical version of rational methods and such methods for this classical version of methods You need really convexity because as I said because in the convex in the in the in the convex case You can replace the actual functional by a good by a good functional. So you have good rational methods But the point is that If you want to generalize these results for more general workflows The first thing you realize is that convexity is not a simple condition. It's not a simple invariant You can take a convex surface And it's easy to construct it is easy to see that you can find a simplectomorph is such a simplectomorph is In order to end such that the image of this convex guy is not convex anymore You can see this in the plane right you can take some You can take a convex A convex region in the plane then you can apply some some simplectomorph is in the plane Which sends this to something something not convex So it's not a simple condition The convex is not simple condition You are using the fine structure of our train so a natural question Is how to generalize the previous results? replace the hypothesis of convexity by a simplect condition And and as I said, this is important to try to to to generalize these results To to more general contact manifolds Using for homology. What is for homology? I will not give details about for homology. I will not discuss about for homology I don't have time for this but remember these action functional is ill behaved if Fleur was the first one that Showed the way to to to use he construct a Morse A Morse homology for this action functional He showed how to how to to to use Morse theory for this function So It it works in it works on general simplect manifolds so So to to get together simplect condition is important if you want to generalize these results for more general contact manifold Contact manifolds, okay for more general rib flows How to understand convexity from the simplect point of view And the first definition in this direction is due to Hofer-Wieselke-Zehnder Is called the dynamical convexity. What is dynamic convexity is the So sigma is called a dynamic convex if every Simple periodic orbit has colonies and the index bigger than or bigger than or equal to n plus one Well, I'll not give you a precise definition definition of the colonies and the index But in the case of closed shard asics, it's the is the Is the usual Morse index, okay? I'll not give you that details about this But the but the colonies and the index is defined in terms of the linear rise of the metronome flow along the periodic orbit Okay, so this is the dynamical convexity Okay, so every periodic orbit has index at least n plus one It's not hard to see that if sigma is convex, it's dynamically convex Okay, this is the second point that about your question What is special in the in the convex case one thing that is special is this is that in the convex case You have dynamical convexity something that something Actually, it's a very good question and we'll talk about this later It's a very important question if there is some example of of Of a very good question if there is some example of dynamically convex sphere Which is not simple to morphic to some convex sphere. It's a very good question. We don't know Of course, you see of course Of course, the name for convexity is a variant by is a variant by simplectomorphisms So if we take if you take some something convex Then the image can be something that's not convex by some simplectomorphism, but it's it's a dynamically convex But the question is if there is some example of a dynamically convex sphere that is not given this way By the image of convex one by simplectomorphism This is an important question Important and difficult in difficult question So clearly the dynamically convexity is a condition variant by simplectomorphism It depends on it. It's it's it's it's defined in terms of the of the index of the periodic cooperative is invariant by simplectomorphism And and why you have this lower bound n plus one, but there is a homological reason for this I don't want to give you details about this, but essentially Associated to this associated to this to the center context sphere You have the what we call the contact homology is a kind of fluoromology for the standard contact sphere And this is a graded group The chain complex generated by the pedagogical orbits and blah blah blah and what you can show is that This contact homology is as a morphic to q. It takes rational coefficients for degrees The the the grade is given by the columns and the index for degrees n plus one n plus three and so on and zero otherwise So any plus one is the lowest no trivial degree In contact homology for the standard for the standard for the standard contacts for you Okay, so now assume that sigma is dynamically convex Which is a simple condition What are the results of for the for the lower bound? of periodic orbits The first one is a joint work with miguel abril. We proved that you have at least two periodic orbits in any dimension It was published in 2007 last year There is another result which is a generalization Of the previous result in the convex case Due to geeseberg url the preprint appeared three years ago Okay, but it was it appeared after or after my work with miguel abril That they proved that you have at least n over to the the sale of function of n over two plus one And good kind proved that in the non-generate case you have at least any periodic orbits And and with abril we prove as in the convex case that if you have finally many Then you have at least one elliptic periodic orbit In in in my work with abril. We actually Have results for more general contact manifolds using a more general definition of dynamical convexity using contact homology using this Using the contact homology of the corresponding contact manifold. Okay So we were able to generalize the results for more general contact manifolds assume some sort of dynamical convexity Okay, so we have these results So remember we had results in the star-shaped case in the convex case in the cement in the convex symmetric case Now the results for in the dynamically convex case. So now assume that's the name you have dynamical convexity and That the end that sigma is invariant by the tip of the metric It's a metric Just like in the convex case, but now assuming only dynamic convexity What we have We have at least one elliptic periodic orbit Like in the other tunnel don't offer you echelon in the convex case And as before we proved actually this we got results for more general contact manifolds Assuming some sort of dynamical convexity And since we are not assuming convexity the proof Are not based on classical variational matters We really need to consider this this action functional, which is you behaved But now we use for homology To do with this to do with this So, okay So remember we want to find the lower bound for for the periodic orbits for the elliptic periodic orbits Assume dynamical convexity and that is invariant by the tip of the map then we have at least one elliptic orbit in the convex case we Assuming in the convex case in the symmetric assuming symmetry and convexity. We also have any periodic orbits so another question is Now suppose that sigma is dynamically convex and symmetric Is it true that you have at least any periodic orbits like in the convex case? That's the question Now we have a simple condition and and we have the symmetry the question is if the if the result in the convex case holds under this assumption that is dynamically convex So far, we don't know how to prove this using only dynamic convexity We need something stronger a bit stronger. That's what we call strong dynamical convexity in order to define strong dynamical convexity, I need some Normal forms for the agent value one Which are the following? So you take a sympathetic matrix suppose that it's totally degenerate it means that All the eigenvalues are equal to one Okay, then you can show that a this the the sympathetic matrix is given by the explanation of jk Where k is a symmetric matrix with our eigenvalues equal to zero And you have the the following Normal forms for k Have four types of k okay k Can be the identically zero quadratic form Can be given by k zero in that in in that way when where d is bigger than or equal to three Or can be given this way you have this sorry three Three types of three normal forms for the agent value one Okay, and we define b plus minus a as a number of This the the number of q of q plus minus that appear in the normal form okay This is technical condition, but It's important okay So b plus b plus minus are the number of Number of times that the third Normal form appear For the for the matrix Okay, and now given a symplatic matrix a general symplatic matrix and now you take the generalized generalize the agent space Of the agent value one Restricted the symplatic matrix to this to this guy and you define b plus minus as As before with p restricted to to this generalized generalized agent space Examples to dimensions Well for the identity of course the quadratic form is zero So b plus minus is zero and for these guys It depends on the nipple on the new potent part Is a computation you can show that in these guys you have this Okay And now given a close orbit you define b plus minus as The b plus minus of the linearized pancay map So you see in particular b plus minus is zero whenever the periodic orbits now degenerate because you don't have any value one It's it's not zero only in the in the in the the generate case so Now what is strong dynamical convex convexity? Suppose that sigma is invariant by the nipple on map Okay And and then you can you then you write the set of simple periodic orbits as the joint union of the symmetric orbits and non-symmetric orbits Symmetric means that the orbit the image of the orbit by the nipple on map is the is the orbit itself And non-symmetric if it's not symmetric If the image is not the orbit And then we have this definition We said that sigma is strongly dynamic convex if If it's a dynamically convex in the sense that every periodic orbit The index of every periodic orbits at least n plus one but now we have this additional condition Namely that for the symmetric orbits you have this inequality Involving the index and b minus and b plus And for the non-symmetric orbits For the non-symmetric orbits you have the same condition, but only for the second iterate Of the orbit Okay It's a technical condition It's important and and and and and and and again Of course in the in the in the in the non-generate case It's it's the same as dynamic convexity because in this case b minus and b plus are both zero Okay, so in the non-generate case b minus minus b plus b minus is the number of of the of the q minus that appear in the Normal form and b plus is the number of q plus and in this example in dimension two you have this Okay It depends on the on the on the normal form of the eigenvalue one It's really important because we are considered the the generate case, which is the hardest one Not really because you see depends also depends on the on the index So as I said as I said when sigma is non degenerates Strongly dynamical convexity is equivalent to dynamical convexity because b minus and b plus are both zero And in general when the when the eigenvalue one is semi-simple namely that geometric and algebraic multiplicities are the same Then these conditions Dynamical convexity is equivalent to strong dynamical convexity Okay, because in the same simple case you have the the norm The quadratic form is the zero form right you have the identity And the strong dynamical convexity is invariant by symptomomorphism that commutes with the tip of the map And then we have this result If sigma is convex and invariant by the tip of the map Then it's strongly dynamic convex So we realize that convexity implies more than at least in the symmetric case Convexity implies more than dynamical convexity You have this slightly stronger condition In the degenerate case It was a very important point And then what we prove is this Suppose that you have you have a symmetric Strongly dynamic convex convex sphere Then you have at least any close orbits And actually we get we got two results That one and this one The second result we don't need to assume symmetry Now instead of assuming symmetry you assume the following condition assume that you have this condition For every simple periodic orbit Okay, then you have at least any close orbits, but notice that this condition is Stronger than than the the first one why because remember strong dynamical convexity means that the conditions that it's That it's the the the star is dynamically convex and you need this this condition for the symmetric orbits And only for the second iterate of the non-symmetric orbits But here in fear in the in the second theorem We are supposing this condition for every simple periodic orbit. So it's a it's a stronger condition So under this condition you get at least any orbits But we don't assume symmetry in the second theorem Okay, so as I said it's important to mention that it departs a theorem too Possibly it's not satisfied when sigma is not convex when sigma is convex in fact It's through the linear in the linear level you can construct a positive sympathetic path which is Which is a it's a it's a candidate of a periodic orbit of a convex sphere that does not Satisfy this condition So probably probably convexity does not imply this condition for every periodic orbit okay, and as a corollary of this result of the of the second result as I said Where you see when when when when when the when the eigenvalue at eigenvalue one is semi-simple B minus and B plus are both zero So in particular we get this corollary suppose a sigma is anemic convex And the eigenvalue and the eigenvalue one of the linearizer punker ham app is semi-simple then you have at least any periodic orbits So we have the result assuming only dynamical convexity, but we need to we also need to assume that The eigenvalue one is semi-simple In the non-symmetric case Okay, very good. So so as I said before A Dynamic convex sphere does not need to be convex right because you can take the image of a convex You can take a convex guy and take the image of this convex guy by some simplectomorphism. It does not need to be convex Right, of course So but dynamical convexity is invariant by simplectomorphism So so this guy is not this guy the image of this convex guy by this simplectomorphism Is not convex, but it's dynamically convex because this guy's dynamic convex in dynamical convexity is invariant by simplectomorphism So an important question in simplectopology is this one Are there examples of dynamically convex spheres? They are not simplectomorphic to convex ones It's a very important question In simplectopology It's a way if if if for instance if it's if it's if every if If the answer is no, then it's a it's a We would get a simplect characterization of convexity, right? We would get that convexity is equivalent to to the dynamic convexity, which is a simplect invariant and We got the following negative For the following positive answer to this question in the symmetric case When n is bigger than 2 We prove the following Given any bigger than 2 then there exists a symmetric dynamic convex sphere That is not equivalent to a convex one, but Is not equivalent by a simplectomorphism that commutes with the antipodal map We need to assume this Unfortunately So we have a we have a positive answer to that question, but in the only the symmetric case Assuming that assuming also that the simplectomorphism is symmetric in the sense that it commutes with the antipodal map In this case we have We have examples of dynamically convex spheres that are not equivalent to convex ones And how we construct this example The idea is the following The example is given by a symmetric dynamic convex that is not strongly dynamic convex More precisely in the example what we have We have we have a symmetric orbit. We have a symmetric orbit in the example That does not satisfy that condition Have some gamma s Symmetric orbit in this example such that That's the point So in particular the example is that this orbit is is is degenerate The example is an empty convex so the orbit the orbit does satisfy this condition So it's dynamically convex But it does not satisfy it does not satisfy that condition that that condition that this this This guy is bigger than or bigger equal to any plus one That's the point and strong dynamical convexity is invariant by simplectomorphism that commute with with the antipodal map So We get the result The construction is very technical. Okay, but the the point is this is to find the We need we need only to find this symmetric We need to find a symmetric orbit satisfying these conditions What you also need to show that all the other all the other periodic orbits Are than empty convex namely the all the other all the other periodic orbits satisfy this condition So all the all the all the all the periodic orbits satisfy this condition This condition, but you have a special one a symmetric special one that that Does that satisfy This condition That's the point So summarizing We have three important questions Which are the following That you have at least any close orbits any simple close orbits at least one elliptic orbits And the question convexity vast versus dynamical convexity In the in the symmetric case. We have good results So for the first for the first inequality, we know that it holds under the assumption of strong dynamical convexity Okay The second inequality holds for assuming dynamical convexity and the third And for the third question, we do have examples of Dynamically convex spheres that are not equivalent to convex ones Via simplectomorphism that commutes with the antipodal map So in the symmetric case these three questions have good results not complete not complete results, but good results And in the non-symmetric case things are worse Okay, we don't know if one and two holds only under the assumption of dynamical convexity Okay Under the assumption of dynamical convexity, we have only for for the low for the number of periodic orbits. We have only that Only this Okay, assuming only dynamical convexity And we have no lower bounds for the for the number of elliptic orbits Even assuming as a dynamical convexity If you don't if you are in the non-symmetric case And we don't know examples of dynamic convex spheres that are not equivalent to convex ones Okay Of course, you can ask why the symmetric case is easier or less hard It turns out that The reasons are different Okay We use the same in the proofs of this of these true inequalities. We use the symmetry We use the symmetry in different ways. I don't have time to give you details, but They appear that we use this in different ways. Okay There is no common reason Why the symmetric case is less hard than the non-symmetric case And and as explained before in for the for the example of dynamic convex Example of of a dynamic convex sphere that is not equivalent to a convex one. The point is to construct A symmetric dynamic convex sphere that's not strongly dynamic convex. That's a point. Okay This is the last slide and further directions Okay, once you Once we have a simple condition to get BD watch quad, it's it's natural. It's natural to ask to turn out to try to generalize these results for more general real flows and so future question future direction is to extend these results for Contact forms or more general contacting variables, which are invariant by some special z2 action free z2 action We already have some partial results in this direction Okay And to finish We have this question It's a very important questions for people in sympathy in sympathy apology Other examples Of dynamic convex spheres that are not equivalent to convex ones via any simplectomorphism Because what we got what we proved is that we got we got examples That cannot be equivalent via simplectomorphism that commutes With the tip of the map But we have no idea how to improve this result For any simplectomorphism How to get an example of dynamic convex sphere that's not equivalent to a convex one All right, and as I said, this is important to To to have a simplect characterization of convexity. What is convexity from the simplect point of view? Okay, that's all. Thank you very much Yes, because then in this case when you The restriction of the sympathetic matrix to the to the generalized agent space is the identity The identity There is no new potent part is just just the identity Yes Yeah, more precisely The way that they appear and the proof it's related to the what we call the splitting numbers it's it's a bit technical, but there is a If you give me one minute, I can try to explain you There is a construction due to bots It comes from the geodesics Well for a sympathetic path, so you have a sympathetic path, okay? And then you have a function which is called the bots function will have a function defined on the circle Such that a function bots function Taking values in z integer values such that this function It's it is it's it's continuous except at except possibly at the eigenvalues of the of the Final point of the sympathetic path Okay So so you have these eigenvalues So in other words, this function is constant except possibly at the at the eigenvalues with with modulus one So and this function has the following remarkable property that if you take the index Of the cave iterates of the orbits It's nothing else that When you take the sun of the of this function of the k roots of the of the unity in the circle So so these functions are constant except at the eigenvalues. So at these eigenvalues you have the have a jump You have this what we call splitting numbers and this b plus and b minus are related to these splitting numbers something technical, but This is a very nice construction due to bots