 I'm happy to welcome you to one more session of the ICTB Math Associated Seminar. Today our speaker is Kali Mbar and he's talking about closures and leaks on surfaces without conflict. Okay, thank you Alex Alejandra. So I'm very happy to speak in this seminar for the first time so I'm not very very knowledgeable about this technology but I will try to do my best so I will be writing live as I'm speaking. So we'll be talking about closed geodesics on surfaces without conjugate points. Basically we are interested in counting closed geodesics as you seen abstract the Margules estimate that's what we recover. So during this talk I will have mg where m here this is a surface it's a two-dimensional closed manifold closed manifold and g is a remanometric. So we all know what geodesics are geodesics are just shortest path that join points like shortest path joining points in the manifold and so geodesics shortest path so shortest respect to the remanometric that is given joining so I'm assuming that this is just very standard and then we are now interested in closed geodesics so the geodesics see so we can think of a geodesic that is parameterized from the whole real line to the manifold is closed if there exists time t such that I mean naturally you have c t plus t equals c t for all t in r. So in here I will be interested in counting closed geodesics and interested in doing this counting in the setting of non conjugate points which is quite general like more general than negative curvature more general than non positive curvature but to count these closed geodesics there is a problem that we are facing that sometimes you might have for instance if you if you if you look at this picture where here you have a surface of genus 2 where here you have curvature the curvature here is negative here you have a flat piece curvature 0 and here you have curvature negative again so you see that here you might have like countably many closed geodesics here all these are closed geodesics so to do this counting we want to count all these geodesic as one so in that sense we we will be counting free homotopy classes because all these closed geodesics are free homotopic so so so we do not for t for t positive so let b t be the number of free homotopy classes with a geodesic of lengths at least t with the closed geodesics sorry of lengths less or equal to t so the length of the closed geodesics is actually in the previous slides is the shortest time for that that equality holds so that's just the period of the lengths so so we will be interested in doing this counting so there is a fact that it is very well known that every free homotopy class contains at least one closed geodesic in negative curvature we know that it has only one closed geodesics but as I said in the previous slides in this flat in this flat bit you have all these geodesics that are free homotopy so one free homotopy class can have multiple closed geodesics okay so there is this very old result for for instance in this setting of surfaces when the curvature when the curvature is just constant minus one okay equals negative one so there is this very well known very like standard results of Haber this was in the 1959 that you have this number of closed geodesics is just asymptotic too so you have p t is just of order of e t over t this is also called a prime geodesic theorem in some by some people so so our result is going along so there are more results that I will I will about this that I will I will list later so our result is more about we will first try to relax I mean the coverage assumption and then also yeah so the coverage assumption we relax it by this what we call no conjugate point so I will just recall the definition point here no conjugate points so if you if you're given a geodesic so given a geodesic c this from r to m if you have a vector field along this along this geodesic so a vector field j along this vector field defined from r to t c m so this is just to say this is along along the geodesic c is a Jacobi field if it satisfies this nice equation which is j double prime t plus k c of t j of t equals zero oops that's not what I want to show equals zero so this is a vector field that satisfies this is called a Jacobi a Jacobi field and now two points on this geodesics are conjugate if there is a non vanishing vector field a non vanishing Jacobi field that vanishes at one point and at the other point so two points ct1 and ct2 are conjugate if there exists j which is a Jacobi field on c so Jacobi field I would just write it like that so that we have j at t1 equals j at t2 equals zero but j prime at t1 is non zero like it's not identically zero in j prime at t1 is different from zero so no having conjugate points is just that definition and you can see that if you have a negative curvature for instance from the equation so if you have a the curvature is negative you see that the Jacobi field they grow exponentially from this from this simple equation so you have exponential growth of Jacobi fields so you do you cannot have Jacobi field that vanish at two points of Jacobi fields and then then no conjugate points similarly also if you have a non positive curvature so it's from the from the equation you can you can see that you cannot have a Jacobi field that is vanishing twice along the same geodesics so this is just saying the picture you would have is if you have this is your geodesic and then this is ct1 this is ct2 you have the Jacobi field that is vanishing here but not identically zero it's growing and start decaying to get zero so these two points will be will be conjugate and this notion of conjugacy it has to do actually with you know the geodesics being minimally in the universal cover for instance if if if you don't have conjugate points you can prove that the geodesics are minimally in the universal cover by using that there is a nice formulation of Jacobi fields by the exponential map and then you can see that it is it is equivalent to say that you have geodesics are globally minimizing in the universal cover which is which is quite general so the theorem our theorem says that so as I said in the title this is joined with Clemenaga and and and and and Knieper yeah Knieper and myself so our results it just it just gives the the generalization of of the results I state I said before the harbor results the one in the 1951 so that mg be a closed surface without conjugate points genius at least two so this is because you we we're just trying to avoid the the torus the flat torus of at least two then we have the following asymptotic estimate so say that the limit when t goes to infinity of the number of homotopic classes free homotopic classes is just as you would expect oops so I should use ht ht over ht here h is the topological entropy of the geodesic flow h is a topological entropy so I will not define it so but topological entropy of the geodesic flow so this is the main result we prove here so you say that if you have a surface without conjugate points of genius at least two you have oops I did not write the limit equals something I should write equals one okay so actually I want to discuss actually you know different type of of of asymptotic bound that people have studied so there is so we have three different bound that I studied here one is we call it the exponential growth so this is just that the limit when t goes to infinity of one over t log of pt equals the topological entropy so this is one bound let's call this eg exponential growth there is another bound which is a little stronger which is a uniform bound the uniform bound says that you have pt is bounded by b over t okay um sorry but I think I need to switch to little t because I think that's what I'm used to doing instead of using capital t so b over t exponential of ht and here a over t exponential of ht so you see that this bound it implies it is a little stronger than the exponential growth so the uniform bound let's call it ub and the last one that one we prove in the theorem which is we call it the multiplicative asymptotic asymptotics which is just to prove that to improve the bound in ub that this constant a and b they can be optimal that when t goes to when t goes to infinity they they converge to one so this is limit when t goes to infinity of pt over eht over ht this is one so as I said here so this three bounds this last one the multiplicative asymptotic is to see that it implies ub which implies eg sorry so let's call this ma as I said before I I go to discuss the main like some points in the proofs I would I want just to to to tell you a little bit about the history like what was done in this in this subject before so as I said before earlier there is a result of harbor in 1959 which was for surfaces and curvature is constant minus one and for surfaces okay so this proves ma and here curvature minus one gives you that the topological entropy is one and that's why in the formula the h was one in the in the formula I wrote before for these results so and this uses the the this uses the cell vectors formula that is that relates the the spectrum of the laplacian to the length spectrum the length spectrum is just it gives you the information about the length of the closed geodesics and there is a nice formula the cell vectors formula that that gives you this relation and yeah so and this this is something that you have in surfaces for surfaces in constant curvature minus one and later there is a result by sin i in 66 this proves the uniform bound okay the uniform bound that I have above so this proves u b for variable curvature but still negative curvature and bound it above and below by some positive some negative constant so variable negative curvature and any dimension this doesn't have a section on dimension any dimension and and it's after in 1969 that that Margules in his thesis proved the general one for not even for the geodesic flow but for topologically mixing analysis of flow so there is Margules in 1969 that is in his thesis so proves m a the multiplicative asymptomatic bound for topologically mixing analysis of flow so um so i'm just showing this word but for those of you who don't know about analysis of flow is just so the the one nice example of analysis of flows are geodesic flow in negative curvature they have they have this nice property of being analysis so in particular it it improved the results of of sin i which was u b and also this also was was generalized by perian polycote in 83 for axiom a flows so perian polycote 83 for axiom a flows and yeah so but you see that all these all these results that I listed they kind of use what we call the uniform hyperbolicity of of of the system like you have uniform hyperbolicity which just means that you have a nice splitting of your tangent space into stable and unstable that are contracting and expanding uniformly and so these are some problems that we're facing in the non conjugate point already in non-positive curvature you have this problem that you don't have uniform hyperbolicity so and and beyond negative curvature actually there is the results of katoch so it's just let me just list a few more a couple of more so beyond negative curvature so there is this results by katoch who proves the exponential growth for sumo flows with positive topological entropy and no fixed point so 1982 so sumo flow and h positive positive topological entropy and no fixed point this gives the exponential growth like the number of closed geodesics they grow like the entropy exponentially right and then and then actually last year there is a results that by lima unilema and omri serig 2019 and they give now the the uniform bound they they improve katoch's results to give actually the uniform bound for three-dimensional flow with positive topological entropy 3d flow with positive topological entropy okay I should just write h and also in the setting of negative curvature there is another result of creeper that proves the proof that proved the exponential growth so creeper in 1983 gives the for non-positive curvature and rank one so this gives the exponential growth so so this is mostly what what is known in terms of dynamics also in terms of in terms of remanion there are also results that do other settings but I'm not going to talk about them and actually I should say that there is a very recent result by racer ricks ricks in 2019 actually our our method is uses uses a lot of techniques from from his paper racer ricks for rank one cut zero space this is beyond the remanion setting so this is rank one cut zero space so in particular it gives it it recover it gives it gives the results for for non-positive curvature in rank one so here he has the actual bound the multiplicative asymptote and yeah there is a preprint for this that is on the archive yes so yeah so I'm halfway so I will the rest of the talk I will mainly give you ideas on how to prove this multiplicative asymptotic bound some ideas so this proof it actually it is base so I would just list two like three points maybe that are very central in the proof and then I will tell later how they are used so if you don't know it yet maybe you just have to skip this and then follow after what I will what I will say but I would just say two three three points that are very essential in the proof so one is that the measure of maximum entropy the measure of maximum entropy has the the product structure so the measure of so if you don't know what measure of maximum entropy is you can just skip this and then I will save some more later measure of maximum entropy has product structure and another point is that the floor is mixing with respect to this measure can you say a little bit what what you mean by product structure so you have the the the measure you can write it as a product of two measures so in the in the in the in the so you can you can see this as so you can think of this measure being disintegrating on this stable and unstable on this unstable and unstable manifold and you can write this measure as a product of two measures there and if you know something about the partisan Sullivan measures this is a typical measure actually that's what we use here that's that's actually the we proved that the measure of maximum entropy is given by partisan Sullivan measure which by construction has a product structure so it's just the measure that you can write it as a product of two measure and these two measure they have they have dynamical meaning like they are associated to stable and unstable which is not true in general for instance if you look at the the for the geodesic flow if you look at the Louisville measure for instance if you don't have in variable curvature variable negative curvature the Louisville measure is not in general like a product measure so this is a this is a very very very you know specific to the measure of maximum entropy that we use here that was also used in the original proof in the Margules and this you have to prove in your case this is not yeah yeah we have to we have to prove that we have to prove that the measure of maximum entropy is built in a way that it has a product structure and then the other point is that the flow is mixing respect to this the measure of maximum entropy let's call it M and the flow is mixing respect to this measure so I will also say later what mixing is if you don't know it so the geodesic flow is mixing with respect to M so this is also something we have to prove we we had it in our in our first in our first joint paper with Klima Naga and Thompson we prove these two properties like the measure of maximum entropy has a product structure and the geodesic flow is mixing with respect to it and then the third property is some type of equity distribution of closed geodesics so I will also say explicitly what I mean by this equity distribution of closed geodesics so this is roughly saying that if I if I take a measure that is supporting along this closed geodesics of length t if I if I take the limit when t goes to infinity I'm going to get the measure of maximum entropy basically these three properties they are the main ingredient of the proof okay so I will now move to more specific on how to how to prove this so we're going to count so just some notation I will denote by M tilde is the universal cover so universal cover universal cover and then the the first fundamental group I denoted by gamma so gamma is the first fundamental group and it is known to be isomorphic to the group of isometrics of M tilde so this is very standard and so the first step of this proof is to I mean what I'm what I'm going to say is so this is just using the this proof is using the main ingredient that that we that people know in the Margielus in the Margielus asymptote proof so which is you find is this box that is foliated by stable and unstable and then you use this box as what what some people call a detector that you detect the number of closed geodesics of a certain lengths that are that are going through these blocks and then and then you use mixing and and and and an equity distribution to have the bound so that's the very rough thing of about Margielus so the first step here the first difficulty would be to find this Margielus box this this this detector that is foliated by stable and unstable so first of all you want to okay I will draw it here directly so first of all you want us kind of a box here let's say B and then you want to know the number of closed geodesics through this set so let's say B so what happened here is from B you can define you can find there is a what we do is we find a nice bijection I mean almost a bijection between the okay so I should be so here I have this box B and I'm looking at the number of closed geodesic through B of lengths less than t and then there is now I want to relate this set to the to a subset of the group of isometrics so what I'm basically saying is that there's a almost an isometry between this set and gamma t which is a set of gamma in gamma so that you have this is for some people who know more than me about dynamic this is kind of a closing lemma so B gamma star phi so phi minus t B is not empty so phi t is a geodesic flow okay geodesic flow this is a geodesic flow and see that okay I'm taking B in the manifold but think of it as lifted in the universal cover and then you have gamma that is acting on the universal cover the elements of isometrics okay so there is a nice bijection between the number of closed geodesic that are going through this set B so what I did not say is how to construct this set B maybe after if I have some more time I can come back to it and tell you how this is constructed but this this was like a main difficulty here like to construct this set B this detector that Margules has in the in the in the in his proof and then so so see that if you have a so I'm just going to tell you roughly why there is a bijection between these two these two set the number of closed geodesic that is going through a set B through this set B and then and then and then this subset of isometric elements so see that if you have a closed geodesic so given closed geodesic C so so every closed geodesic you have a you have an elements that is invariant by this so this there exists invariant there is gamma in gamma says that gamma of C equals C like there's an element in the isometric group that leaves invariant is C for it being closed so you have every closed geodesics will correspond to an element here because if you see if the geodesics go through B you will see that here if the element go through B so this guy will come back to B for instance if this is C you flow it by C so closed geodesics we apply to isometry it will bring it back to the set C so this this way is kind of easy what what requires a proof is the other way like which is more like proving a closing lemma in this setting and the proof is kind of it's not very difficult actually let me just say a few words about that so so meaning that if I have an element here in this set gamma T how to find a closed geodesic so what you have to know is that if I have here the usual compactification of my universal cover which you can do again here in non-conjugate points there is a there is a way to do it there is a yeah there is a nice way to define the boundary as a topological manifold by Ibaland by just taking the geodesic asymptotic to each other and then so think of my set B being somewhere here okay this is B that I'm detecting so if I take a gamma in in that set I said that means that this gamma if you take it you flow this set by the by the geodesic flow it comes here let's say this is phi minus T B and then gamma will bring it back and it will intersect so this is gamma star phi minus T B so basically what I'm saying is that you will you would see that so this set B you can project it to the boundary and it defines you two subsets here and these two subsets they will be invariant by by gamma here so they so you can have here this guy this interval here let's call it C minus and then C plus so you will see that this gamma applied to gamma at C minus will be in C minus and gamma minus one in C plus will be in C plus again oops subset and this you would find a fixed point in the boundary and this fixed point you can find the geodesics joining them and that geodesics will be fixed by the by the by this gamma and that gives you the close so find fixed point in the boundary so you have two points psi and eta that are here psi and eta that are here that are fixed and then you can join these two points by by by geodesics so and the geodesic the geodesic joining psi to eta is invariant under under this gamma is invariant under gamma and therefore it descends to a close geodesics downstairs in the manifold and then you have a so this is a closing lemma that we use here and then another ingredient that we need now so we are relating the number of closed geodesics to this detector with some elements of the of the of the fundamental group a subset of the fundamental group that is defined like that now if you take let nu t be the the distribution along the distribution so i'm just taking here distribution along closed geodesic of lengths at least t so this is saying that i'm just averaging along this closed geodesic and i find this measure nu t and the the the bijection that i said before so the previous bijection gives you that the measure of of this detector b is of order of the cardinality of gamma t over the cardinality of ct so i'm putting here some bound so this epsilon is the the thickness of b in the flow direction so see that here the set b it has certain thickness in the flow direction so you take stable and stable and you flow it so this this is a this is epsilon here this size here is epsilon okay so so this gives you this bound easily from the from the what i said before from the bijection that you have so this hasn't used the ingredient that i was telling but it will use it later in a sense that you have so now we we use mixing to measure the components no sorry before we use the the product structure of the measure so really the way that it is given by partisan Sullivan measures the product structure the product structure to see that these sets so this the measure of b intersects with gamma star phi minus tb these are the the components of the set i was having before this guy will be roughly exponential of minus ht so this is this has to do with the construction of the measure given by the partisan Sullivan it's not very difficult it uses a trick that that racerix has in his paper to measure these components these sets times the measure of b so remember here the measure is a measure of maxima entropy m is a measure of maximum entropy so if you see i'm trying to measure these components here so i'm trying to measure i'm trying to measure these components here okay and these components the the measure time the cardinality should give you the total measure and then yeah this is something i won't go into details on this and then there is the mixing part of the measure so this is very crucial and this uses the product structure this what i just wrote here it uses the product start of the measure and basically it uses the fact that the measure is given by the partisan Sullivan construction which is which has a product structure and then we use mixing here we use mixing to have that the measure of b intersects with phi minus tb this guy is converging to the measure of b square so as i said what i'm but i'm sketching here is is i mean this is very sketchy already in the paper if you if you if you have a look in that now preprint there is a main technicality that i'm not saying here that is due to here we cannot do the exact margillus b and b in the paper we need for some technicality we need to use a subset that is much smaller here but just for the presentation here i'm using b and b okay so this is just the mixing this is a like you can take this as definition of mixing of a measure so the measure is this we prove it in a previous preprint and then and and now if you have this mixing property this gives you now that if you if you couple this with the inequality i have above this gives you so this would imply that the cardinality of gamma t goes like e ht times the measure of b right because previously i have the i have the other bound which is here so from from this step i combined the the measure of the measures so these are the measures of the components so i should have written something before let me just let me just erase this and write one more line before i say this so from here you can have that the measure of this full set measure of of b intersect with phi minus t b this goes like the cardinality of gamma t times e h minus ht times the measure of b right because these are just one components and these are disjoint okay for different gamma we choose b some more so that for different gamma we have the addition so the measure is just the union the sum of the measures so and and this implied what i raised before that the number of these elements is like e ht times the measure of b because you use mixing here and use this bound these two yeah from these two you you have you have this so this so far i have used the product structure in the mixing property of the measure and then the last the last bit is to use equidistribution for instance to say that this quantity that i have here this measure here equidistribution implied that this measure is converging to the measure of maxima entropy this measure so when t goes to infinity this is like mb okay this is mb and this will be epsilon over t times this so now we use equidistribution equidistribution equidistribution to have that the number of pt is like epsilon over t times the cardinality so this was just from this from this definition here okay because this is converting to the measure so this is what why did i say see it should be pt sorry this should be p and you use the previous bound that you have in the in the in the cardinality so this is very sketchy again this is not like a rigorous proof but it just gives you the ideas of the main ingredient that we use here and one thing that i'm hiding i did not say is the construction of b which is a bit technical but i don't think there would be time to say it but so this is uh like approximation so this will be epsilon like epsilon e ht over t and from here to the to the to the final we just do an integration because here we are looking at at at close geodesic that are going through b and and then we we do an integration to get that to get that down the number of guys that are in the in my full manifold is ht over ht so this is basically the main ingredients in the proof here so one thing i said i am hiding i did not say is like the construction of this set b it's not very trivial because you don't have transversality between stable and unstable manifold but you need to be used like the hub map that is defined the boundary to define this set yeah so i think i will stop here and i'm open for questions thank you gaden thanks gaden for your talk so are there any questions you can now admit your microphones or if you want to write it on the shelf and i can also read them please go ahead well i i had actually very general questions so i don't know maybe you want to answer it's it's uh maybe the other people have more specific questions because when we're discussing at some point you talked about how these kind of counting results have relevance to other areas of mathematics except can you say something about the motivation for these results and what are the i don't actually i don't know the the precise the precise maybe some people in the urgence will know this more than me should know but i just know it has to do with this it can give the i mean there's a prime number theorem like the it is a prime number theorem that's in number theory but the way we use it is the way it's very different this is just going through dynamics but already that prime number theorem you can have it from the negative curvature from the constant negative curvature one but it's it's it's related and yeah it has also relation with zeta function but i don't know very much about this so yeah and how much of a restriction is this not having conjugate points is that a very general condition or is it something yeah i think that's the that's the very that's very generalized thing that's the um so does it not hold the results not hold if you have conjugate points it's a very degenerate situation or it's not known for instance you have you have flow with conjugate points that have positive topological entropy so you have this uniform bound by seric but the exact asymptote is not known it's not known if yeah if you assume but you can have you have the the uniform bound if you just have positive entropy no are there any other questions i have question yeah and the first theorem you have stated what's the problem that happens with the torus why the genius is at least two what happens in genius one yeah for instance if if you have genius one so you just have two uh the if you have genius one without conjugate points it is flat so you just have two orbits oh okay it is flat so and yeah and everything i'm talking about it should be it should have even positive entropy and the flat torus have zero topological entropy and yeah you just have two close orbit if the if you are in the flat torus okay thank you any other questions so no thank you again thank you so much thank you thank you God