 Thank you very much, I would like to thank the organizers and especially Stefano for the invitation and so I want to thank Keith for He stopped yesterday because he defined it almost all the things I have to define here So he will spare me some time and then so let me state the theorem the main theorem That's right readable, okay, so Let us consider a compact surface of genus greater than two Without focal points. I will define this in a moment. This for me means C infinity surface G is a remanometric on the surface a symmetry minus metric and then And Then under exist homeomorphism H well, let's say Sigma from T1 to T1. This is the unit tangent bundle of the surface and an expensive Continuous expensive, let's say From this is no need. It is not on homeomorphism. It's a continuous objective. Okay continuous such that so Sigma By T is sight the sigma where fight T Is the genetic flow? okay, so this is a theorem and the main theorem and and so Some definitions first I will Be more precise in some of the definitions Keith gave yesterday. So He won is the unit tangent bundle. There is this set of Points in the surface and vectors tangent to the point such that the norm is one So genetic flow, I mean, okay, so first The up here PV is cold. I mean it's in canonical coordinates in the tangent bundle. So this is P in the manual fold and This is V in the tangent space Surface so This is the surface This is P and this is V and So the genetic flow is From acts in the unit tangent bundle and it sends the point PV to the point in canonical coordinates gamma Gamma with the prime T where gamma PV there is the unique you desic such that gamma PV zero is being and gamma The derivative of digitistic at zero easy. So this is We have another there are no colors. Okay, so This is something like a okay, this is gamma. Okay And what else so Let me define first no conjugate points first after before no for all points. So, thank you. So Mg has no no point points. Yes Exponential map is no singular for every P The manifold and then No focal points is a subset of manifolds with that with a concrete point satisfying What is the best thing the shortest perhaps not not the best But the shortest way of defining this is the following Every Jacobi field j such that J of zero is equal to zero and the derivative I mean non trivial Jacobi field with Which vanishes at a point satisfies The derivative is always different from zero for every Cases is perhaps The best way to define. I mean there is a more geometric way to define Equivalently Sears our convex our strictly convex in the universal covering Endowed with the pullback of the metric G Okay, so the G is pullback of G by covering map, so this is the okay, so They're very close to many falls we know we know we know with non positive curvature But there are six examples of many for without focal points Admitting some regions of positive curvature. So this is I mean, this is a category. So what we have is that non positive curvature It's subset of no focal points and this is No quandary points and there is well, I will talk later if I have time there is an intermediate category here Interesting for the talk that I mean for the subject that is called the many falls will bounded aspect Okay, there is sort of in the middle here. So Okay, I think I Define it Everything and then okay, so let me let me comment briefly the theorem here when When you look at the at the statement here, you have a semi conjugacy. I mean a continuous objective map Preserving the orbits of the geodesic flow and an expansive flow Preserving time This is somehow rigid. You see Because and the I mean what it is known is That let's say a famous paper by Tignis It is known that every Every and also flow in a cypher fiber bundle is topological equivalent geodesic flow of a hyperbolic surface I mean a surface with constant negative curve and this equivalence Does not preserve time Okay in general and moreover you have rigidity Associated to the length of spectrum. Okay, what does it mean? I Mean many people involved in this in this in this object or Tarzan perotal for Many falls of negative compact many falls of negative curvature then crow Christopher crow baffati and Feldman for surfaces Okay, what they prove they're following so suppose at mg Me Compact surface genus very than two without good points and and G zero The same surface another metric With an opposite career metric or in my instructor with non-positive curves then if mg and Mg zero have the same length Spectrum market then the horizon metric so there is I mean if When you see and a statement like this you immediately think of the result So the conclusion here is that this flow is not the geodesic flow Offender remain a metric. Okay It cannot be possible. Okay, so And Then I will I will try to okay. So this I will sort of D-value talking to parts the first part is sort of topological part When I'm going to do is I mean in a few words is to define a quotient in the unit tangent bundle to kill the Non-expansivity of the flow to get an expensive flow. This is a sort of You will see is somehow the most I mean perhaps It would be considered the most naive approach to prove the thing and so you would You would be you know willing to not to believe that it would that it works okay, but it works and And then and the second part I will Talk about some consequences in the in the I mean regarding the theory of measures of maximum entropy and This is I mean this and then we say this is a joint work with Catherine Guilford and This is more or less in the same spirit of Keith talk We were trying we're starting looking at the get our nippers work about the you know the the uniqueness of the measures of maximum entropy for rank one many faults, you know positive character and We thought that for surfaces It should be a simpler proof okay, and and Simple proof in the in the sense that it should exist something closer to Bowen's approach To prove the same thing Okay, and this is what we get so let me explain now. I'm going to Want it to the details of the proof so some ideas Some main ideas so I have to talk a little bit about the I mean horror spheres, so I mean They were mentioned yesterday, but I will be more precise. So If you consider Let me draw the universal covering here This is well with me put back of the metric and then you consider a jadesic here that is let's say VV let me Shorten the notation so theta is a point in the unit tangent bundle of Universal covering that you take a jadesic here, and then you define debuse man function of this jadesic is was associated to the point is the following is The limit as T goes to infinity of distance from gamma theta the X minus T Okay, so you pick Gamma theta T X is here, and then you measure the distance from X to here, and then T is the distance from Gamma theta 0 to here, and then this I mean it is proved that this limit exists and gives SC1 L function meaning that the C1 derivatives are lip sheets. I mean there are lots of properties of this function and they were I Mean, I'm going to mention Many results of many basic results of the theorem of manifold and the conjugate points mainly Stated by Eberlein and Pessin in the 70s. Okay, so this is sort of Well, then me perhaps I have What is it? Okay, perhaps I write a big theorem with the basic things so theorem a good reference for this is Pessin and so So with theta is C1 lip sheets Level sets, but this is whole spheres which I denote by H theta T. I mean level sets You get a level you get a foliation Of the inner circle covering by sort of spheres and infinity called our spheres because if you Forget about this limit, then the level sets of this function are just the spheres centered at this point. So if you let T go to infinity it's like you If like you would be letting the center of the spheres to go to infinity and then it is proven I mean the limit of these spheres Exists and is the whole sphere and depends on the point of the geodextric So this this dependence is expressed here by T. They are equidistant spheres they give an equidistant foliation of and children so if Let me define the So a little different notation. This is H with a little H. Oh, no perhaps so here This is the set of points P minus gradient at P of the theta Where P is in? H theta zero This is a set in the unit tangent bundle Of the inner circle covering that I know how to call this or a sphere too Because this is this the set of the a whole sphere endowed with the normals because if the Boozman function is the because if the horror spheres at the level sets of the movement function then the gradient is Perpendicular and the field of I mean the I didn't I didn't put here by the norm of the gradient is one and Then the set of unit normals inner unit normals is the set of is the field of Gradient okay of the gradient of the Newton function around the whole sphere and then the you have a Flow that is tangent to this gradient here and this flow is formed by Asymptotic to this for where asymptotic, okay So now I'm talking in the for the case of surfaces in the case of surfaces compact surfaces at the point And genus so the flow tangent the gradient is flow by for where asymptotic and The same thing well This the collection I'm going to define another thing fcs of I mean this is the union of theta in the unit engine bundle of Fcs theta which is is the union t by T of H theta is a continuous Okay, so what I'm saying is if you take this set but here a tilde and You saturate by the flow then you get a Surface and this collection of surfaces is a continuous relation in the unit engine Okay, so you can make the same thing for the past and then you get sort of a Get the same thing. I mean if you this for the Asian defines Fcs and H H That's a union of theta Define for the Asians for the Asians in T one of them too. I mean you just The quotient those variations by the natural quotient between the Tangent the unit engine bundles in the rest of covering and unit engine bundle. This is more or less the Horticycle for the Asian and then you can make the same thing for the past so Let us define Let H Well H with a H minus here of Well, let me let be minus theta define saying This is the limit As it goes to minus infinity this is between Our thing is the contrary. I mean this is minus mod of those T. So you make the same thing for the past so Gamma and then you may take X here then you take gamma TV dish here Okay, it's level sets are let's say H minus T let me define H I Will put a I will put here perhaps an S and then an S and then I put here at you That is the collection of points P radiant of This This is in I H minus the zero and then F See you is the collection of I mean Union of D in our fight Okay, so Let me draw the picture In the Let me draw the picture of this. I mean you have all these objects can be defined in the universal covering of the unit tangent bundle and then the picture will be something like this You have here The orbit of theta this is this would be the geodesic flow restricted to FCS CS stands for center stable Yeah, and then the FCU Would be something like this Let me perhaps use colors here So this table sets is something like this and the unstable set is something like this This will be the picture in the case of an awesome. Okay, this is an awesome case if the geodesic flow is an awesome then then this construction gives the invariant some manifolds of each orbit but in the case of Many photosurface is with no focal points with no conduit points What you get is in general something Well something like this I mean instead of just getting one geodesic here an intersection of both some manifolds and you get typically a set where they coincide, I mean they are tangent and this set might might be a Strip a whole strip of bi asymptotic orbits of the flow Okay, this is the let's say intersection of CS theta and CU theta is a strip asymptotic geodesics and this strip is endowed with a Foliation that is invariant by the flow consistent in geodesics okay, I Mean in the case of the manifold with no focal points, of course if MG has no focal points This strip is flat each strip is flat because you may have many I mean in each, okay strips are flat with a sort of invariant foliation by geodesics so Imagine that you have a flat strip in our tool at say and then You have this picture here Sort of the geodesic is here Now this is universal covering It's I mean better for pictures and then in each for each T you have as a geodesic in intersection of the This whole sphere and this horse here, okay, and this foliation here. Let me call this Let me call I theta intersection of HS Theta and H So I'm looking at this I'm looking at the at the lift of this strip in the tangent bound Okay, and then what I'm going to do to Kill okay, so This is clearly I mean this the strips are clearly an obstruction for To expansivity, okay, so what I'm going to do is to define an equivalence relation in the unit tangent bundle to kill strips okay, and then Equivalence relation to kill Expansivity non-expansivity and the equivalent relation is just Let's say let me call sigma the equivalent relation Okay Two points let's say theta and and see and psi. I don't know and our equivalent if if and only if Psi is in the same I Theta or theta is in the same I See or I saw I don't know So I Define an equivalent relation where a class is just Each segment of this soil, okay each of this is a class and then define the same extents to the unit tangent bundle and then let X be the unit tangent bundle Quotient by the equivalent relation and then the theorem is that this is a manifold and Is I mean it's not obvious because When you think a little bit about the structure of strips Then I will construct something like a cross-section of this picture here So Let's consider following cross-section section so I will pick theta here and then I will take a Subset of H s theta large enough to contain the class of theta, okay this is I Okay, let us I will consider a piece of the vertical fiber through theta and Make the same thing for every point here. So since the the The center stable set gave a falliation then It's all this is this is a sort of fun, you know trivializing picture for the falliation for the center stable falliation and then this is the Okay, this is the center stable falliation and then the center unstable falliation in this Trivializing picture for this center stable would be something like this See, I mean that but a lot lots of tangencies or Coincidences non-trivial coincidences between stable and unstable falliations May appear Okay, so what I'm what I'm claiming here is that if you quotient all these Coincidences Then just to get a manifold This is here and this is not not obvious Okay so And okay, so Okay, in five minutes, let me at least say that I Give you a minute an idea of the proof of this Once I mean even if you don't if you don't know what is I mean if this space has a nice structure Nice topological or metric structure, then you observe that there is Endowed with a flow That is the quotient flow because the classes are preserved by the genetic flow So it is so remark there is There exists a quotient flow. This is let's say psi T Okay, and if X If you get us something like a Metric space structure for X then we would expect psi T to be Expensive Okay, that's that is what what happens and then and moral more more than that if it is true then this Expensive thing this expensive flow would be something like like the expensive flow of of our dreams or I might say of Baldwin's dreams because This expensive flow would inherit all the good properties of the genetic flow of the surface You know in everland in the 70s proved that the genetic flow Of I mean I won't define visibility many falls but Visibility many falls include compact surfaces With no conjugate points and genus greater than one Then everland proved mg compact I will state with Like everline visibility manifold then The genetic flow Is transitive. I mean there is a dance orbit is topologically mixing meaning that for every UV open set there exists Let's say R in R such that Phi s u intersects V for every s Bigger than our This is topologically mixing then I mean Horsesicles are minimal In lots of Lots of properties Okay, and then since If okay, if we get some structure for this flow and It is expensive V we would expect it to have invariant sets Because the quotient of invariant sets would play the role of the invariant sets of the expensive thing so You would expect to have a local product property for this role, okay? And this would imply that there is shadowing lemma and there is and seen combining the shadowing lemma with the Where the topologically mixing property in the transitivity we get the specification by close orbits and then ball wins result says then so So then the state theorem for for now for the sighting site the Has local product structure Specification my close orbits and it will get all these properties here topologically mixing transitivity everything and then ball wins implies that the measure theorem and this is the second part of the proof of the talk, sorry the measure of maximal entropy is Unique okay, this is a nice application Of the thing because you get it straightforward from Bowens I mean from what you already know from the theory of digitistic flow Visibility manifolds and ball wins, you know, classical work about, you know variational principles and Then how to get the uniqueness of because this was I saw what it what was the The the the main let's say application of this is that the the same happens for digitistic flow of the surface so Perhaps to finish Time goes fast indeed so they are fight the How they need and this is a combination of? the previous result and And of course proof previous result plus Remark a very nice remark made by busy Fisher a lot of people here Sambarino Vasquez saying essentially that I Mean what they they look at at flows they call an extension of an expensive flow and They give some conditions to to show that if The an expensive flow has a unique measure of maximal entropy and the extension have some properties then The measure of maximal entropy for the extension would be also unique and these conditions are Fulfilled here for the flow and that's all this is ketchup So I hope my time is over so Thank you very much